harmonic analysis w. schlag - department of mathematicsschlag/harmonicnotes.pdf · chapter 1...

Harmonic Analysis

W. Schlag

Contents

Chapter 1. Fourier Series: Convergence and Summability 5

Chapter 2. Harmonic Functions on D and the Poisson Kernel 25

Chapter 3. Analytic h1pDq Functions, F. & M. Riesz Theorem 35

Chapter 4. The Conjugate Harmonic Function 43

Chapter 5. Fourier Series on L1pTq: Pointwise Questions 53

Chapter 6. Fourier Analysis on Rd 63

Chapter 7. Calderon-Zygmund Theory of Singular Integrals 79

Chapter 8. Almost Orthogonality 101

Chapter 9. The Uncertainty Principle 111

Chapter 10. Littlewood-Paley theory 125

Chapter 11. Fourier Restriction and Applications 145

Bibliography 165

3

CHAPTER 1

Fourier Series: Convergence and Summability

We begin this book with one of the most basic objects in analysis, namely theFourier series associated with a function or a measure on the circle. To be specific,let T RZ be the one-dimensional torus (in other words, the circle). We considervarious function spaces on the torus T, namely the space of continuous functionsCpTq, the space of Holder continuous functions CαpTq where 0 α ¤ 1, and theLebesgue spaces LppTq where 1 ¤ p ¤ 8. The space of complex Borel measureson T will be denoted by MpTq. Any µ P MpTq has associated with it a Fourierseries

µ 8

n8µpnqepnxq (1.1)

where we let epxq : e2πix and

µpnq :» 1

0epnxq µpdxq

»T

epnxq µpdxq

The symbol in (1.1) is formal, and simply means that the series on the right-handside is associated with µ. If µpdxq f pxq dx where f P L1pTq, then we also writef pnq instead of µpnq.

Of course, the central question which we wish to explore in this chapter iswhen µ equals the right-hand side in (1.1) or represents f in a suitable sense. Notethat if we start from a trigonometric polynomial

f pxq 8

n8anepnxq

where all but finitely many an are zero, then we see that

f pnq an @n P Z (1.2)

In other words, we have the pointwise equality

f pxq 8

n8f pnqenpxq

with enpxq : epnxq. What enters (1.2) is of course the basic orthogonality relation»T

enpxqempxq dx δ0pn mq (1.3)

where δ0p jq 1 if j 0 and δ0p jq 0 otherwise.

5

6 1. FOURIER SERIES: CONVERGENCE AND SUMMABILITY

It is therefore natural to explore the question of convergence of the Fourier se-ries for more general functions. Of course, the sense of convergence of the infiniteseries needs to be specified before this fundamental question can be answered. Itis fair to say that much of modern analysis (including functional analysis) aroseout of the struggle with this question. For example, the notion of Lebesgue integralwas developed in order to overcome deficiencies of the older Riemannian definitionof the integral which had been revealed in the study of Fourier series. The readerwill also note the recurring theme of convergence of Fourier series throughout thisbook.

It is natural to start from the most classical notion of convergence, namely thatof pointwise convergence in case the measure µ is of the form µpdxq f pxq dxwith f pxq continuous or better. The partial sums of f P L1pTq are defined as

S N f pxq N

nN

f pnqepnxq N

nN

»T

epnyq f pyq dy epnxq

»T

N

nN

epnpx yqq f pyq dy »T

DNpx yq f pyq dy

where DNpxq °N

nN epnxq is the Dirichlet kernel. In other words, we haveshown that the partial sum operator S N is given by convolution with the Dirichletkernel DN :

S N f pxq pDN f qpxq (1.4)

In order to understand basic properties of this convolution, we first sum the geo-metric series defining DNpxq so as to have an explicit expression for the Dirichletkernel.

Exercise 1.1. Verify that for each integer N ¥ 0

DNpxq sinpp2N 1qπxqsinpπxq (1.5)

and draw the graph of DN for several different values of N, say N 2 and N 5.Prove the bound

|DNpxq| ¤ C minN,

1|x|

(1.6)

for all N ¥ 1 and some absolute constant C. Finally, prove the bound

C1 log N ¤ DNL1pTq ¤ C log N (1.7)

for all N ¥ 2 where C is another absolute constant.

The growth of the bound in (1.7) as well as the oscillatory nature of DN as givenby (1.5) may indicate that it is in general very delicate to understand pointwise oralmost everywhere convergence properties of S N f . This will become more clearas we develop the theory.

1. FOURIER SERIES: CONVERGENCE AND SUMMABILITY 7

In order to study (1.4) we need to establish some basic properties of the con-volution of two functions f , g on T. If f and g are continuous, say, then define

p f gqpxq :»T

f px yqgpyq dy »T

gpx yq f pyq dy (1.8)

It is helpful to think of f g as an average of translates of f by the measure gpyq dyor vice versa. In particular, convolution commutes with the translation operator τzwhich is defined for any z P T by the action on functions: pτz f qpxq f px zq.Indeed, one immediately verifies that

τzp f gq pτz f q g f pτzgq (1.9)

In passing, we mention the important relation between the Fourier transform andtranslations: zpτzµqpnq epznqµpnq @ n P ZIn what follows, we abbreviate almost everywhere or almost every by a.e.

Lemma 1.1. The operation of convolution as defined above satisfies the follow-ing properties:

(1) Let f , g P L1pTq. Then for a.e. x P T one has that f px yqgpyq is L1

in y. Thus, the integral in (1.8) is well-defined for a.e. x P T (but notnecessarily for every x), and the bound f g1 ¤ f 1g1 holds.

(2) More generally, f gp ¤ f pg1 for all 1 ¤ p ¤ 8, f P Lp, g P L1.This is called Young’s inequality.

(3) If f P CpTq, µ PMpTq then f µ is well-defined. Show that, for 1 ¤ p ¤8,

f µp ¤ f pµwhich allows one to extend f µ to arbitrary f P Lp.

(4) If f P LppTq and g P Lp1pTq where 1 ¤ p ¤ 8, and 1p 1

p1 1 thenf g, originally defined only a.e., extends to a continuous function on Tand

f g8 ¤ f pgp1 (1.10)

(5) For f , g P L1pTq show that for all n P Zzf gpnq f pnqgpnqProof. (1) is an immediate consequence of Fubini’s theorem since f pxyqgpyq

is jointly measurable on T T and belongs to L1pT Tq. (2) can be obtained byinterpolating between the cases p 1 just convered and the easy bound for p 8.Alternatively, one can use Minkowski’s inequality:

f gp ¤»T f p yqp|gpyq| dy ¤ f pg1

which also implies (3). The bound (1.10) is just Holder’s inequality. Part (4)follows from the density of CpTq in LppTq for 1 ¤ p 8 and the translationinvariance (1.9). Indeed, one verifies from the uniform continuity of functionsin CpTq and (1.9) that f µ P CpTq for any f P CpTq and µ P MpTq. Since


uniform limits of continuous functions are continuous, (4) now follows from theaforementioned density of CpTq and (1.10).

Finally, (5) is a consequence of Fubini’s theorem and the character property ofthe exponentials epnpx yqq epnxqepnyq.

The following exercise introduces the convolution on the Fourier side in thecontext of the largest class of functions where the respective series are absolutelyconvergent. This class of functions is necessarily a subalgebra of CpTq called theWiener algebra.

Exercise 1.2. Let µ PMpTq have the property that¸nPZ

|µpnq| 8 (1.11)

Show that µpdxq f pxq dx where f P CpTq. Denote the space of all measures withthis property by ApTq and identify these measure with their respective densities.Show that ApTq is an algebra under multiplication, and thatxf gpnq

¸mPZ

f pmqgpn mq @ n P Z

where the sum on the right-hand side is absolutely convergent for every n P Z, anditself is absolutely convergent over all n. Moreover, f gA ¤ f AgA where f A : f `1 . Finally, verify that if f , g P L2pTq, then f g P ApTq.

Note that the Wiener algebra has a unit, namely the constant function 1. It isclear that if f has an inverse in ApTq, then it is 1

f which in particular require thatf , 0 everywhere on T. It is a remarkable theorem due to Norbert Wiener that theconverse here holds, too. I.e., if f P ApTq does not vanish anywhere on T, then1f P ApTq.

One of the most basic as well as oldest results on the pointwise convergence ofFourier series is the following theorem. We shall see later that if fails for functionswhich are merely continuous.

Theorem 1.2. If f P CαpTq with 0 α ¤ 1, then S N f f 8 Ñ 0 as N Ñ8.

Proof. One has, with δ P p0, 12q to be determined,

S N f pxq f pxq » 1

0p f px yq f pxqqDNpyq dy

»|y|¤δ

p f px yq f pxqqDNpyq dy

»

12¡|y|¡δ


(1.12)

Here one exploits the fact that »T

DNpyq dy 1


We now use the bound from (1.6), i.e.,

|DNpyq| ¤ C minN,

1|y|

Here and in what follows, C will denote a numerical constant that can change fromline to line. The first integral in (1.12) can be estimated as follows»

|y|¤δ| f pxq f px yq| 1

|y| dy ¤ r f sα»|y|¤δ

|y|α1 dy ¤ Cr f sαδα (1.13)

with the usual Cα semi-norm:

r f sα supx,y

| f pxq f px yq||y|α

To bound the second term in (1.12) one needs to invoke the oscillation of DNpyq.In fact,

B :»

12¡|y|¡δ


»

12¡|y|¡δ

f px yq f pxqsinpπyq sinpp2N 1qπyq dy

»

12¡|y|¡δ

hxpyq sinp2N 1qπpy 1

2N 1q

dy

where hxpyq : f pxyq f pxqsinpπyq .

Therefore, with all integrals being understood to be in the interval p 12 ,

12q,

2B »|y|¡δ

hxpyq sinpp2N 1qπyq dy

»|y 1

2N1 |¡δhxy 1

2N 1

sinpp2N 1qπyq dy

»|y|¡δ

hxpyq hxpy 1

2N 1q

sinpp2N 1qπyq dy

»rδ,δ 1

2N1 shxy 1

2N 1

sinpp2N 1qπyq dy

»rδ,δ 1

2N1 shxy 1

2N 1

sinpp2N 1qπyq dy

These integrals are estimated by putting absolute values inside. To do so we usethe bounds

|hxpyq| C f 8δ

|hxpyq hxpy τq| C |τ|αr f sα

δ f 8

δ2 |τ|

if |y| ¡ δ ¡ 2τ.


In view of the preceding one checks

|B| ¤ C

Nαr f sαδ

N1 f 8δ2

(1.14)

provided δ ¡ 1N . Choosing δ Nα2 one concludes from (1.12), (1.13), and

(1.14) that

|pS N f qpxq f pxq| ¤ C

Nα22 Nα2 N1α

(1.15)

for any function with f 8 r f sα ¤ 1, which proves the theorem.

The reader is invited to optimize the rate of decay that was derived in (1.15).In other words, the challenge is to obtain the largest β ¡ 0 in terms of α so that thebound in (1.15) becomes CNβ for any f with f 8 r f sα ¤ 1.

The difficulty with the Dirichlet kernel, such as its slow 1|x| -decay, can be re-

garded as a result of the “discontinuity” of xDN χrN,Ns; in fact, this indicatorfunction on the lattice Z jumps at N. Therefore, we may hope to obtain a kernelwhich is easier to analyze — in a sense that will be made precise by means of thenotion of approximate identity below — by substituting DN with a suitable averagewhose Fourier transform does not exhibit such jumps.One elementary way of carrying this out is given by the Cesaro means, i.e.,

σN f : 1N

N1

n0

S n f

Setting KN : 1N

°N1n0 Dn, which is called the Fejer kernel, one therefore has

σN f KN f .

Exercise 1.3. Let KN be the Fejer kernel with N a positive integer.

Verify that pKN looks like a triangle, i.e., for all n P ZpKNpnq

1 |n|

N

(1.16)

Show that

KNpxq 1N

sinpNπxqsinpπxq

2

(1.17)

Conclude that

0 ¤ KNpxq ¤ C N1 minN2, x2 (1.18)

We remark that the square and thus the positivity in (1.17) are not entirelysurprising, since the triangle in (1.16) can be written as the convolution of tworectangles (convolution is now on the level of the Fourier coefficients on the lat-tice Z). Therefore, we should expect KN to look like the square of a version of DMwith M about half the size of N.

The properties established in Exercise 1.3 ensure that KN are what one calls anapproximate identity.


Definition 1.3. tΦNu8N1 L8pTq form an approximate identity provided

A1)³1

0 ΦNpxq dx 1 for all NA2) supN

³10 |ΦNpxq| dx 8

A3) for all δ ¡ 0 one has³|x|¡δ |ΦNpxq| dx Ñ 0 as N Ñ8.

The name here derives from the fact that ΦN f Ñ f as N Ñ 8 in anyreasonable sense, see Proposition 1.5. In other words, ΦN á δ0 in the weak-sense of measures. Clearly, the so-called box kernels

ΦNpxq Nχr|x|N 12 s, N ¥ 1

satisfy A1)–A3) and can be considered as the most basic example of an approx-imate identity. Note that tDNuN¥1 are not an approximate identity. Finally, weremark that Definition 1.3 has nothing to do with the torus T, it applies equallywell to the line R, tori Td, or Euclidean spaces Rd.

Next, we verify that KN belong to this class.

Lemma 1.4. The Fejer kernels tKNu8N1 form an approximate identity.

Proof. We clearly have³1

0 KNpxq dx 1 and KNpxq ¥ 0 so that A1) and A2)hold. A3) follows from the bound (1.18).

Now we establish the basic convergence property of such families.

Proposition 1.5. For any approximate identity tΦNu8N1 one has

(1) If f P CpTq, then ΦN f f 8 Ñ 0 as N Ñ8(2) If f P LppTq where 1 ¤ p 8, then ΦN f f p Ñ 0 as N Ñ8(3) For any measure µ PMpTq, one has

ΦN µá µ N Ñ8in the weak- sense.

Proof. We begin with the uniform convergence. Since T is compact, f is uni-formly continuous. Given ε ¡ 0, let δ ¡ 0 be such that

supx

sup|y| δ

| f px yq f pxq| ε

Then, by A1)–A3),

|pΦN f qpxq f pxq| »Tp f px yq f pxqqΦNpyq dy

¤ sup

xPTsup|y| δ

| f px yq f pxq|»T|ΦNptq| dt

»|y|¥δ

|ΦNpyq|2 f 8 dy

Cε

provided N is large.


Fix any f P LppTq and let g P CpTq be such that f gp ε. Then

ΦN f f p ¤ ΦN p f gqp f gp ΦN g gp

¤ sup

NΦN1 1

f gp ΦN g g8

where we have used Young’s inequality, see Lemma 1.1, to obtain the first termon the right-hand side. Using A2), the assumption on g, and Young’s inequalityfinishes the proof.

The final statement is immediate from (1) and duality.

This simple convergence result applied to the Fejer kernels implies some basicanalytical properties. A trigonometric polynomial is a series

°n anepnxq in which

only finitely many an are non-zero. Clearly, this class forms a vector space overthe complex numbers. In fact, they form an algebra under both multiplication andconvolution.

Corollary 1.6. The exponential family tepnxqunPZ satisfies the following prop-erties:

(1) The trigonometric polynomials are dense in CpTq in the uniform topology,and in LppTq for any 1 ¤ p 8.

(2) For any f P L2pTq f 2

2 ¸nPZ

| f pnq|2

This is called the Plancherel theorem.(3) The exponentials tepnxqunPZ form a complete orthonormal basis in L2pTq.(4) For all f , g P L2pTq one has Parseval’s identity»

Tf pxqgpxq dx

¸nPZ

f pnqgpnq

Proof. By Lemma 1.4, tKNu8N1 form an approximate identity and Proposi-tion 1.5 applies. Since σN f KN f is a trigonometric polynomial, we are donewith (1).

Properties (2)–(4) are equivalent by basic Hilbert space theory. We begin fromthe orthogonality relations (1.3). One thus has Bessel’s inequality¸

| f pnq|2 ¤ f 22

and equality is equivalent to the linear span of tenu being dense in L2pTq. That,however, is guaranteed by part (1).

We remark that the previous corollary is only possible with the Lebesgue inte-gral, since unlike the Riemann integral it guarantees that L2pTq is a complete spaceand thus a Hilbert space. We record two further basic facts which are immediateconsequences of the approximate identity property of the Fejer kernels.


Corollary 1.7. One has the following uniqueness property: If f P L1pTq andf pnq 0 for all n P Z, then f 0. More generally, if µ PMpTq satisfies µpnq 0for all n P Z, then µ 0.

Proof. One has σN f 0 for all N by assumption and now apply the conver-gence property of Proposition 1.5.

The following is the Riemann-Lebesgue lemma.

Corollary 1.8. If f P L1pTq, then f pnq Ñ 0 as n Ñ8

Proof. Given ε ¡ 0, let N be such that σN f f 1 ε. Then | f pnq| |yσN f pnq f pnq| ¤ σN f f 1 ε for all |n| ¡ N.

We now turn our attention to the issue of convergence of the partial sums S N fin the sense of LppTq or CpTq. Observe that it makes no sense to ask about uniformconvergence of S N f for general f P L8pTq because uniform limits of continuousfunctions are continuous.

Proposition 1.9. The following statements are equivalent for any 1 ¤ p ¤ 8:

a) For every f P LppTq por f P CpTq if p 8q one has

S N f f p Ñ 0 as N Ñ8b) supN S NpÑp 8

Proof. The implication b)ùñ a) follows from the fact that trigonometric poly-nomials are dense in the respective norms, see Corollary 1.6. The implication a)ùñ b) can be deduced immediately from the uniform boundedness principle offunctional analysis. Alternatively, can also prove this directly by the method of the“gliding hump”: suppose supN S NpÑp 8. For every positive integer ` one cantherefore find a large integer N` such that

S N` f`p ¡ 2`

where f` is a trigonometric polynomial with f` p 1. Now let

f pxq 8

`1

1`2 epM`xq f`pxq

with some integers tM`u to be specified. Notice that

f p ¤8

`1

1`2 f`p 8

Now choose tM`u tending to infinity so rapidly that the Fourier support of

epM jxq f jpxq


lies to the right of the Fourier support ofj1

`1

1`2 epM`xq f`pxq

for every j ¥ 2 (here Fourier support means those integers for which the corre-sponding Fourier coefficients are non-zero). We also demand that N` M` Ñ 8as `Ñ8. Then

pS M`N` S M`N`1q f p 1`2 S N` f`p ¡ 2`

`2

which Ñ 8 as ` Ñ 8. On the other hand, since N` M` Ñ 8 andM` N` 1 Ñ 8, the left-hand side Ñ 0 as ` Ñ 8. This contradiction fin-ishes the proof.

Exercise 1.4. Let tcnunPZ be an arbitrary sequence of complex numbers andassociate to it formally a Fourier series

f pxq ¸

n

cn epnxq

Show that there exists µ P MpTq with the property that µpnq cn for all n P Zif and only if tσn f un¥1 is bounded in MpTq. Discuss the case of LppTq with1 ¤ p 8 and CpTq as well.

We can now settle the question of uniform convergence of S N on functionsin CpTq.

Corollary 1.10. Fourier series do not converge on CpTq and L1pTq, i.e., thereexists f P CpTq so that S N f f 8 9 0 and g P L1pTq so that S Ng g1 9 0as n Ñ8.

Proof. By Proposition 1.9 it suffices to verify the limitssup

NS N8Ñ8 8

supNS N1Ñ1 8 (1.19)

Both properties follow from the fact that

DN1 Ñ8as N Ñ8, see (1.7). To deduce (1.19) from this, notice that

S N8Ñ8 sup f 81

DN f 8

¥ sup f 81

|pDN f qp0q| DN1

We remark that the inequality sign here can be replaced with an equality in viewof the translation invariance (1.9). Furthermore, with tKMu8M1 being the Fejerkernels,

S N1Ñ1 ¥ DN KM1 Ñ DN1

as M Ñ8.


Exercise 1.5. The previous proof was indirect. Construct f P CpTq and g PL1pTq so that S N f does not converge to f uniformly as N Ñ8, and such that S Ngdoes not converge to g in L1pTq.

In the positive direction, we shall see below that for 1 p 8sup

NS NpÑp 8

so that by Corollary 1.10 for any f P LppTq with 1 p 8S N f f p Ñ 0 as N Ñ8 (1.20)

The case p 2 is clear, see Corollary 1.6, but p , 2 is a somewhat deeper result.We will develop the theory of the conjugate function to obtain it, see Chapter 4.Note that unlike Theorem 1.2 there is no explicit rate of convergence here in termsof some expression involving N. Clearly, this cannot be expected given only thesize of f p since gpxq : epmxq f pxq has the same Lp-norm as f but gpnq f pn mq for all n P Z. This suggests that we may hope to obtain such a rateprovided we remove this freedom of translation on the Fourier side. One way ofaccomplishing this is by imposing a little regularity, as expressed for example interms of the standard Sobolev spaces. Since we do not yet have the Lp-convergencein (1.20) for general p at our disposal, we need to restrict ourselves to p 2 whichis particularly simple.

Exercise 1.6. For any s P R define the Hilbert space HspTq by means of thenorm

f 2Hs : | f p0q|2

¸nPZ

|n|2s| f pnq|2 (1.21)

Derive a rate of convergence for S N f f 2 in terms of N alone assuming that f Hs ¤ 1 where s ¡ 0 is fixed.

We now investigate the connection between regularity of a function and itsassociated Fourier series somewhat further. The following estimate, which buildsupon the fact that d

dx epnxq 2πinepnxq, goes by the name of Bernstein’s inequal-ity.

Proposition 1.11. Let f be a trigonometric polynomial with f pkq 0 for all|k| ¡ n. Then

f 1p ¤ Cn f p

for any 1 ¤ p ¤ 8. The constant C is absolute.

Proof. LetVnpxq : p1 epnxq epnxqqKnpxq (1.22)

be de la Vallee Poussin’s kernel. We leave it to the reader to check thatpVnp jq 1 if | j| ¤ n

as well asV 1

n1 ¤ Cn


Then f Vn f and thus f 1 V 1n f so that by Young’s inequality

f 1p ¤ V 1n1 f p ¤ Cn f p

as claimed.

It is interesting to note that the following converse holds, due to Bohr.

Exercise 1.7. Suppose that f P C1pTq satisfies f p jq 0 for all j with | j| n.Then

f 1p ¥ Cn f p

for all 1 ¤ p ¤ 8, where C is independent of n P Z, the choice of f and p.

Now we will turn to the question how smoothness of a function reflects itselfin the decay of its Fourier coefficients. We begin with the easy observation, basedon integration by parts, that for any f P C1pTq one haspf 1pnq 2πin f pnq @n P Z (1.23)

This not only shows that f pnq Opn1q but also that C1pTq ãÑ H1pTq where H1

is the Sobolev space defined in (1.21). If f possesses more derivatives, say k ¥ 2,then we may iterate this relation to obtain decay of the form Opn2q. The followingexercise establishes the connection between rapid decay of the Fourier coefficientsand infinite regularity of the function.

Exercise 1.8. Let f P L1pTq. Show that f P C8pTq if and only if f decays rapidly, i.e., for every M ¥ 1

one has f pnq Op|n|Mq as |n| Ñ 8. Show that f pxq Fpepxqq where F is analytic on some neighborhood oft|z| 1u if and only if f pnq decays exponentially, i.e., f pnq Opeε|n|qas |n| Ñ 8 for some ε ¡ 0.

As far as less than one derivative is concerned, i.e., for Holder continuousfunctions, one has the following facts. Starting from

f pnq »T

epnpy 12nqq f pyq dy

»T

epnyq fy 1

2n

dy

whence

f pnq 12

»T

epnyq f pyq f py 12nq dy

which implies that if f P CαpTq, then

f pnq Opnαq as |n| Ñ 8 (1.24)

Note that (1.23) is strictly stronger than this bound, since it allows for square sum-mation and thus the embedding into H1pTq, whereas (1.24) does not. However,since we already observed that C1pTq ãÑ H1pTq and since clearly C0pTq ãÑH0pTq L2pTq, one can invoke some interpolation machinery at this point toconclude that CαpTq ãÑ HαpTq for all 0 ¤ α ¤ 1; however, we shall not make useof this fact.


Next, we ask how much regularity on f it takes so that8

n8| f pnq| 8 (1.25)

In other words, we ask for a sufficient condition for f to lie in the Wiener algebraApTq of Exercise 1.2. There are a number of ways to interpret the requirement of“regularity”. It is easy to settle this imbedding question on the level of the spacesHspTq. In fact, applying Cauchy-Schwarz yields¸

n,0

| f pnq| ¤ ¸n,0

| f pnq|2|n|1ε 12¸

n,0

|n|1ε 12

so that8

n8| f pnq|2|n|1ε 8 (1.26)

is a sufficient condition for (1.25) to hold. In other words, we have shown thatHspTq ãÑ ApTq for any s ¡ 1

2 . We leave it to the reader to check that this fails fors 1

2 .Next, we would like to address the more challenging question as to the minimal

α ¡ 0 so that CαpTq ãÑ ApTq. In fact, we first ask which α have the property thatCαpTq ãÑ HspTq for some s ¡ 1

2 . We already observed that α ¡ 12 suffices for

this, but it required some interpolation theory.Instead, we prefer to give a direct proof of this fact in the following theorem.

It introduces an important idea that we shall see repeatedly throughout this book,namely the grouping of the Fourier coefficients f pnq into blocks of the same sizeof n; more precisely, we introduce the partial sums

pP j f qpxq :¸

2 j1¤|n| 2 j

f pnqepnxq (1.27)

for all j ¥ 1.

Theorem 1.12. For any 1 ¥ α ¡ 0 one has CαpTq ãÑ HβpTq for arbitrary0 β α. In particular, for any f P CαpTq with α ¡ 1

2 one has¸nPZ

| f pnq| 8

and thus CαpTq ãÑ ApTq for any α ¡ 12 .

Proof. Let r f sα ¤ 1. We claim that for every j ¥ 0,¸2 j¤|n| 2 j1

| f pnq|2 ¤ C22 jα (1.28)

The reader should note the similarity between this bound and the estimate (1.24),with (1.28) being of course strictly stronger. What allows for this improvement is


the use of orthogonality or in other words, the Plancherel theorem. If (1.28) is true,then for any β α,¸

n,0

| f pnq|2|n|2β ¤ C8

j0

22 jβ¸

2 j¤|n| 2 j1

| f pnq|2 ¤ C8

j0

22 jpαβq 8

To prove (1.28) we choose a kernel ϕ j so thatpϕ jpnq 1 if 2 j ¤ |n| ¤ 2 j1 (1.29)

and pϕ jpnq 0 if |n| ! 2 j or |n| " 2 j1 (1.30)where ! and " mean “much smaller” and “much bigger”, respectively. The pointis of course that (1.29) implies that¸

2 j¤|n| 2 j1

| f pnq|2 ¤ ϕ j f 22 (1.31)

so that it remains to bound the right-hand side for which (1.30) will be decisive— at least implicitly. The explicit properties of ϕ j that will allow us to bound theright-hand side are cancellation meaning that pϕ jp0q 0 and bounds on the kernelϕ jpxq which resemble those of the Fejer kernel KN with N 2 j.

There are various ways to construct ϕ j. We use de la Vallee Poussin’s kernelfrom above for this purpose. Set

ϕ jpxq : V2 j1pxq pepp3 2 j1 1qxq epp3 2 j1 1qxqq (1.32)

We leave it to the reader to check thatpϕ jpnq 1 for 2 j ¤ |n| ¤ 2 j1

and that pϕ jp0q 0 which is the same as»Tϕ jpxq dx 0

Moreover, since the ϕ j are constructed from Fejer kernels one has the bounds

|ϕ jpxq| ¤ C 2 j min22 j , |x|2

Therefore,

|pϕ j f qpxq| |»Tϕ jpyqp f px yq f pxqq dy|

¤»T|ϕ jpyq|| f px yq f pxq| dy

¤ C» 1

0|ϕ jpyq||y|α dy

¤ C2 j»|y|¡2 j

|y|α2 dyC2 j»|y|¤2 j

|y|α dy

¤ C2α j


as claimed. In summary, we see that for any 0 ¤ β α ¤ 1

f Hβ ¤ Cpα, βq f Cα

and the theorem follows.

Next, we show that Theorem 1.12 is optimal up to the fact that one can takeα β in the first statement.

Exercise 1.9. Consider lacunary series of the form

f pxq :¸k¥1

k22αkep2kxq

to show that CαpTq does not embed into Hβ for any β ¡ α.

The second statement of Theorem 1.12 concerning ApTq is more difficult.

Proposition 1.13. There is no embedding from C12 pTq into ApTq. In fact, there

exists a function f P C12 pTqzApTq.

Proof. We claim that there exists a sequence of trigonometric polynomialsPnpxq

°2n1`0 an,` ep`xq such that with N 2n,

Pn8 ?

N

xPn`1 N(1.33)

for each n ¥ 1. Here a b means C1a ¤ b ¤ Ca where C is an absoluteconstant, and xPn refers to the sequence of Fourier coefficients.

Assuming such a sequence for now, we set

Tnpxq : 2nep2nxqPnpxq

f :8

n1

Tn

Note that this series converges uniformly to f P CpTq since Tn8 ¤ C2n2 .

Moreover, the Fourier supports of the Tn are pairwise disjoint for distinct n. Thus, f ApTq 8 and f < ApTq. Finally, let h 2m for some positive integer m.Then

| f px hq f pxq| ¤m

n1

|Tnpx hq Tnpxq| ¸

n¡m

Tn8

¤m

n1

C|h|T 1n8

¸n¡m

C 2n2

¤ Cpm

n1

2n2 |h| 2

m2 q ¤ C|h| 1

2

and thus f P C12 pTq as desired. To pass to the last line, we used Bernstein’s

inequality, i.e., Proposition 1.11.


It thus remains to establish the existence of polynomials as in (1.33). Definethem inductively by P0pxq Q0pxq 1 and

Pn1pxq Pnpxq ep2nxqQnpxqQn1pxq Pnpxq ep2nxqQnpxq

for each n ¥ 0. Since

|Pn1pxq|2 |Qn1pxq|2 2p|Pnpxq|2 |Qnpxq|2q 2n1

we see that Pn18 is of the desired size. Furthermore, all coefficients of both Pnand Qn are either1 or1, and the only exponentials with nonzero coefficients areep`xq with 1 ¤ ` ¤ 2n 1. Hence tPnu8n0 is the desired family of polynomials.

Exercise 1.10. Show that any trigonometric polynomial P with P`1 Nand the property that the cardinality of its Fourier support is at most N satisfies?

N ¤ P2 ¤ P8 ¤ N. Hence, the polynomials constructed in the previousproof are “extremal” for the lower bound on P8, whereas the Dirichlet kernel(or Fejer kernel) are extremal for the upper bound.

We conclude this chapter with a brief discussion of Fourier series associatedwith functions on Td RdZd with d ¥ 2. The exponential basis in this case istepν xquνPZd , and this is an orthonormal family in the usual L2pTdq sense. Thus,to every measure µ PMpTdq we associate a Fourier series¸

νPZd

µpνq epν xq, µpνq :»Td

epν xq µpdxq

As before, a special role is played by the trigonometric polynomials¸νPZd

aν epν xq

where all but finitely many aν vanish. In contrast to the one-dimensional torus, inhigher dimensions we face a very nontrivial ambiguity in the definition of partialsums. In fact, we can pose the convergence problem relative to any exhaustion ofZd by finite sets Ak, which are increasing and whose union is all of Zd. Possiblechoices here are of course the squares rk, ksd, but one can also take more generalrectangles, or balls relative to the Euclidean metric or other shapes. Even thoughthis distinction may seem innocuous, it gave rise to very important developmentsin harmonic such as Fefferman’s ball multiplier theorem, and the still unresolvedBochner-Riesz conjecture. We will return to these matters in a later chapter.

For now, we merely content ourselves with some very basic results.

Proposition 1.14. The space of trigonometric polynomials is dense in CpTdq,and one has the Plancherel theorem for L2pTdq, i.e.,

f 22

¸νPZd

| f pνq|2 @ f P L2pTdq


If f P C8pTdq, then the Fourier series associated to f converges uniformly to firrespective of the way in which the partial sums are formed.

Proof. We base the proof on the fact that the products of Fejer kernels withrespect to the individual coordinate axes form an approximate identity. In otherwords, the family

KN,dpxq :d¹

j1

KNpx jq @ N ¥ 1

forms an approximate identity on Td. Here x px1, . . . , xdq. This follows imme-diately from the one-dimensional analysis above. Hence, for any f P CpTdq onehas

KN,d f f 8 Ñ 0 as N Ñ8By inspection, KN,d f is a trigonometric polynomial which implies the claimeddensity. This implies all assertions about the L2 case.

Suppose that f P C8pTdq. Repeated integration by parts yields the estimatef pνq Op|ν|mq as |ν| Ñ 8 for arbitrary but fixed m ¥ 1. The convergencestatement now follows from KN,d f Ñ f in CpTdq as N Ñ 8, and the rapiddecay of the coefficients.

A useful corollary to the previous result is the density of tensor functions inCpTdq. A tensor function on Td is a linear combination of functions of the form±d

j1 f jpx jq where f j P CpTq. In particular, trigonometric polynomials are tensorfunctions, whence the claim.

Notes

An encyclopedic treatment of Fourier series is the classic reference by Zygmund [58].A less formidable but still comprehensive account of many classical results is Katznel-son [30]. A standard reference for interpolation theory is the book by Bergh and Lofstrom [4].For the construction in Proposition 1.13, see [30, page 36, Exercise 6.6]. For gap seriessee the timeless paper by Rudin [41] which introduced the Λp-set problem, later solved byBourgain [5].

Problems

Problem 1.1. Suppose that f P L1pTq and that tS n f u8n1 (the sequence of partial sumsof the Fourier series) converges in LppTq to g for some p P r1,8s and some g P Lp. Provethat f g. If p 8 conclude that f is continuous.

Problem 1.2. Let T pxq °Nn0ran cosp2πnxq bn sinp2πnxqs be an arbitrary trigono-

metric polynomial with real coefficients a0, . . . , aN , b0, . . . , bN . Show that there is a poly-nomial Ppzq °2N

`0 c` z` P Crzs so that

T pxq e2πiNxPpe2πixqand such that Ppzq z2N Ppz1q. How are the zeros of P distributed in the complex plane?


Problem 1.3. Suppose T pxq °Nn0

an cosp2πnxqbn sinp2πnxq is such that T ¥ 0

everywhere on T and an, bn P R for all n 0, 1, . . . ,N. Show that there are c0, . . . , cN P Csuch that

T pxq N

n0

cne2πinx2 (1.34)

Find the cn for the Fejer kernel.

Problem 1.4. Suppose that T pxq a0 °H

h1 ah cosp2πhxq satisfies T pxq ¥ 0 for allx P T and T p0q 1. Show that for any complex numbers y1, y2, . . . , yN ,

N

n1

yn2 ¤ pN Hq

a0

N

n1

|yn|2 H

h1

|ah| Nh

n1

ynhyn

Hint: Write (1.34) with°

n cn 1. The apply Cauchy-Schwartz to°

n yn °

m,n yn cm.

Problem 1.5. Suppose°8

n1 n |an|2 8 and°8

1 an is Cesaro summable. Showthat

°8n1 an converges. Use this to prove that any f P CpTq X H

12 pTq satisfies S n f Ñ f

uniformly. Note that H12 pTq does not embed intoApTq, so this convergence does not follow

by trivial means.

Problem 1.6. Show that there exists an absolute constant C so that

C1¸n,0

|n| | f pnq|2 ¤»T2

| f pxq f pyq|2sin2pπpx yqq dxdy ¤ C

¸n,0

|n| | f pnq|2

for any f P H12 pTq.

Problem 1.7. Use the previous two problems to prove the following theorem of Pal-Bohr: For any real function f P CpTq there exists a homeomorphism ϕ : T Ñ T suchthat

S np f ϕq ÝÑ f ϕuniformly. Hint: Without loss of generality let f ¡ 0. Consider the domain defined interms of polar coordinates by means of rpθq f pθ2πq. Then apply the Riemann mappingtheorem to the unit disc.

Problem 1.8. Show that

f g2L2pTq ¤ f f L2pTqg gL2pTq

for all f , g P L2pTq.Problem 1.9. Given N disjoint arcs tIαuN

α1 T, set f °Nα1 χIα . Show that¸

|ν|¡k

| f pνq|2 . Nk.

Problem 1.10. Given any function ψ : Z Ñ R so that ψpnq Ñ 0 as n Ñ 8, showthat you can find a measurable set E T for which

lim supnÑ8

|xχEpnq|ψpnq 8.

Problem 1.11. The problem discusses some well-known partial differential equationson the level of Fourier series.


Solve the heat equation ut uθθ 0, up0q u0 (the data at time t 0) on Tvia Fourier series. Show that if u0 P L2pTq, then upt, θq is analytic in θ for everyt ¡ 0 and solves the heat equation. Prove that uptq u02 Ñ 0 as t Ñ 0. Writeuptq Gt u0 and show that Gptq is an approximate identity for t ¡ 0. Concludethat uptq Ñ 0 as t Ñ 0 in the Lp or CpTq sense. Repeat for higher-dimensionaltori.

Solve the Schrodinger equation iut uθθ 0, up0q u0 on T with u0 P L2pTq.In what sense can you say that this Fourier series “solves” the equation? Showthat uptq2 u02 for all t. Discuss the limit uptq as t Ñ 0. Repeat forhigher-dimensional tori.

Solve the wave equation uttuθθ 0 on T by Fourier series. Discuss the Cauchyproblem as in the previous items. Show that if utp0q 0, then with up0q f ,

upt, θq 12p f pθ tq f pθ tqq

CHAPTER 2

Harmonic Functions on D and the Poisson Kernel

There is a close connection between Fourier series and analytic or harmonicfunctions on the disc D : tz P C | |z| ¤ 1u. In fact, at least formally, Fourierseries can be viewed as the “boundary values” of a Laurent series

8

n8anzn (2.1)

which can be seen by setting z x iy epθq. Alternatively, suppose we aregiven a continuous function f on T and wish to find the harmonic extension u of finto D, i.e., a solution to

4u 0 in Du f on BD T (2.2)

“Solution” here refers to a classical one, i.e., a function f P C2pDq XCpDq. How-ever, as we shall see it is also very important to investigate other notions of solu-tions of (2.2) with rougher f .

Note that we cannot use negative powers of z as in (2.1) in order to write anansatz for u. However, we can use complex conjugates instead. Indeed, since4zn 0 and 4zn 0 for every integer n ¥ 0, we are lead to defining

upzq 8

n0

f pnqzn 1

n8f pnqz|n| (2.3)

which at least formally satisfies upepθqq °8n8 f pnqepnθq f pθq. Inserting

z repθq and

f pnq »T

epnϕq f pϕq dϕ

into (2.3) yields

uprepθqq »T

¸nPZ

r|n|epnpθ ϕqq f pϕq dϕ

This resembles the derivation of the Dirichlet kernel in the previous chapter, andwe now ask the reader to find a closed form for the sum.

Exercise 2.1. Check that, for 0 ¤ r 1,

Prpθq :¸nPZ

r|n|epnθq 1 r2

1 2r cosp2πθq r2

This is the Poisson kernel.

25

26 2. HARMONIC FUNCTIONS ON D AND THE POISSON KERNEL

Based on our formal calculation above, we therefore expect to obtain the har-monic extension of a “nice enough” function f on T by means of the convolution

uprepθqq »T

Prpθ ϕq f pϕq dy pPr f qpθq

for 0 ¤ r 1.Note that Prpθq, for 0 ¤ r 1, is a harmonic function of the variables x iy

repθq. Moreover, for any finite measure µ PMpTq the expression pPr µqpθq is notonly well-defined, but defines a harmonic function on D.

The remainder of this chapter will therefore be devoted to analyzing the bound-ary behavior of Pr µ. Clearly, Proposition 1.5 will play an important role in thisinvestigation, but the fact that we are dealing with harmonic functions will enter ina crucial way (such as through the maximum principle).

In the following, we use the notion of approximate identity in a more generalform than in the previous chapter. However, the reader will have no difficultytransferring this notion to the present context, including Proposition 1.5.

Exercise 2.2. Check that tPru0 r 1 is an approximate identity. The role ofN P Z in Definition 1.3 is played here by 0 r 1 and N Ñ8 is replaced withr Ñ 1.

An important role is played by the kernel Qrpθq which is the harmonic conju-gate of Prpθq. Recall that this means that Prpθq iQrpθq is analytic in z repθqand Q0 0. In this case it is easy to find Qrpθq since

Prpθq Re

1 z1 z

and therefore

Qrpθq Im

1 z1 z

2r sinp2πθq

1 2r cosp2πθq r2 .

Exercise 2.3.

(1) Show that tQru0 r 1 is not an approximate identity.(2) Check that Q1pθq cotpπθq. Draw the graph of Q1pθq. What is the

asymptotic behavior of Q1pθq for θ close to zero?

We will study conjugate harmonic functions later. First, we clarify in whatsense the harmonic extension Pr f of f attains f as its boundary values.

Definition 2.1. For any 1 ¤ p ¤ 8 define

hppDq :!

u : DÑ C harmonic sup

0 r 1

»T|uprepθqq|p dθ 8

)These are the “little” Hardy spaces with norm

|||u|||p : sup0 r 1

uprepqqLppTq

2. HARMONIC FUNCTIONS ON D AND THE POISSON KERNEL 27

It is important to observe that Prpθq P h1pDq. This function has “boundaryvalues” δ0 (the Dirac mass at θ 0q since Pr Pr δ0.

Theorem 2.2. There is a one-to-one correspondence between h1pDq andMpTq,given by µ PMpTq ÞÑ Frpθq : pPr µqpθq. Furthermore,

µ sup0 r 1

Fr1 limrÑ1

Fr1 , (2.4)

and

(1) µ is absolutely continuous with respect to Lebesgue measure pµ ! dθqif and only if tFru converges in L1pTq. If so, then dµ f dθ wheref L1-limit of Fr.

(2) The following are equivalent for 1 p ¤ 8: dµ f dθ with f P LppTqðñ tFru0 r 1 is Lp- bounded

ðñ tFru converges in Lp if 1 p 8 and in the weak- sense in L8 ifp 8 as r Ñ 1

(3) f is continuous ô F extends to a continuous function on D ô Fr con-verges uniformly as r Ñ 1.

This theorem identifies h1pDq withMpTq, and hppDq with LppTq for 1 p ¤8. Moreover, h8pDq contains the subclass of harmonic functions that can be ex-tended continuously onto D; this subclass is the same as CpTq. Before provingthe theorem we present two simple lemmas. In what follows we use the notationFrpθq : F

repθq.

Lemma 2.3.

a) If F P CpDq and 4F 0 in D, then Fr Pr F1 for any 0 ¤ r 1.b) If 4F 0 in D, then Frs Pr Fs for any 0 ¤ r, s 1.c) As a function of r P p0, 1q the norms Frp are non-decreasing for any

1 ¤ p ¤ 8.

Proof.

a) Let uprepθqq : pPr F1qpθq for any 0 ¤ r 1, θ. Then 4u 0 in D.By Proposition 1.5 and Exercise 2.2, ur F18 Ñ 0 as r Ñ 1. Hence,u extends to a continuous function on D with the same boundary valuesas F. By the maximum principle, u F as claimed.

b) Rescaling the disc sD to D reduces b) to a).c) By b) and Young’s inequality

Frsp ¤ Pr1Fsp Fsp

as claimed.

Lemma 2.4. Let F P h1pDq. Then there exists a unique measure µ P MpTqsuch that Fr Pr µ.


Proof. Since the unit ball of MpTq is weak- compact there exists a subse-quence r j Ñ 1 with Fr j Ñ µ in weak- sense to some µ P MpTq. Then, for any0 r 1,

Pr µ limjÑ8

pFr j Prq limjÑ8

Frr j Fr

by Lemma 2.3, b). Let f P CpTq. Then xFr, f y xPrµ, f y xµ, Pr f y Ñ xµ, f yas r Ñ 1 where we again use Proposition 1.5. This shows that in the weak- sense

µ limrÑ1

Fr (2.5)

which implies uniqueness of µ.

Proof of Theorem 2.2. If µ P MpTq, then Pr µ P h1pDq. Conversely, givenF P h1pDq then by Lemma 2.4 there is a unique µ so that Fr Pr µ. This givesthe one-to-one correspondence. Moreover, (2.5) and Lemma 2.3 c) show that

µ ¤ lim suprÑ1

Fr1 sup0 r 1

Fr1 limrÑ1

Fr1

Since clearly alsosup

0 r 1Fr1 ¤ sup

0 r 1Pr1µ µ

(2.4) follows. If f P L1pTq and µpdθq f pθq dθ, then Proposition 1.5 shows thatFr Ñ f in L1pTq. Conversely, if Fr Ñ f in the sense of L1pTq, then because of(2.5) necessarily µpdθq f pθq dθ which proves p1q. p2q and p3q are equally easyand we skip the details (simply invoke Proposition 1.5 again).

Next, we turn to the issue of almost everywhere convergence of Pr f to f asr Ñ 1. By the previous theorem, we have pointwise convergence if f is continuous.If f P L1pTq, then we write f limn gn as a limit in L1 with gn P CpTq. But thenwe face two limits, namely then one as n Ñ 8 and that as r Ñ 1, and wewish to interchange them. It is a common feature that such an interchange requiressome form of uniform control. To be specific, we will use the Hardy-Littlewoodmaximal function M f associated to f in order to dominate the convolution with thePoisson kernel Pr f uniformly in 0 r 1. This turns out to be an instance ofthe general fact that radially bounded approximate identities are dominated by theHardy-Littlewood maximal function, see condition A4) below.

The Hardy-Littlewood maximal function is defined as

M f pθq supθPIT

1|I|

»I| f pϕq| dϕ

where I T is an open interval and |I| denotes the length of I. The basic fact aboutM f that we shall need in this context is given by the following result. We say thatg belongs to weak-L1 if

|tθ P T | |gpθq| ¡ λu| ¤ Cλ1g1

for all λ ¡ 0. Henceforth, | | applied to sets denotes Lebesgue measure. Note thatany g P L1 belongs to weak-L1. This is an instance of Markov’s inequality

|tθ P T | | f pθq| ¡ λs ¤ λp f pp


for any 1 ¤ p 8 and any λ ¡ 0.

Proposition 2.5. The Hardy-Littlewood maximal function satisfies the follow-ing bounds:

a) M is bounded from L1 to weak-L1, i.e.,

|tθ P T | M f pθq ¡ λs ¤ 3λ f 1

for all λ ¡ 0.b) For any 1 p ¤ 8, M is bounded on Lp.

Proof. Fix some λ ¡ 0 and any compact

K tθ P T | M f pθq ¡ λu (2.6)

There exists a finite cover tI juNj1 of T by open arcs I j such that»

I j

| f pϕq| dϕ ¡ λ|I j| (2.7)

for each j. We now apply Wiener’s covering lemma to pass to a more convenientsub-cover: Select an arc of maximal length from tI ju; call it J1. Observe that anyI j such that I j X J1 , H satisfies I j 3 J1 where 3 J1 is the arc with the samecenter as J1 and three times the length (if 3 J1 has length larger than 1, then set3 J1 T). Now remove all arcs from tI juN

j1 that intersect J1. Let J2 be one ofthe remaining ones with maximal length. Continuing in this fashion we obtain arcstJ`uL

`1 which are pair-wise disjoint and so that

N¤j1

I j L¤`1

3 J`

In view of (2.6) and (2.7) therefore

|K| ¤ L¤`1

3 J` ¤ 3

L

`1

|J`|

¤ 3λ

L

`1

»J`| f pϕq| dϕ ¤ 3

λ f 1

as claimed.To prove part b), one interpolates the bound from a) with the trivial L8 bound

M f 8 ¤ f 8by means of Marcinkiewicz’s interpolation theorem.

We remark that the same argument based on Wiener’s covering lemma yieldsthe corresponding statement for Mµ when µ PMpTq. In other words, if we define

pMµqpθq supθPIT

|µ|pIq|I|


then for all λ ¡ 0|tθ P T | pMµqpθq ¡ λu| ¤ 3

λµ

where µ is the total variation of µ.

Exercise 2.4. Find within a multiplicative constant Mδ0, where δ0 is the Diracmeasure at 0. Use this to prove that the weak-L1 bound on Mµ cannot be improvedin general, even when µ is absolutely continuous.

For a considerable strengthening of the previous exercise, the reader shouldconsult Problem 3.4 in the next chapter. We now introduce the class of approximateidentities which are dominated by the maximal function.

Definition 2.6. Let tΦnu8n1 be an approximate identity as in Definition 1.3.We say that it is radially bounded if there exist functions tΨnu8n1 on T so that thefollowing additional property holds:

A4) |Φn| ¤ Ψn, Ψn is even and decreasing, i.e., Ψnpθq ¤ Ψnpϕq for 0 ¤ ϕ ¤θ ¤ 1

2 , for all n ¥ 1. Finally, we require that supn Ψn1 8.

All basic examples of approximate identities which we have encountered sofar (Fejer, Poisson, and box kernels) are of this type. The following lemma estab-lishes the aforementioned uniform control of the convolution with radially boundedapproximate identities in terms of the Hardy-Littlewood maximal function.

Lemma 2.7. If tΦnu8n1 satisfies A4q, then for any f P L1pTq one has

supn|pΦn f qpθq| ¤ sup

nΨn1 M f pθq

for all θ P T.

Proof. It clearly suffices to show the following statement: let

K : r12,

12s Ñ R Y t0u

be even and decreasing. Then for any f P L1pTq we have

|pK f qpθq| ¤ K1M f pθq (2.8)

Indeed, assume that (2.8) holds. Then

supn|pΦn f qpθq| ¤ sup

npΨn | f |qpθq ¤ sup

nΨn1 M f pθq

and the lemma follows. The idea behind (2.8) is to show that K can be written asan average of box kernels, i.e., for some positive measure µ

Kpθq » 1

2

0χrϕ,ϕspθq µpdϕq (2.9)

We leave it to the reader to check that dµ dK K1

2

δ 1

2is a suitable choice.

Notice that (2.9) implies that» 1

0Kpθq dθ

» 12

02ϕ dµpϕq


Moreover, by (2.9),

|pK f qpθq| » 1

2

0

12ϕχrϕ,ϕs f

pθq 2ϕ µpdϕq

¤ M f pθq» 1

2

02ϕ µpdϕq

M f pθq K1

which is (2.8).

The following theorem establishes the main almost everywhere convergenceresult of this chapter. The reader should note how the maximal function entersprecisely in order to interchange the aforementioned double limit.

Theorem 2.8. If tΦnu8n1 satisfies A1)–A4), then for any f P L1pTq one hasΦn f Ñ f almost everywhere as n Ñ8.

Proof. Pick ε ¡ 0 and let g P CpTq with f g1 ε. By Proposition 1.5,with h f g one has

|tθ P T | lim supnÑ8

|pΦn f qpθq f pθq| ¡ ?εu|

¤|tθ P T | lim supnÑ8

|pΦn hqpθq| ¡ ?ε2u| |tθ P T | |hpθq| ¡ ?

ε2u|

¤|tθ P T | supn|pΦn hqpθq| ¡ ?

ε2u| |tθ P T | |hpθq| ¡ ?ε2u|

¤|tθ P T | CMhpθq ¡ ?ε2u| |tθ P T| | hpθq| ¡ ?

ε2u|¤C

?ε

To pass to the final inequality we used Proposition 2.5 as well as Markov’s inequal-ity (recall h1 ε).

As a corollary we not only obtain the classical Lebesgue differentiation theo-rem (by considering the box kernel), but also almost everywhere convergence ofthe Cesaro means σN f (via the Fejer kernel), as well as of the Poisson integralsPr f to f for any f P L1pTq. A theorem of Kolmogoroff states that this fails forthe partial sums S N f on L1pTq. We will present this example in Chapter 5.

Exercise 2.5. It is natural to ask whether there is an analogue of Theorem 2.8for measures µ PMpTq. Prove the following:

a) If µ P MpTq is a positive measure which is singular with respect toLebesgue measure m (in symbols, µKm), then for a.e. θ P T with respectto Lebesgue measure we have

µprθ ε, θ εsq2ε

Ñ 0 as εÑ 0


b) Let tΦnu8n1 satisfy A1)–A4), and assume that the tΨnu8n1 from Defini-tion 2.1 also satisfy

supδ |θ| 1

2

|Ψnpθq| Ñ 0 as n Ñ8

for all δ ¡ 0. Under these assumptions show that for any µ PMpTqΦn µÑ f a.e. as n Ñ8

where µpdθq f pθq dθ νspdθq is the Lebesgue decomposition, i.e.,f P L1pTq and νsKm.

Notes

Standard references on everything here are Hoffman [25], Garnett [20], and Koosis [34].

Problems

Problem 2.1. Let pX, µq be a general measure space. We say a sequence t fnu8n1 PL1pµq is uniformly integrable if for every ε ¡ 0 there exists δ ¡ 0 such that

µpEq δ ùñ supn

»E

fn du ε

Suppose µ is a finite measure. Let φ : r0,8q Ñ r0,8q be a continuous increasing functionwith limtÑ8

φptqt 8. Prove that

supn

»φp| fnpxq|q µpdxq 8

implies that t fnu is uniformly integrable.

Problem 2.2. Suppose t fnu8n1 is a sequence in L1pTq. Show that there is a subse-

quence t fn ju8j1 and a measure µ with fn j

σÝÑ µ provided supn fn1 8. Here σ isthe weak-star convergence of measures. Show that in general µ < L1pTq. However, if weassume that, in addition, t fnu81 is uniformly integrable, then µpdθq f pθq dθ for somef P L1pTq. Can we conclude anything about strong convergence (i.e., in the L1-norm) oft fnu? Consider the analogous question on LppTq, p ¡ 1.

Problem 2.3. Let µ be a positive finite Borel measure on Rd or Td. Set

Mµpxq : supr¡0

µpBpx, rqqmpBpx, rqq

where m is the Lebesgue measure. This problem examines the behavior of Mµ when µ is asingular measure, cf. Problem 3.4 for more on this case.

(1) Show that µ K m implies µptx | Mµpxq 8uq 0(2) Show that if µ K m, then

lim suprÑ0

µpBpx, rqqmpBpx, rqq 8

for µ a.e. x. Also show that this limit vanishes m-a.e.


Problem 2.4. For any f P L1pRdq and 1 ¤ k ¤ d let

Mk f pxq supr¡0

rk»

Bpx,rq| f pyq| dy

Show that for every λ ¡ 0

mLptx P L | Mk f pxq ¡ λuq ¤ Cλ f L1pRdq

where L is an arbitrary affine k-dimensional subspace and “mL” stands for Lebesgue mea-sure (i.e., k-dimensional measure) on this subspace. C is an absolute constant.

Problem 2.5. Prove the Besicovitch covering lemma on T: Suppose tI ju are finitelymany arcs with |I j| 1. Then there is a sub-collection tI jku such that the followingproperties hold:

(1) YkI jk Y jI j

(2) No point belongs to more than C of the I jk ’s where C is an absolute constant.What is the best value of C?

What can you say about higher dimensions?

Problem 2.6. Let F ¥ 0 be a harmonic function onD. Show that there exists a positivemeasure µ on T with Fprepθqq pPr µqpθq for 0 ¤ r 1 and with µ Fp0q.

Problem 2.7. A sequence of complex numbers tanunPZ is called positive definite if

an an for all n P Z °

n,k ank znzk ¥ 0 for all complex sequences tznunPZ

Show that any positive definite sequence satisfies |an| ¤ a0 for all n P Z. Show that ifµ is a positive measure, then an ³

T epnθqµpdθq is such a sequence. Now prove thatevery positive definite sequence is of this form. Hint: apply the previous problem to theharmonic function

°n an rnepnθq.

CHAPTER 3

Analytic h1pDq Functions, F. & M. Riesz Theorem

In the previous chapter, we mainly dealt with functions harmonic in the disksatisfying various boundedness properties. We now turn to functions F u iv Ph1pDq which are analytic in D. This is well-defined since analytic functions arecomplex-valued harmonic ones. These functions form the class H1pDq, the “big”Hardy space. We have shown there that Fr Pr µ for some µ P MpTq. Itis important to note that by analyticity µpnq 0 if n 0. A theorem by F. &M. Riesz asserts that such measures are absolutely continuous. From the example

Fpzq : 1 z1 z

Prpθq iQrpθq, z repθqone sees an important difference between the analytic and the harmonic cases. In-deed, while Pr P h1pDq, clearly F < h1pDq. The boundary measure associated withPr is δ0, whereas F is not associated with any boundary measure in the sense ofthe previous chapter.

An important technical device that will allow us to obtain more information inthe analytic case is given by subharmonic functions. Loosely speaking, this devicewill allow us to exploit algebraic properties of analytic functions that harmonicones do not have (such as the fact that F2 is again analytic if F is analytic, whichdoes not hold in the harmonic category).

Definition 3.1. Let Ω R2 be a region (i.e., open and connected) and letf : Ω Ñ R Y t8u where we extend the topology to R Y t8u in the obviousway. We say that f is subharmonic if

(1) f is continuous(2) for all z P Ω there exists rz ¡ 0 so that

f pzq ¤» 1

0fz repθq dθ

for all 0 r rz. We refer to this as the (local) “submean value prop-erty”.

It is helpful to keep in mind that in one dimension “harmonic = linear” and“subharmonic = convex”. Subharmonic functions derive their name from the factthat they lie below harmonic ones, in the same way that convex functions lie be-low linear ones. We will make this precise by means of the important maximumprinciple which subharmonic functions obey. We begin by deriving some basicproperties of this class.

35

36 3. ANALYTIC h1pDq FUNCTIONS, F. & M. RIESZ THEOREM

Lemma 3.2. Subharmonic functions satisfy the following properties:

1) If f and g are subharmonic, then f _ g : maxp f , gq is subharmonic.2) If f P C2pΩq then f is subharmonic ðñ 4 f ¥ 0 in Ω

3) F analytic ùñ log |F| and |F|α with α ¡ 0 are subharmonic4) If f is subharmonic and ϕ is increasing and convex, then ϕ f is subhar-

monic pwe set ϕp8q : limxÑ8 ϕpxqq.Proof. 1) is immediate. For 2) use Jensen’s formula»

Tf pz repθqq dθ f pzq

"Dpz,rq

logr

|w z|4 f pwqmpdwq (3.1)

where m stands for two-dimensional Lebesgue measure and

Dpz, rq tw P C | |w z| ruThe reader is asked to verify this formula in the following exercise. If 4 f ¥ 0, thenthe submean value property holds. If 4 f pz0q 0, then let r0 ¡ 0 be sufficientlysmall so that 4 f 0 on Dpz0, r0q Since log r0

|wz0| ¡ 0 on this disk, Jensen’sformula implies that the submean value property fails. Next, we verify 4) by meansof Jensen’s inequality for convex functions:

ϕp f pzqq ¤ ϕ »T

f pz repθqq dθ¤

»Tϕp f pz repθqq dθ

The first inequality sign uses that ϕ is increasing, whereas the second uses convex-ity of ϕ (this second inequality is called Jensen’s inequality). If F is analytic, thenlog |F| is continuous with values in R Y t8u. If Fpz0q , 0, then log |Fpzq| isharmonic on some disk Dpz0, r0q. Thus, one has the stronger mean value propertyon this disk. If Fpz0q 0, then log |Fpz0q| 8, and there is nothing to prove.To see that |F|α is subharmonic, apply 4) to log |Fpzq| with ϕpxq exppαxq.

Exercise 3.1. Prove Jensen’s formula (3.1) for C2 functions.

Now we can derive the aforementioned domination of subharmonic functionsby harmonic ones.

Lemma 3.3. Let Ω be a bounded region. Suppose f is subharmonic on Ω,f P CpΩq and let u be harmonic on Ω, u P CpΩq. If f ¤ u on BΩ, then f ¤ u onΩ.

Proof. We may take u 0, so f ¤ 0 on BΩ. Let M maxΩ f and assumethat M ¡ 0. Set

S tz P Ω | f pzq MuThen S Ω and S is closed in Ω. If z P S , then by the submean value propertythere exists rz ¡ 0 so that Dpz, rzq Ω. Hence S is also open. Since Ω is assumedto be connected, one obtains S Ω. This is a contradiction.

The following lemma shows that the submean value property holds on any diskin Ω. The point here is that we upgrade the local submean value property to a truesubmean value property using the largest possible disks.

3. ANALYTIC h1pDq FUNCTIONS, F. & M. RIESZ THEOREM 37

Lemma 3.4. Let f be subharmonic in Ω, z0 P Ω, Dpz0, rq Ω. Then

f pz0q ¤»T

f pz0 repθqq dθ

Proof. Let gn maxp f ,nq, where n ¥ 1. Without loss of generality z0 0.Define unpzq to be the harmonic extension of gn restricted to BDpz0, rq where r ¡ 0is as in the statement of the lemma. By the previous lemma,

f p0q ¤ gnp0q ¤ unp0q »T

unprepθqq dθ

the last equality being the mean value property of harmonic functions. Since

max|z|¤r

unpzq ¤ max|z|¤r

f pzq

the monotone convergence theorem for decreasing sequences yields

f p0q ¤»T

f prepθqq dθ

as claimed.

Corollary 3.5. If g is subharmonic on D, then for all θ

gprsepθqq ¤ pPr gsqpθqfor any 0 r, s 1.

Proof. If g ¡ 8 everywhere on D, then this follows from Lemma 3.3. Ifnot, then set gn g_ n. Thus

gprsepθqq ¤ gnprsepθqq ¤ pPr pgnqsqpθqand consequently

gprsepθqq ¤ lim supnÝÑ8

pPr pgnqsqpθq ¤ pPr gsqpθq

where the final inequality follows from Fatou’s lemma (which can be applied in the“reverse form” here since the gn have a uniform upper bound).

Note that if gs < L1pTq, then g 8 on Dp0, sq and so g 8 on Dp0, 1q.We now introduce the radial maximal function associated with any function on thedisk. It, and the more general nontangential maximal function where the supremumis taken over a cone in D, are of central importance in the analysis of this chapter.

Definition 3.6. Let F be any complex-valued function on D then F : TÑ Ris defined as

Fpθq sup0 r 1

|Fprepθqq|


We showed in the previous chapter that any u P h1pDq satisfies u ¤ CMµwhere µ is the boundary measure of u, i.e., ur Pr µ. In particular, one has

|tθ P T | upθq ¡ λu| ¤ Cλ1|||u|||1We shall prove the analogous bound for subharmonic functions which are L1-bounded.

Proposition 3.7. Suppose g is subharmonic on D, g ¥ 0 and g is L1-bounded,i.e.,

|||g|||1 : sup0 r 1

»T

gprepθqq dθ 8

Then

(1) |tθ P T | gpθq ¡ λu| ¤ 3λ |||g|||1 for @λ ¡ 0.

(2) If g is Lp bounded, i.e.,

|||g|||pp : sup0 r 1

»T

gprepθqqp dθ 8

with 1 p ¤ 8, then

gLppTq ¤ Cp|||g|||pProof.

(1) Let grn á µ PMpTq in the weak- sense. Then µ ¤ |||g|||1 and

gs ÐÝ grn s ¤ grn Ps ÝÑ Ps µThus, by Lemma 2.7,

g ¤ sup0 s 1

Ps µ ¤ Mµ

and the desired bound now follows from Proposition 2.5.(2) If |||g|||p 8, then dµ

dθ P LppTq with dµdθ p ¤ |||g|||p and thus

g ¤ CMdµ

dθ

P LppTqby Proposition 2.5, as claimed.

We now present three versions of a theorem due to F. & M. Riesz. It is impor-tant to note that the following result fails without analyticity.

Theorem 3.8 (First Version of the F. & M. Riesz Theorem). Suppose F P h1pDqis analytic. Then F P L1pTq.

Proof. |F| 12 is subharmonic and L2-bounded. By Proposition 3.7 therefore

|F| 12 P L2pTq. But |F| 1

2 |F| 12 and thus F P L1pTq.


Let F P h1pDq. By Theorem 2.2, Fr Pr µ where µ PMpTq has a Lebesguedecomposition µpdθq f pθq dθ νspdθq, νs singular and f P L1pTq. By Exer-cise 2.5 b) one has Pr µÑ f a.e. as r Ñ 1. Thus, limrÑ1 Fprepθqq f pθq existsfor a.e. θ P T. This justifies the statement of the following theorem.

Theorem 3.8 (Second Version). Assume F P h1pDq and F analytic. Let f pθq limrÑ1 F

repθq. Then Fr Pr f for all 0 r 1.

Proof. We have Fr Ñ f a.e. and |Fr| ¤ F P L1 by the previous theorem.Therefore, Fr Ñ f in L1pTq and Theorem 2.2 finishes the proof.

Theorem 3.8 (Third Version). Suppose µ P MpTq, µpnq 0 for all n 0.Then µ is absolutely continuous with respect to Lebesgue measure.

Proof. Since µpnq 0 for n P Z one has that Fr Pr µ is analyticon D. By the second version above and the remark preceding it, one concludes thatµpdθq f pθq dθ with f limrÑ1 F

repθq P L1pTq, as claimed.

The logic of this argument shows that if µKm (where m is Lebesgue measure),then the harmonic extension uµ of µ satisfies uµ < L1pTq. It is possible to give amore quantitative version of this fact. Indeed, suppose that µ is a positive measure.Then for some absolute constant C,

C1Mµ uµ CMµ (3.2)

where the upper bound is Lemma 2.7 (applied to the Poisson kernel) and the lowerbound follows from the assumption µ ¥ 0 and the fact that the Poisson kerneldominates the box kernel. Part a) of Problem 3.4 below therefore implies the quan-titative non-L1 statement

|tθ P T | uµpθq ¥ λu| ¥ Cλµ

for µKm , where µ is a positive measure.The F. & M. Riesz theorem raises the following question: Given f P L1pTq,

how can one decide ifPr f iQr f P h1pDq ?

We know that necessarily uf pPr f q P L1pTq. A theorem by Burkholder,Gundy, and Silverstein states that this is also sufficient (they proved this for thenon-tangential maximal function). It is important to note the difference from (3.2),i.e., that this is not the same as the Hardy-Littlewood maximal function satisfyingM f P L1pTq due to possible cancellation in f . In fact, it is known that

M f P L1pTq ðñ | f | logp2 | f |q P L1

, see Problem 7.6.

We conclude this chapter with another theorem due to the Riesz brothers,which can be seen as a generalization of the uniqueness theorem for analytic func-tions. In fact, it says that if an analytic function F on the disk does not have too wildgrowth as one approaches the boundary (expressed through the L1-boundedness


condition), then the boundary values are well-defined and cannot vanish on a set ofpositive measure (unless F vanishes identically).

Theorem 3.9 (Second F. & M. Riesz Theorem). Let F be analytic on D andL1-bounded, i.e., F P h1pDq. Assume F 0 and set f limrÑ1 Fr. Thenlog | f | P L1pTq. In particular, f does not vanish on a set of positive measure.

Proof. The idea is that if Fp0q , 0, then»T

log | f pθq| dθ ¥ log |Fp0q| ¡ 8

Since log | f | ¤ | f | P L1pTq by Theorem 3.8, we should be done. Some careneeds to be taken, though, as F attains the boundary values f only in the almosteverywhere sense. This issue can easily be handled by means of Fatou’s lemma.First, F P L1pTq, so log |Fr| ¤ F implies that log | f | P L1pTq by Lebesguedominated convergence. Second, by subharmonicity,»

Tlog |Frpθq| dθ ¥ log |Fp0q|

so that »T

log | f pθq| dθ »T

limrÑ1

log |Frpθq| dθ

¥ lim suprÑ1

»T

log |Frpθq| dθ ¥ log |Fp0q|

If Fp0q , 0, then we are done. If Fp0q 0, then choose another point z0 P Dfor which Fpz0q , 0. Now one either repeats the previous argument with thePoisson kernel instead of the submean value property, or one composes F with anautomorphism of the unit disk that moves 0 to z0. Then the previous argumentapplies.

Theorem 3.9 generalizes the following fact from complex analysis, which isproved by Schwarz reflection and Riemann mapping: Let F be analytic in Ω andcontinuous up to Γ BΩ which is an open Lipschitz arc. If F 0 on Γ, thenF 0 in Ω.

Notes

For all of this see Hoffman’s classic [25], Garnett [20], and especially Koosis [34]. TheBurkholder-Gundy-Silverstein theorem, which is proved in [34], can be seen as a precursorto the Fefferman-Stein grand maximal function which leads to the real-variable theory ofHardy space, see [46]. There are some glaring omissions in this chapter such as inner andouter functions. However, we shall not require that aspect of Hp theory in this book.

Problems


Problem 3.1. Establish the maximum principle for subharmonic functions:a) Show that if u is subharmonic on Ω and satisfies upzq ¤ upz0q for all z P Ω and

some fixed z0 P Ω, then u is a constant.b) Show that if Ω is bounded and u is continuous on Ω, then upzq ¤ maxBΩ u for all

z P Ω with equality being possibly only if u is a constant.

Problem 3.2. Let u be subharmonic on a domain Ω. Show that there exist a uniquemeasure µ on Ω such that µpKq 8 for every K Ω (i.e., K is a compact subset of Ω)and so that

upzq "

log |z ζ| dµpζq hpzqwhere h is harmonic on Ω. This is called “Riesz’s representation of subharmonic func-tions”.

Problem 3.3. With u and µ as in the previous exercise, show that»T

upz repθqq dθ upzq » r

0

µpDpz, tqqt

dt

for all Dpz, rq Ω (this generalizes “Jensen’s formula” (3.1) to functions which are nottwice continuously differentiable). Recover the classical Jensen’s formula from complexanalysis by setting upzq : log | f pzq| where f is analytic.

Problem 3.4. This problem explores the behavior of the Hardy-Littlewood maximalfunction of measures which are singular relative to Lebesgue measure.

a) Prove that if µ PMpTqzt0u satisfies µ K m, then Mµ < L1. In fact, show that

|tθ P T | Mµpθq ¡ λu| ¥ cλµ

provided λ ¡ µ with an absolute constant c ¡ 0.b) Prove that there is a numerical constant C such that if µ P MpTq is a positive

measure and F the associated harmonic function, then Mµ ¤ CF. Concludethat if µ is singular, then F < L1.

Problem 3.5. Show that the class of analytic function Fpzq in the unit disk with posi-tive real part is in one-to-one relation with the class of functions of the form

Fpζq »T

1 ζepθq1 ζepθq µpdθq ic, |ζ| 1

where µ is a positive Borel measure on T and c is a real constant.

CHAPTER 4

The Conjugate Harmonic Function

While the previous two chapters focused on the Poisson kernel Pr, this chapteris devoted to the study of the kernel Qr conjugate to Pr which we introduced inChapter 2. This turns out to have far-reaching consequences, for example it willallow us to answer the question of Lp-convergence of Fourier series with 1 p 8which was raised in Chapter 1. We begin by recalling the definition of conjugatefunction.

Definition 4.1. Let u be real-valued and harmonic inD. Then we define u to bethat unique real-valued and harmonic function in D for which u iu is analytic andup0q 0. If u is complex-valued and harmonic, then we set u : pRe uqr ipIm uqr.

The following lemma presents some simple but useful properties of the har-monic conjugate u.

Lemma 4.2. Let u be the harmonic conjugate as above.

1) If u is constant, then u 0.2) If u is analytic in D and up0q 0, then u iu. If u is co-analytic

(meaning that u is analytic), then u iu.3) Any harmonic function u can be written uniquely as u c f g with

c constant, f , g analytic, and f p0q gp0q 0.

Proof. 1) and 2) follow immediately from the definition, whereas 3) is givenby

u up0q 12pu up0q iuq 1

2pu up0q iuq

Uniqueness of c, f , g is also clear.

There is a simple relation between the Fourier coefficients of ur and that ofits harmonic conjugate. This relation will be the key to expressing the partial sumoperator for Fourier series in terms of the harmonic conjugate. Henceforth, weshall occasionally use F to denote the Fourier transform for notational reasons.

Lemma 4.3. Suppose u is harmonic on D. Then for all n P Z, n , 0

F purqpnq i signpnqpurpnq (4.1)

Proof. By Lemma 4.2 it suffices to consider u constant, analytic, co-analytic.We present the case u analytic, up0q 0. Then u iu so that F purqpnq ipurpnq for all u P Z. But purpnq 0 for u ¤ 0 and thus (4.1) holds in this case.

43

44 4. THE CONJUGATE HARMONIC FUNCTION

Together with the Plancherel theorem, the previous lemma immediately allowsus to conclude the following relation between the L2-norms.

Proposition 4.4. Let u P h2pDq be real-valued. Then u P h2pDq. In fact,

ur22 ur2

2 |up0q|2

Proof. As already mentioned, this follows from Lemma 4.3 and the Planchereltheorem. Alternatively, by Cauchy’s theorem,»

Tpur iurq2pθq dθ pu iuq2p0q u2p0q

Since the right-hand side is real-valued, the left-hand side is also necessarily realand thus

u2p0q »T

u2r pθq dθ

»T

u2r pθq dθ

as claimed.

The previous L2-boundedness result allows us to determine the boundary val-ues of the conjugates of h2 functions.

Corollary 4.5. If u P h2pDq, then limrÑ1 uprepθqq exists for a.e. θ P T as anL2pTq function, and the convergence is also in the sense of L2.

Proof. By Proposition 4.4 we know that u P h2pDq and the result of Chapter 2lead to the desired conclusion.

We now consider the case of h1pDq. From the example of the Poisson kerneland its conjugate we see that there is no hope to obtain a result as strong as inthe case of h2. However, we shall now show that the radial maximal function (asdefined in the previous chapter) of the conjugate to any h1pDq-function is boundedin weak-L1pTq.

We emphasize, however, that this has nothing to do with the weak-L1 bound-edness of the Hardy-Littlewood maximal function. In fact, it is not possible tocontrol the convolutions with Qr by means of that maximal function uniformlyin 0 r 1.

Nevertheless, the following theorem due to Besicovitch and Kolmogoroff showsthat the weak-L1 property still holds. The proof presented here is based on the no-tion of harmonic measure.

Theorem 4.6. Let u P h1pDq. Then

|tθ P T | |upθq| ¡ λu| ¤ Cλ|||u|||1

with some absolute constant C.

4. THE CONJUGATE HARMONIC FUNCTION 45

Proof. By Theorem 2.2, ur Pr µ. Splitting µ into real and imaginary parts,and then each piece into its positive and negative parts, we reduce ourselves to thecase u ¥ 0. Let

Eλ tθ P T | upθq ¡ λuand set F u iu. Then F is analytic and Fp0q iup0q. Define a function

ωλpx, yq : 1π

»p8,λqYpλ,8q

ypx tq2 y2 dt

which is harmonic for y ¡ 0 and non negative. The following two properties of ωλwill be important:

(1) ωλpx, yq ¥ 12 if |x| ¡ λ

(2) ωλp0, yq ¤ 2yπλ .

For the first property it suffices to note that» 8

0

1π

yx2 y2 dx 1

2@y ¡ 0

For the second property compute

ωλp0, yq 1π

»p8,λqYpλ,8q

yt2 y2 dt ¤ 2

π

» 8

λydt

1 t2 ¤2yπλ,

as claimed.Observe now that ωλ F is harmonic and that θ P Eλ implies

pωλ Fqprepθqq ¥ 12

for some 0 r 1. Thus

|Eλ| ¤ |tθ P T | pωλ Fqpθq ¥ 12u| ¤ 312 |||ωλ F|||1 (4.2)

by Proposition 3.7. Since ωλ F ¥ 0, the mean value property implies that

|||ωλ F|||1 pωλ Fqp0q ωλpiup0qq ¤ 2π

up0qλ

2π

|||u|||1λ

.

Combining this with (4.2) yields

|Eλ| ¤ 12π

|||u|||1λ

and we are done.

Exercise 4.1. Show that ωλpx, yq equals an angle measured from z x iy.Which angle? Use this to show that ωλpx, yq ¥ 1

2 provided px, yq lies outside thesemi-circle with radius λ and center 0. Furthermore, show that ωλ is the uniqueharmonic function in the upper half plane which equals 1 on Γ : p8,λs Yrλ,8q, equals 0 on pλ, λq RzΓ, and which is globally bounded. It is called theharmonic measure of p8,λs Y rλ,8q, and can be defined in the same fashionon general domains Ω and any open Γ Ω (in fact, more general Γ). It turns outthat the harmonic measure of Γ relative to Ω equals the probability that Brownian


motion starting at z P Ω hits the boundary for the first time at Γ rather than in BΩzΓ.Of particular importance is the conformal invariance of this notion, but we makeno use of this probabilistic connection in this book.

The following result introduces the Hilbert transform and establishes a weak-L1 bound for it. Formally speaking, the Hilbert transform Hµ of a measure µ PMpTq is defined by

µ ÞÑ uµ ÞÑ ruµ ÞÑ limrÑ1

p ruµqr : Hµ

i.e., the Hilbert transform of a function on T equals the boundary values of theconjugate function of its harmonic extension. By Corollary 4.5 this is well definedif µpdθq f pθq dθ, f P L2pTq. We now consider the case f P L1pTq.

Corollary 4.7. Given u P h1pDq the limit limrÑ1 uprepθqq exists for a.e. θ.With u Pr µ, µ P MpTq, this limit is denoted by Hµ. There is the weak-L1

bound

|tθ P T | |Hµpθq| ¡ λu| ¤ Cλµ

Proof. If µpdθq f pθq dθ with f P L2pT, then limrÑ1

ru f prepθqq exists for a.e. θ

by Corollary 4.5. If f P L1pTq and ε ¡ 0, then let g P L2pTq such that fg1 ε.Denote, for any δ ¡ 0,

Eδ : tθ P T | lim supr,sÑ1

| ru f prepθqq ru f psepθqq| ¡ δu

andFδ : tθ P T | lim sup

r,sÑ1| ruhprepθqq ruhpsepθqq| ¡ δu

where h f g. In view of the preceding theorem and the L2-case,

|Eδ| |Fδ| ¤ |tθ P T | p ruhqpθq ¡ δ2u|¤ C

δ|||uh|||1 ¤ C

δ f g1 Ñ 0

as ε Ñ 0. This finishes the case where µ is absolutely continuous relative toLebesgue measure.

To treat singular measures, we first consider measures µ ν which satisfy|supppνq| 0. Here

supppνq : Tz Y tI T | νpIq 0uI being an arc. Observe that for any θ < supppνq the limit limrÑ1 rurpθq existssince the analytic function u iu can be continued across that interval J on Tfor which µpJq 0 and which contains θ. Hence limrÑ1 rur exist a.e. by theassumption |supppνq| 0. If µ PMpTq is an arbitrary singular measure, then useinner regularity to say that for every ε ¡ 0 there is ν P MpTq with µ ν εand |supppνq| 0. Indeed, set νpAq : µpA X Kq for all Borel sets A whereK is compact and |µ|pTzKq ε. The theorem now follows by passing from the


statement for ν to that for µ by means of the same argument that was used in theabsolutely continuous case above.

It is now easy to obtain the Lp boundedness of the Hilbert transform on 1 p 8. This result is due to Marcel Riesz, who gave a different proof, see thefollowing exercise for his original argument.

Theorem 4.8. If 1 p 8, then H f p ¤ Cp f p. Consequently, if u PhppDq with 1 p 8, then u P hppDq and |||u|||p ¤ Cp|||u|||p

Proof. By Proposition 4.4, Hu2 ¤ u2 with equality if and only if»T

upθq dθ 0

Interpolating this with the weak-L1 bound from Corollary 4.7 by means of theMarcinkiewicz interpolation theorem finishes the case 1 p ¤ 2. If 2 p 8,then we use duality. More precisely, if f , g P L2pTq, then

x f ,Hgy ¸nPZ

f pnqxHgpnq ¸nPZ

isignpnq f pnqgpnq

¸nPZ

xH f pnqgpnq xH f , gy

This shows that H H. Hence, if f P LppTq L2pTq and g P L2pTq Lp1pTq,then

|xH f , gy| |x f ,Hgy| ¤ f pHgp1 ¤ Cp1 f pgp1

and thus H f p ¤ Cp1 f p as claimed.

Exercise 4.2. Give a complex variable proof of the L2mpTq-boundedness of theHilbert transform by applying the Cauchy integral formula to pu iuq2m when mis a positive integer, cf. the proof of Proposition 4.4. Obtain the previous theoremfrom this by interpolation and duality.

Consider the analytic mapping F u iv that takes D onto the strip

tz | |Re z| 1uThen u P h8pDq but clearly v < h8pDq so that Theorem 4.8 fails on L8pTq. Byduality, it also fails on L1pTq. The correct substitute for L1 in this context is thespace of real parts of functions in H1pDq. This is a deep result that goes muchfurther than the F. & M. Riesz theorem. The statement is that

H f 1 ¤ Cuf 1 (4.3)

where uf is the non-tangential maximal function of the harmonic extension u f off . Recall that by the Burkholder-Gundy-Silverstein theorem the right-hand side in(4.3) is finite if and only if f is the real part of an analytic L1-bounded function.A more modern approach to (4.3) is given by the real-variable theory of Hardyspaces, which subsume statements such as (4.3) by the boundedness of singularintegral operators on such spaces.


Next, we turn to the problem of expressing H f in terms of a kernel. By Exer-cise 2.3, it is clear that one would expect that

pHµqpθq »T

cotpπpθ ϕqq µpdϕq (4.4)

for any µ P MpTq. This, however, requires justification as the integral on theright-hand side is not necessarily convergent.

Proposition 4.9. If µ PMpTq, then

limεÑ0

»|θϕ|¡ε

cotpπpθ ϕqq µpdϕq pHµqpθq (4.5)

for a.e. θ P T. In other words, (4.4) holds in the principal value sense.

Exercise 4.3. Verify that the limit in (4.5) exists for all dµ f dθ dν wheref P C1pTq and |supppνq| 0. Furthermore, show that these measures are dense inMpTq.

Proof of Proposition 4.9. The idea is to represent a general measure as a limitof measures of the kind given by the previous exercise. As we have seen in theproof of the a.e. convergence result Theorem 2.8, the double limit appearing insuch an argument requires a bound on an appropriate maximal function. In thiscase the natural bound is of the form

mesθ P T | sup

0 ε 12

»|θϕ|¡ε

cotpπpθ ϕqq µpdϕq ¡ λ

¤ C

λµ (4.6)

for all λ ¡ 0. We leave it to the reader to check that (4.6) implies the theorem.In order to prove (4.6) we invoke our strongest result on the conjugate function,namely Theorem 4.6. More precisely, we claim that

sup0 r 1

pQr µqpθq »|θϕ|¡1r

cotpπpθ ϕqq µpdϕq ¤ CMµpθq (4.7)

where Mµ is the Hardy-Littlewood maximal function. Since

sup0 r 1

|pQr µqpθq| ruµpθq(4.6) follows from (4.7) by means of Theorem 4.6 and Proposition 2.5. To verify(4.7) write the difference inside the absolute value signs as pKr µqpθq, where

Krpθq "

Qrpθq cotpπθq if 1 r |θ| 12

Qrpθq if |θ| ¤ 1 r

By means of calculus one checks that

|Krpθq| ¤#

C p1rq2

|θ|3 if |θ| ¡ 1 rCp1 rq1 if |θ| ¤ 1 r

This proves that tKru0 r 1 form a radially bounded approximate identity and (4.7)therefore follows from Lemma 2.7.


Proposition 4.9 raises the question whether Theorem 4.8 can be proved basedon the representation (4.5) alone, i.e., without using complex variables or harmonicfunctions. Going even further, we may ask if the L2-boundedness of the Hilberttransform can be proved without using the Fourier transform. We shall answerall of these questions later in the context of Calderon-Zygmund theory where theyplay an important role.

Exercise 4.4. Show by means of (4.5) that H is not bounded on L8pTq. Forexample, consider Hχr0, 1

2 q. Also, deduce from the fact that cotpπθq < L1pTq that H

is not bounded on L1.

The following proposition shows that the Hilbert transform H f is exponen-tially integrable for bounded functions f . In Exercise 4.4, you should find thatHχr0, 1

2 s has logarithmic behavior at 0 and 12 , which shows that the following result

is best possible.

Proposition 4.10. Let f be a real-valued function on T with | f | ¤ 1. Then forany 0 ¤ α π

2 »T

eα|H f pθq| dθ ¤ 2cosα

Proof. Let u u f be the harmonic extension of f to D and set F u iu.Then |u| ¤ 1 by the maximum principle and hence cospαuq ¥ cosα. Therefore,

Re peαFq Reeαu eiαu cospαuqeαu ¥ cospαqeαu (4.8)

By the mean value property,»T

Re eαFrpθq dθ Re eαFp0q Re eiαpup0qq cospαup0qq ¤ 1

Combining this with (4.8) yields»T

eαurpθq dθ ¤ 1cosα

and by Fatou’s lemma therefore»T

eαpH f qpθq dθ ¤ 1cosα

Since this inequality also holds for f , the proposition follows.

In later chapters we will develop the real-variable theory of singular integralswhich contains the results on the Hilbert transform obtained above. The basictheorem, due to Calderon and Zygmund, states that singular integrals are boundedon LppRnq for 1 p 8 thus generalizing Theorem 4.8.

The analogue of Proposition 4.10 for singular integrals is given by the factthat they are bounded from L8 to BMO (the space of bounded mean oscillation)and that BMO functions are exponentially integrable (this is called John-Nirenberginequality). The dual question of what happens on L1 leads into the real variabletheory of Hardy spaces. The analogue of (4.3) is then that singular integrals are


bounded on H1pRnq. The dual space of H1 is BMO, a classical result of CharlesFefferman. While the real-variable theory will not be developed in full in this text,a later chapter will present some elements of it by means of a dyadic model.

We conclude the theory of the conjugate function by returning to the issue ofLppTq convergence of Fourier series. Recall from Chapter 1 that this convergencefails for p 1 and p 8 but we will now deduce from Theorem 4.8 that it doeshold for the full range 1 p 8.

Theorem 4.11. Let S N denote the partial sum operator of Fourier series. Thenfor any 1 p 8 the S N are uniformly bounded on LppTq, i.e.,

supNS NpÑp 8

By Proposition 1.9 this implies convergence of S N f Ñ f in LppTq with 1 p 8for any f P LppTq.

Proof. The point is simply that S N can be written in terms of the Hilbert trans-form. Indeed, recall that xH f pnq i signpnq f pnqso that, with enpxq epnxq,

T f : 12p1 iHq f

¸n

χp0,8qpnq f pnqen

In other words, on the Fourier side T is multiplication by χp0,8q whereas S N ismultiplication by χrN,Ns. It remains to write χrN,Ns as the difference of twoshifted χp0,8q, i.e.,

χrN,Ns χpN1,8q χpN,8qor in terms of H and T ,

S N f eN1 T peN1 f q eN T peN f qHence, for 1 p 8

S NpÑp ¤ 2TpÑp ¤ 1 HpÑp

uniformly in N, as claimed.

We invite the reader to investigate the analogous result for higher-dimensionalFourier series.

Exercise 4.5. Let N pN1, . . . ,Ndq denote a vector with positive integer en-tries. Prove that

supNS NpÑp 8 @ 1 p 8

where S N is the partial sum operator for Fourier series on Td defined as follows:

S N f pxq ¸νPZd

d¹j1

χr|ν j| N js f pνqepν xq


Conclude that S N f converges in the Lp sense for any f P LppTdq and 1 p 8provided each component of N Ñ8.

We remark that the corresponding result in which expanding rectangles arereplaced with Euclidean balls in Zd is false; in fact, Charlie Fefferman’s ball multi-plier theorem states that the analogue of the previous exercise only holds for p 2in that case. This is delicate, and relies on the existence of Kakeya sets. We shallreturn to these matters in a later chapter.

Notes

The conjugate function and the Hilbert transform provide the link between the complex-variable based theory of Hardy spaces and the real-variable Calderon-Zygmund theorywhich we turn to in a later chapter. Koosis [34] proves the Burkholder-Gundy-Silversteintheorem. This reference also presents some elements of the modern real-variable theory ofH1 spaces, such as the atomic decomposition of H1. The full real-variable Hp theory inhigher dimensional is developed in Stein’s book [46]. We shall discuss some of the basicideas of that theory by means of a dyadic model in a later chapter.

Problems

Problem 4.1. Find an expression for the harmonic measure of an arc on T relative tothe disk D. Relate it to the harmonic measure of an interval relative to the upper half planeby means of a conformal mapping.

Problem 4.2. This problem provides a weak form of a.e. convergence of Fourier serieson L1pTq. In the following chapter we will see via Kolmogoroff’s construction that the fullform of the a.e. convergence fails for L1 functions.

a) Show that, for any λ ¡ 0

supN|tθ P T | |pS N f qpθq| ¡ λu| ¤ C

λ f 1

with some absolute constant C.b) What would such an inequality mean with the supN inside, i.e.,

|tθ P T | supN|pS N f qpθq| ¡ λu| ¤ C

λ f 1

for all λ ¡ 0? Can this be true?c) Using a) show that for every f P L1pTq there exists a subsequence tN ju ÝÑ 8

depending on f such that

S N j f ÝÑ f a.e.

as j Ñ8.

Problem 4.3. Prove the following multiplier estimate. Let m : tmnunPZ be a se-quence in C satisfying ¸

nPZ

|mn mn1| ¤ B, limnÑ8

mn 0


Define the multiplier operator Tm by the rule

Tm f pθq ¸

n

mn f pnq epnθq

for trigonometric polynomials f . Prove that for any such f one has

Tm f p ¤ CppqB f p

for any 1 p 8.

Problem 4.4. Let ϕ P C8pTq and denote by Aϕ the multiplication operator by ϕ. WithH the Hilbert transform on T, show that

rAϕ,Hs Aϕ H H Aϕ

is a smoothing operator, i.e., if µ PMpTq is an arbitrary measure, then

rAϕ,Hsµ P C8pTqProblem 4.5. The complex variable methods developed in Chapters 2, 3, as well as

this one equally well apply to the upper half plane instead of the disk. For example, thePoisson kernel is

Ptpxq 1π

tx2 t2 x P R, t ¡ 0

and its harmonic conjugate is

Qtpxq 1π

xx2 t2

Observe that Ptpxq iQtpxq iπ

1xit i

πz with z x it, which should remind the readerof Cauchy’s formula. The Hilbert transform now reads

pH f qpxq 1π

» 8

8

f pyqx y

dy

in the principal value sense, the precise statement being just as in Proposition 4.9.In this problem, the reader is asked to develop the theory of the upper half-plane in the

context of the previous chapters. The relation between the disk and upper half-plane givenby conformal mapping should be considered as well.

Problem 4.6. By considering F pu iuq logp1 u iuq show that if u ¥ 0 andu P h1pDq, then

³T u logp1 uq dθ 8.

CHAPTER 5

Fourier Series on L1pTq: Pointwise Questions

This chapter is devoted to the question of almost everywhere convergence ofFourier series for L1pTq functions. From the previous chapters we know that thisis not an elementary question, since the partial sums S N are given by convolutionwith the Dirichlet kernels DN which do not form an approximate identity. Note alsothat we encountered an a.e. convergence question which was not associated withan approximate identity, namely for the Hilbert transform, see Proposition 4.9.However, the needed uniform control was provided by the weak-L1 bound on themaximal function u, see Theorem 4.6.

By a natural analogy, one sees that a similar bound on the maximal functionassociated with S N f , i.e., supN |S N f |, would lead to an a.e. convergence result.

What Kolmogoroff realized is essentially that no bound on this maximal func-tion is possible for f P L1pTq. This has to do with the oscillatory nature of theDirichlet kernel DN and the growth of DNL1 . In fact, by a careful choice off P L1 we may arrange the convolution pDN f qpxq so that only the positive peaksof DN contribute at many points x.

What Stein [44] realized is that a weak-type bound on supN |pS N f qpxq| is notonly sufficient but also necessary for the a.e. convergence. In view of this factwe will conclude indirectly that S N f fails to converge a.e. for some f P L1pTq.Kolmogoroff’s original argument involved the construction of a function f P L1pTqfor which a.e. convergence failed. However, what we do here is very close to hisoriginal approach even though it is somewhat more streamlined.

Later in this book we will take up to more challenging question of a.e. conver-gence of S N f for f P L2pTq or LppTq for 1 p 8. This is Carleson’s theorem,and the proof introduces some essentially new and far-reaching ideas.

Next to presenting Kolomogoroff’s theorem, this chapter also serves to intro-duce ideas based on the use of randomness. This turns out to be a very powerfuldevice for many purposes. We begin the actual argument with such a probabilis-tic construction, which is a version of the Borel-Cantelli lemma (but the followingproof is self-contained).

Lemma 5.1. Let tEnu8n1 be a sequence of measurable subsets of T such that

8

n1

|En| 8

53

54 5. FOURIER SERIES ON L1pTq: POINTWISE QUESTIONS

Then there exists a sequence txnu8n1 P T so that

8

n1

χEnpx xnq 8 (5.1)

for almost every x P T.

Proof. View Ω : ±8n1 T as a probability space equipped with the infinite

product measure. Given x P T, let Ax Ω be the event characterized by (5.1). Weclaim that

PpAxq 1 @x P T (5.2)

By Fubini, it then follows that for almost every txnu8n1 P Ω, the event (5.1) holdsfor almost every x P T. Hence fix an arbitrary x P T. Then

Ax !txnu8n1 | x belongs to infinitely many Enp xnq

)

!txnu8n1 | x P

8£N1

8¤mN

pxm Emq)

8£

N1

AN

where

AcN :

!txnu8n1 | x P

8£mN

pxm Ecmq)

!txnu8n1 | xm P x Ec

m @m ¥ N)

By definition of the product measure on Ω, it therefore follows that

PpAcNq

8¹mN

p1 |Em|q 0

by assumption (5.1). Hence (5.2) holds, and the lemma follows.

Lemma 5.1 will play an important role in the proof of the following theo-rem, which establishes the necessity of a weak-type maximal function bound fora.e. convergence. In addition to the previous probabilistic lemma, we shall alsorequire the basic Kolmogoroff 0, 1 law which we now recall: let tXku8k1 be inde-pendent random variables on a probability space pΩ,Pq. Assume that E Ω ismeasurable with respect to the tail σ-algebra. This means that E is measurablewith respect to the σ-algebra generated by tXku8kK for every positive integer K.Then either PpEq 0 or PpEq 1. Such an E is called a tail event. The proof ofthis law proceeds by showing that E is independent of itself whence

0 PpE X Ecq PpEqp1 PpEqqyielding the desired conclusion.

5. FOURIER SERIES ON L1pTq: POINTWISE QUESTIONS 55

In our case, the Carleson maximal operator

C f pxq : supN|S N f pxq| (5.3)

is the main object that needs to be bounded. However, we shall formulate thefollowing “abstract” result in much greater generality. In fact, the reader will easilysee that it has nothing to do with any special properties of T, either, although werestrict ourselves to that case for simplicity.

Theorem 5.2. Let Tn be a sequence of linear operators, bounded on L1pTq,and translation invariant. Define

M f pxq : supn¥1

|Tn f pxq|

and assume that M f 8 8 for every trigonometric polynomial f on T.If for any f P L1pTq,

|tx P T |M f pxq 8u| ¡ 0

then there exists a constant A so that

|tx P T |M f pxq ¡ λu| ¤ Aλ f 1

for any f P L1pTq and λ ¡ 0.

Proof. We will prove this by contradiction. Thus, assume that there exists asequence t f ju8j1 L1pTq with f j1 1 for all j ¥ 1, as well as λ j ¡ 0 so that

E j : tx P T |M f jpxq ¡ λ jusatisfies

|E j| ¡ 2 j

λ j

for each j ¥ 1. Be definition ofM we then also have

limmÑ8 |tx P T | sup

1¤k¤m|Tk f jpxq| ¡ λ ju| ¡ 2 j

λ j

for each j ¥ 1. Hence there are M j 8 with the property that

|tx P T | sup1¤k¤M j

|Tk f jpxq| ¡ λ ju| ¡ 2 j

λ j

for each j ¥ 1. Let σN denote the Nth Cesaro sum, i.e., σN f KN f , where KNis the Fejer kernel. Since each T j is bounded on L1, we conclude that

limNÑ8

|tx P T | sup1¤k¤M j

|TkσN f jpxq| ¡ λ ju| ¡ 2 j

λ j

Hence, we assume from now on that each f j is a trigonometric polynomial. Let m jbe a positive integer with the property that

m j ¤λ j

2 j m j 1


Then8

j1

m j|E j| 8

by construction. Counting each of the sets E j with multiplicity m j, the previouslemma implies that there exists a sequence of points x j,`, j ¥ 1, 1 ¤ ` ¤ m j, sothat

8

j1

m j

`1

χE jpx x j,`q 8 (5.4)

for almost every x P T. Let

δ j : 1j2m j

and define

f pxq :8

j1

m j

`1

δ j f jpx x j,`q

where the signs will be chosen randomly. First note that irrespective of thechoice of these signs,

f 1 ¤8

j1

m jδ j 8

The point is now to choose the signs so that

M f pxq 8for almost every x P T. For this purpose, select x P T such that x P E j x j,` forinfinitely many j and ` `p jq (just pick one such `p jq if there are more than one).Denote the set of these j by J . Since the Tn are translation invariant, so is M.Hence

M f jpx x j,`q ¡ λ j

for all j P J . We conclude that for those j there is a positive integer np j, xq so that

|Tnp j,xq f jpx x j,`q| ¡ λ j

At the cost of removing another set of measure zero we may assume that

Tn f pxq 8

j1

m j

`1

δ jTn f px x j,`q

for all positive integers n. In particular, we have that for all j P J

Tnp j,xq f pxq 8

k1

mk

`1

δkTnp j,xq f px xk,`q

which implies that

Pr|Tnp j,xq f pxq| ¡ δ jλ js ¥ 12


where the probability measure is with respect to the choice of signs . Sinceδ jλ j Ñ8, we obtain that

PrM f pxq 8s ¥ 12

We claim that the event on the left-hand side is a tail event. Indeed, this holds since

M f pxq ¤8

j1

M f jpxq

and each summand on the right-hand side here is finite ( f j is a trigonometric poly-nomial and we are assuming thatM is uniformly bounded on trigonometric poly-nomials). By Kolmogoroff’s zero-one law we therefore have

PrM f pxq 8s 1

It follows from Fubini’s theorem that almost surely (in the choice of )

M f pxq 8 for a.e. x

This would contradict our main hypothesis, and we are done.

The previous lemma reduces Kolmogoroff’s theorem on the failure of almosteverywhere convergence of Fourier series of L1 functions to disproving a weak-L1 bound for the Carleson maximal operator. More precisely, we arrive at thefollowing corollary. It will be technically more convenient later on to formulatethis result in terms of measures rather than on L1pTq.

Corollary 5.3. Suppose tS N f u8N1 converges almost everywhere for everyf P L1pTq. Then there exists a constant A such that

|tx P T | Cµpxq ¡ λu| ¤ Aλµ (5.5)

for any complex Borel measure µ on T and λ ¡ 0, where C is as in (5.3).

Proof. By the previous theorem, our assumption implies that there exists aconstant A such that

|tx P T | C f pxq ¡ λu| ¤ Aλ f 1

for all λ ¡ 0. This is the same as

|tx P T | max1¤n¤N

|S n f pxq| ¡ λu| ¤ Aλ f 1

for all N ¥ 1 and λ ¡ 0. If µ is a complex measure, then we set f Vm µ, whereVm is de la Vallee-Poussin’s kernel, see (1.22). It follows that for all N ¥ 1

|tx P T | max1¤n¤N

|S nrVm µspxq| ¡ λu| ¤ Aλµ

for all m ¥ 1 and λ ¡ 0. Note that for m ¥ N the kernel Vm can be removed fromthe left-hand side, and we obtain our desired conclusion.


The idea behind Kolmogoroff’s theorem is to find a measure µ which wouldviolate (5.5). This measure will be chosen to create “resonances”, i.e., so that thepeaks of the Dirichlet kernel all appear with the same sign. More precisely, forevery positive integer N we will choose

µN : 1N

N

j1

δx j,N (5.6)

where the x j,N are close to jN . Then

pS nµNqpxq 1N

N

j1

sinpp2n 1qπpx x j,Nqqsinpπpx x j,Nqq . (5.7)

If x is fixed, we will then argue that there exists n so that the summands on theright-hand have the same sign (for this, we will need to make a careful choice ofthe x j,N). Thus, the size of the entire sum will be about

N

j1

1j log N

because of the denominators in (5.7), which clearly contradicts (5.5).The choice of the points x j,N is based on the following lemma due to Kro-

necker.

Lemma 5.4. Assume that pθ1, . . . , θdq P Td is incommensurate, i.e., that for anypn1, . . . , ndq P Zdzt0u one has that

n1θ1 . . . ndθd < Z.

Then the orbit pnθ1, . . . , nθdq mod Zd | n P Z( Td (5.8)

is dense in Td.

Proof. It will suffice to show that for any smooth function f on Td

1N

N

n1

f pnθ1, . . . , nθdq Ñ»Td

f pxq dx. (5.9)

Indeed, if the orbit (5.8) is not dense, then we could find f ¥ 0 so that the settTd| f ¡ 0u does not intersect it. Cleary, this would contradict (5.9). To prove (5.9),expand f into a Fourier series, cf. Proposition 1.14. Then

1N

N

n1

f pnθ1, . . . , nθdq 1N

N

n1

¸νPZd

f pνqe2πinθν

f p0q ¸

νPZdzt0uf pνq 1

N1 e2πipN1qθν

1 e2πiθν


where the ratio on the right-hand side is well-defined by our assumption. Clearly,f pνq is rapidly decaying in |ν| since f P C8pTdq. Thus, since 1

N1 e2πipN1qθν

1 e2πiθν

¤ 1

for all N ¥ 1, ν , 0, it follows that

limNÑ8

¸νPZdzt0u

f pνq 1N

1 e2πipN1qθν

1 e2πiθν 0

and we are done.

We can now carry out our construction of the µN .

Lemma 5.5. There exists a sequence µN of probability measures on T with theproperty that

lim supnÑ8

plog Nq1|S nµNpxq| ¡ 0

for almost every x P T.

Proof. For every N ¥ 1 and 1 ¤ j ¤ N choose x j,N P T which satisfy

|x j,N jN| ¤ N2

and so that tx j,NuNj1 P TN is an incommensurate vector. This can be done since

the commensurate vectors have measure zero. Clearly, the set of x P T such thatt2px x j,NquN

j1 P TN is a commensurate vector is at most countable. It followsthat for almost every x P T, t2npx x j,NquN

j1 mod ZN | n P Z(is dense in TN . Hence, for almost every x P T, tp2n 1qpx x j,NquN

j1 mod ZN | n P Z(is also dense in TN . It follows that for almost every x P T there are infinitely manychoices of n ¥ 1 so that

sinpp2n 1qπpx x j,Nqq ¥ 12

for all 1 ¤ j ¤ N. In particular, for those n the sum in (5.7) satisfies

|S nµNpxq| ¥ 12N

N

j1

1| sinpπpx x j,Nqq| ¥ C

1N

N

j1

1jN ¥ C log N,

as desired.

Finally, combining Lemma 5.5 with Corollary 5.3 yields Kolmogoroff’s theo-rem. The second statement in the following theorem follows from Theorem 5.2.


Theorem 5.6. There exists f P L1pTq so that S n f does not converge almosteverywhere. In fact, there exists f P L1pTq such that

tx P T | C f pxq 8uhas measure 0.

It is known that this statement also holds with everywhere divergence. In fact,Konyagin [35] showed the following stronger result: Let φ : r0,8q Ñ r0,8q benon-decreasing and satisfy φpuq opualog ualog log uq as u Ñ8. Then thereexists f P L1pTq so that »

Tφp| f pxq|q dx 8

and lim supmÑ8 |S m f pxq| 8 for every x P T.

Notes

Kolmogoroff’s original reference is [33]. Much more on pointwise and a.e. conver-gence questions can be found in Zygmund [58] and Katznelson [30].

Problems

Problem 5.1. Consider the following variants of Theorem 5.2:

1) Let tµnu8n1 be a sequence of positive measures on Rd supported in a commoncompact set. Define

M f pxq : supn¥1

|p f µnqpxq|

Let 1 ¤ p 8 and assume that for each f P LppRdqM f pxq 8 on a set of positive measure.

Then show that f ÞÑM f is of weak-type pp, pq.2) Now suppose µn are complex measures of the form µnpdxq Knpxq dx, but again

with common compact support. Show that the conclusion of part 1) holds, butonly for 1 ¤ p ¤ 2.

Problem 5.2. If ω is an irrational number, show that 1N

N

n1

f p nωq »T

f pθq dθ

L2Ñ 0

for any f P L2pTq. In particular, if f P L2 is such that f px ωq f pxq for a.e. x, then f= const.

Problem 5.3. Let txnu8n1 be an infinite sequence of real numbers. Show that thefollowing three conditions are equivalent:

a) For any f P CpTq,1N

N

n1

f pxnq Ñ»T

f pxq dx


b) 1N

°Nn1 epkxnq Ñ 0 for all k P Z

c) limNÑ8

supIT

1N #t1 ¤ j ¤ N | x j P I mod 1u |I| 0

If these conditions hold we say that txnu8n1 is uniformly distributed modulo 1 (abbreviatedby u.d.). The quantity

Dtx juN

j1

: sup

IT

#t1 ¤ j ¤ N | x j P I mod 1u N|I|is called the discrepancy of the finite sequence tx juN

j1.

Problem 5.4. Using Problem 1.4 with a suitable choice of T , prove the following: iftxnu8n1 is a sequence for which txnk xnu8n1 is u.d. modulo 1 for any k P Z, thentxnu81 is also u.d. mod 1. In particular, show that tndωu8n1 is u.d. mod 1 for any irrationalω and d P Z.

Problem 5.5. Let p ¥ 2 be a positive integer. Show that for a.e. x P T the sequencetpnxu8n1 is u.d. modulo 1. Can you characterize those x which have this property? Hint:Consider expansions to power p.

Problem 5.6. Let tθ juNj1 T be N distinct points and consider

Ppzq N¹

j1

pz epθ jqq

Assume that sup|z|1 |Ppzq| ¤ eτ. We may assume that 1 ¤ τ ¤ N log 2. Prove thefollowing bound on the discrepancy of tθ juN

j1 as defined above:

∆N : Dtθ juN

j1

¤ C?τN

Conclude that|tθ P T | |Ppepθqq| eANu| ¤ Ce

AN∆N ¤ Ce

A?

Nτ

for any A ¥ 0. The constants C are absolute but may differ. Hint: Consider the functionFpθq µprθ0, θqq Npθ θ0q with µ °N

j1 δθ j where θ0 P p 12 ,

12 q is chosen such that³ 1

2

12

Fpθq dθ 0. Relate F to to upθq : log |Ppepθqq| via the Hilbert transform. Thenwrite F K µ with a suitable kernel K.

CHAPTER 6

Fourier Analysis on Rd

It is natural to extend the definition of the Fourier transform for the circle givenin Chapter 1 to Euclidean spaces. Thus, let µ PMpRdq be a complex-valued Borelmeasure and set

µpξq »Rd

e2πixξ µpdxq @ ξ P Rd

which is called the Fourier transform of µ. Clearly, µ8 ¤ µ, the total variationof µ. Moreover, it follows immediately from the dominated convergence theoremthat µ is continuous. If µpdxq f pxq dx, then we also write f pξq. In particular, theFourier transform is well-defined on L1pRdq and defines a contraction into CpRdq,the space of continuous functions. Many elementary properties from Chapter 1carry over to this setting, such as convolutions, Young’s inequality, and the actionof the Fourier transform on convolutions and translations. To be precise, we collecta few facts in the following lemma.

Lemma 6.1. Let µ PMpRdq and let τy denote translation by y P Rd: pτyµqpEq µpE yq. Then xτyµpξq e2πiξyµpξq @ ξ P Rd

Let eηpxq : e2πixη. Then xeηµpξq µpξ ηq. If f , g P L1pRdq, then the integral»Rd

f px yqgpyq dy

is absolutely convergent for a.e. x P Rd and is denoted by p f gqpxq. One hasf g P L1pRdq, and zf g f g.

Finally, if g P Lp for 1 ¤ p ¤ 8 and f P L1, then f g P Lp and f gp ¤ f 1gp.

We leave the elementary proof to the reader. We record one more very impor-tant fact, namely the change of variables formula for the Fourier transform: let Abe an invertible d d-matrix with real entries, and denote by f A the functionf pAxq. Then for any f P L1pRdq one haszf A | det A|1 f At (6.1)

where At is the transpose of the inverse of A. A special case of this is the dilationidentity. For any λ ¡ 0 let fλpxq f pλxq. Thenpfλpξq λd f pξλq (6.2)

63

64 6. FOURIER ANALYSIS ON Rd

In analogy with the Fourier transform on the circle, we expect that L2pRdqplays a special role. The be more precise, we expect the Plancherel theorem f 2 f 2 to hold. Notice that as it stands this is only meaningful for f P L1 X L2pRdq,since otherwise the defining integral is not absolutely convergent. In fact, it isconvenient to work with a much smaller space called Schwartz space. In whatfollows, we use the standard multi-index notation: for any α pα1, α2, . . . , αdq PZd

0 (where Z0 are the nonnegative integers)

xα :d¹

j1

xα jj , x px1, x2, . . . , xdq

Bα :d¹

j1

Bα j

Bxα jj

Moreover, |α| °dj1 α j denotes the length of the multi-index.

Definition 6.2. The Schwartz spaceSpRdq is defined as all functions in C8pRdqwhich decay rapidly together with all derivatives. In other words, f P SpRdq if andonly if f P C8pRdq and

xαBβ f pxq P L8pRdq @ α, βwhere α, β are arbitrary multi-indices. We introduce the following notion of con-vergence in SpRdq: a sequence fn P SpRdq converges to g P SpRdq if and onlyif

xαBβp fn gq8 Ñ 0 n Ñ8for all α, β.

For readers familiar with Frechet spaces, we remark that the semi-norms

pαβp f q : xαBβ f 8 α, β P Zd0

turn SpRdq into such a space. Since Frechet spaces are metrizable, it follows thatthe notion of convergence introduced in Definition 6.2 completely characterizesthe topology in SpRdq. However, we make no use of this fact, and only work withthe sequential characterization.

One has that SpRdq is dense in LppRdq for all 1 ¤ p 8 since this is alreadythe case for smooth functions with compact support. It is evident that the mapf ÞÑ xαBβ f pxq is continuous on SpRdq. Somewhat less obvious is that the Fouriertransform has the same property.

Proposition 6.3. The Fourier transform is a continuous operation from theSchwartz space into itself.

Proof. Fix any f P SpRdq. The proof hinges on the pointwise identitiesyBα f pξq p2πiq|α| ξα f pξqBβ f pξq p2πiq|β| yxβ f pξq

6. FOURIER ANALYSIS ON Rd 65

valid for all multi-indices α, β. The first one follows by integration by parts,whereas the second is obtained by differentiating under the integral sign. We leavethe formal verification of these identities to the reader (introduce difference quo-tients, justify the limits by the dominated convergence theorem etc.). This is easy,due to the rapid decay and smoothness of f .

We first note from these identities that f P C8 and, moreover, that

ξαBβξ f P L8pRdq @ α, β P Zd0

which implies by definition that f P SpRdq. For the continuity, let fn be a sequencein SpRdq so that fn Ñ 0 as n Ñ8. Then

ξαBβξ fn8 ¤ Cα,β Bαx xβ fn1 Ñ 0

as n Ñ8.

In view of the preceding fact, it is most natural to ask if the Fourier transformtakes SpRdq onto itself. We will now prove that this is indeed the case, which is aspecial case of the Fourier inversion theorem. The latter is the analogue the Fouriersummation problem for Fourier series. The Schwartz space is a most convenientsetting in which to formulate the inversion.

Proposition 6.4. The Fourier transform takes the Schwartz space onto itself.In fact, for any f P SpRdq one has

f pxq »Rd

e2πixξ f pξq dξ @ x P Rd (6.3)

Proof. We would like to proceed by inserting the expression

f pξq »Rd

e2πiyξ f pyq dy

into (6.3) and show that this recovers f pxq. The difficulty here is of course thatinterchanging the order of integration leads to the formal expression»

Rde2πipxyqξ dξ

which needs to equal the δ0px yq measure for the inversion formula (6.3) tohold. Although this is “essentially” the case, the problem here is of course that theprevious integral is not well-defined. We shall therefore need to be more careful.

The standard procedure is to introduce a Gaussian weight since the Fouriertransform has an explicit form on such weights. In fact, in Exercise 6.1 below weask the reader to check the following identityeπ|x|2pξq eπ|ξ|

2(6.4)

Taking this for granted, we conclude from this and (6.2) thateπε2|x|2pξq εdeπ|ξ|2

ε2

for any ε ¡ 0.


Now fix f P SpRdq. We commence the proof of (6.3) with the observation that»Rd

e2πixξ f pξq dξ limεÑ0

»Rd

e2πixξ eπε2|ξ|2 f pξq dξ

for each x P Rd. This follows from the dominated convergence theorem sincef P L1. Next, one has»

Rde2πixξ eπε

2|ξ|2 f pξq dξ

»Rd

»Rd

e2πipxyqξ eπε2|ξ|2 dξ f pyq dy

»Rdεd eπε

2|xy|2 f pyq dy

The final observation is thatεd eπε

2|x|2

is an approximate identity in the sense of Chapter 1 with respect to the limit εÑ 0.Hence, since f P CpRdq X L8pRdq one sees that

limεÑ0

»Rdεd eπε

2|xy|2 f pyq dy f pxq

for all x P Rd, which concludes the proof.

Exercise 6.1. Prove the identity (6.4). Hint: Use contour integration in thecomplex plane.

Let us now return to the dilation identity (6.2) with f a smooth bump function.We note that this identity expresses an important normalization principle (at leastheuristically), namely that the Fourier transforms maps an L1-normalized bumpfunction onto an L8-normalized one, and vice versa.

The dual of S, denoted by S1 is the space of tempered distributions. In otherwords, any u P S1 is a continuous functional on S, and we write xu, φy for uapplied to φ P S. The space S1 is equipped with the weak- topology. Thus,un Ñ u in S1 if and only if xun, φy Ñ xu, φy as n Ñ 8 for every φ P S. Naturally,LppRdq ãÑ S1pRdq by the rule

x f , φy »Rd

f pxqφpxq dx, f P LppRdq, φ P SpRdq

By a little functional analysis we may see that a linear functional u on S is contin-uous if and only if there exists N such that

|xu, φy| ¤ C¸

|α|,|β|¤N

xαBβφ8 @ φ P S

It follows that tempered distributions can be differentiated any number of times,and multiplied by any element of C8pRdq of at most polynomial growth. One has

xBαu, φy p1q|α|xu, Bαφy @ φ P SpRdq


Any measure of finite total variation is an element of S1, in particular the Diracmeasure δ0 is.

The Fourier transform of u P S1 is defined by the relation xu, φy : xu, φyfor all φ P SpRdq. Since the Fourier transform is an isomorphism on SpRdq, thisdefinition is meaningful and establishes the distributional Fourier transform as anisomorphism on S1. For example, 1 δ0 since

x1, φy »Rdφpξq dξ φp0q

In particular, we can meaningfully speak of the Fourier transform of any functionin LppRdq with 1 ¤ p ¤ 8.

The following exercise presents a very useful fact, namely that an element ofS1pRdq which is supported at a point, say t0u is necessarily a linear combination ofδ0 and finitely many derivatives thereof. “Support” here refers to the property thatxu, ϕy 0 for all ϕ P SpRdq that vanish near 0.

Exercise 6.2. Let u, v P S1 with xu, ϕy xv, ϕy for all ϕ P S with supppϕq Rdzt0u. Then

u v P

for some polynomial P.

The following exercise introduces an important example of a distributionalFourier transform. The functions |x|α go by the name of Riesz potentials.

Exercise 6.3. For any 0 α d compute the distributional Fourier transformof |x|α in Rd. Hint: Show that the Fourier transform equals a smooth functionof ξ , 0. Use dilation and rotational symmetry to determine this smooth function.Nonzero multiplicative constants do not need to be determined. Finally, excludeany contributions coming from ξ 0, see the previous exercise.

An interesting example of a distributional identity is given by the Poisson sum-mation formula.

Proposition 6.5. For any f P SpRdq one has¸nPZd

f pnq ¸νPZd

f pνq

Equivalently, with the convergence being in the sense of S1,¸νPZd

e2πixν ¸

nPZd

δnpxq

where δn denotes the Dirac distribution at n.


Proof. Given f P SpRdq, define Fpxq : °nPZd f pxnqwhich lies in C8pTdq.

The Fourier coefficients of F are with epθq e2πiθ as usual,

Fpνq ¸

nPZd

»r0,1sd

f px nqepx νq dx

¸

nPZd

»r0,1sdn

f pxqepx νq dx »Rd

f pxqepx νq dx

f pνqFrom Proposition 1.14 we therefore conclude that

Fpxq ¸νPZd

f pνqepx νq

which implies the first identity. The second simply follows by chasing definitions.

Henceforth, we shall use the notation f for the integral in (6.3). It is now easyto establish the Plancherel theorem.

Corollary 6.6. For any f P SpRdq one has f 2 f 2. In particular, theFourier transform extends unitarily to L2pRdq.

Proof. We start from the following identity which is of interest in its own right:for any f , g P SpRdq one has»

Rdf pξqgpξq dξ

»Rd

f pxqgpxq dx

The proof is an immediate consequence of the definition of the Fourier transformand Fubini’s theorem. Therefore, one also has»

Rdf pξqgpξq dξ

»Rd

f pxq ˆgpxq dx

However,ˆg ˇg g

by the previous proposition, whence we obtain Parseval’s indentity»Rd

f pξqgpξq dξ »Rd

f pxqgpxq dx

which is equivalent to the Plancherel theorem and the unitarity of the Fourier trans-form.

Exercise 6.4. Show that the Schwartz space is an algebra under both multipli-cation and convolution. In particular, prove that xf g f g for any f , g P SpRdq.

We now take another look at the proof of Proposition 6.4. Denote

Γεpxq : εd eπε2|x|2


Then the proof makes use of the identity»Rd

e2πixξ eπε2|ξ|2 f pξq dξ p f Γεqpxq (6.5)

valid pointwise for all Schwartz functions f and any ε ¡ 0. Let Iεp f q be the oper-ator defined by the previous line. Since Γε is an approximate identity as εÑ 0, weconclude the following useful observation which parallels results we encounteredfor Fourier series on the circle.

Corollary 6.7. For any 1 ¤ p 8 and f P LppRdq one has Iεp f q Ñ f inLppRdq. If f P CpRdq and vanishes at infinity, then Iεp f q Ñ f uniformly.

In particular, one has uniqueness.

Corollary 6.8. Suppose f P L1pRdq satisfies f 0 everywhere. Then f 0.

Exercise 6.5. Let µ be a compactly supported measure in Rd. Show that µextends to an entire function in Cd. Conclude that µ cannot vanish on an open setin Rd.

Finally, we note the following basic estimate which is called Hausdorff-Younginequality.

Lemma 6.9. Fix 1 ¤ p ¤ 2. Then one has the property that f P Lp1pRdq forany f P LppRdq. Moreover, there is the bound f p1 ¤ f p.

Proof. Note that f P LppRdq can be written as f f1 f2 where

f1 fχr| f | 1s P L2pRdq, f2 fχr| f |¥1s P L1pRdqHence f1 P L2 is well-defined by the L2-extension of the Fourier transform, andf2 P L8 by integration. The estimate follows by interpolating between p 1 andp 2.

As in the case of the circle, one can easily express all degrees of smoothnessvia the Fourier transform. For example, one can do this on the level of the Sobolevspaces Hs and 9Hs which are defined in terms of the norms

f Hs xξys f 2, f 9Hs |ξ|s f 2 (6.6)

for any s P R where xξy a

1 |ξ|2. These spaces are the completion of SpRdqunder the norms in (6.6). For the homogeneous space 9HspRdq one needs the re-striction s ¡ d

2 since otherwise S is not dense in it.Perhaps the most basic question relating to the Sobolev spaces concerns their

embedding properties. An example is given by the following result.

Lemma 6.10. For any f P HspRdq one has

f p ¤ Cpsq f HspRdq @ 2 ¤ p ¤ 8provided s ¡ d

2 . In fact HspRdq ãÑ CpRdq X LppRdq for any such s and all2 ¤ p ¤ 8.


Proof. By Cauchy-Schwarz

f 8 ¤ f 1 ¤ xξys2 xξys f pξq2

for any s ¡ d2 . The continuity of f follows by approximation by Schwartz func-

tions. Interpolation with L2 implies the lemma.

By duality, one obtains for any s d2

f HspRdq ¤ Cps, dq f L1pRdq (6.7)

Exercise 6.6. Show that Lemma 6.10 fails at s d2 . Hint: Argue by contra-

diction and use duality. Thus, show that (6.7) cannot hold at s d2 .

A simple generalization of Lemma 6.10 is the following result, which is anexample of what is called a trace lemma.

Lemma 6.11. Let V Rd be an affine subspace of dimension d k where1 ¤ k ¤ d. Then one has

f L2pVq ¤ Cpsq f HspRdq

where s ¡ k2 .

Proof. This follows by choosing coordinates parallel toM as well as perpen-dicular to it. For the parallel ones, one uses Plancherel onM, whereas for the othersCauchy-Schwarz leads to the desired conclusion as in the previous proof.

An analogous statement also holds when V is not flat but a smooth manifold(or rather, a compact piece thereof).

It is also natural to ask about embeddings 9HspRdq ãÑ LppRdq when s ¥ 0.Of course, when s 0 this is tautologically true with p 2, but when s ¡ 0we expect an “improvement” in terms of a better regularity statement expressed bysome p ¡ 2. Note that the corresponding estimate

f LppRdq ¤ C f 9HspRdq (6.8)

is necessarily scaling invariant. Thus, if we replace f by f pλq, then the right-handside scales with λs d

2 , whereas the left-hand side scales like λdp . Hence we arrive

at the relation12 1

p s

d(6.9)

This excludes the case d2 by Exercise 6.6, and leaves open the possibility that for

any s P p0, d2 q and p as in (6.9) the estimate (6.8) does hold.

A good place to begin in this case is to consider f P L2pRdq with suppp f q tR ¤ |ξ| ¤ 2Ru and some R ¡ 0. In that case, the following Bernstein inequalityholds. The basic idea is the same as in Theorem 1.12, viz. functions with boundedFourier support are arbitrarily smooth and this smoothness can be quantified in a


number of ways, for example on the Lp-scale. We remark that (6.10) is scalinginvariant.

Lemma 6.12. Let f P SpRdq satisfy suppp f q t|ξ| ¤ Ru. Then

f q ¤ C Rdp 1p 1

q q f p (6.10)

1 ¤ p ¤ q ¤ 8.

Proof. Since (6.10) is scaling invariant, it suffices to consider R 1. Thenone has f χ f where χ is smooth, compactly supported and χ 1 on the unitball. It follows that f χ f and by Young’s inequality therefore

f q ¤ χr f p

with 1 1q 1

r 1p , which is the same as 1

p1 1q 1

r . Note that 1 ¤ p ¤ q ¤ 8guarantees that 1 ¤ r ¤ p1.

Exercise 6.7. Formulate and prove the general Young’s inequality that wasused in the previous proof.

In particular, if f is supported on tR ¤ |ξ| ¤ 2Ru, then

f p ¤ CRdp 12 1

p q f 2 C f 9HspRdq

with s as in (6.9). This proves (6.8) for such functions, and the challenge is nowto obtain the general case which is indeed true. The approach is to write f °

jPZ P j f where P j f has Fourier support roughly on dyadic shells of size 2 j. While¸jPZP j f 2

9Hs f 29Hs

by definition, the problem is to show that

f 2p ¤ C

¸j

P j f 2p (6.11)

This is indeed true for finite p ¥ 2, but requires some machinery. We shall developthe necessary tools in the Littlewood-Paley chapter below. There are alternativeroutes leading to (6.8), such as fractional integration.

We now take up a topic that has proven to be very important for various ap-plications, especially in partial differential equations: the decay properties of mea-sures with smooth compactly supported densities that live on smooth submanifoldsof Rd. Let us start with a concrete example, namely the Cauchy problem for theSchrodinger equation

iBtψ ∆ψ 0, ψp0q ψ0 (6.12)

where ψ0 is a fixed function, say a Schwartz function, and ψp0q refers to the func-tion ψpt, xq evaluated at t 0. Applying the Fourier transform converts (6.12)into

iBtψpξq 4π2|ξ|2ψpξq 0, ψp0q xψ0


where ψ is the Fourier transform with respect to the space variable alone. Thesolution to the previous ordinary differential equation with respect to time is givenby

ψpt, ξq e4itπ2|ξ|2xψ0pξqwhence the actual solution is after Fourier inversion

ψpt, xq »Rd

e2πixξ e4itπ2|ξ|2 xψ0pξq dξ (6.13)

This formula is of course very relevant as it gives a solution to (6.12). For ourpurposes, we would like to re-interpret the integral on the right-hand side in thefollowing form. Consider the paraboloid

P : pξ,4π2|ξ|2q | ξ P Rd( Rd1

and place the measure µ onto P by the formula (for given ψ0 as above)

µpdξ, dτq xψ0pξq dξ

via the relation»Rd1

Fpξ, τqµpdξ, dτq »Rd

Fpξ,4π2|ξ|2qxψ0pξq dξ

valid for any continuous and bounded F. Then we see that (6.13) can be interpretedas

ψpt, xq »Rd1

e2πipxξtτq µpdξ, dτqIn other words, the solution is given by the inverse Fourier transform of the measureµ which lives on the paraboloid P. For a variety of reasons it has turned out tobe of fundamental importance to understand the decay properties of such Fouriertransforms. Before investigating this question, we ask the reader to verify a fewproperties of (6.13).

Exercise 6.8. Let ψ0 P SpRdq. Prove that (6.13) defines a function ψpt, xqwhich is smooth in t and belongs to the Schwartz class for all fixed times. More-over, show that ψ solves (6.12) and satisfies

ψptqL2 ψ02 @ t P RFinally, verify that for every t ¡ 0

ψpt, xq cd td2

»Rd

ei π2|xy|2

4t ψ0pyq dy (6.14)

with a suitable constant cd. Conclude that

ψptq8 ¤ cd td2 ψ01

for all t ¡ 0.

Now letM Rd be a smooth manifold of dimension k d with Riemannianmeasure µ. Fix some smooth, compactly supported function φ onM. The problemis to describe the asymptotic behavior of the Fourier transform xφ µpξq for large ξ.For example, consider M to be a plane of dimension d 1. By translation and


rotational symmetry we may takeM Rd1, the subspace given by the first d 1coordinates and φ is a Schwartz function of x1 : px1, . . . , xd1q, whereas µ issimply Lebesgue measure on Rd1. Therefore, with ξ pξ1, ξdq,xφµpξq »

Rd1e2πix1ξ1φpx1q dx1

which evidently is a Schwartz function in ξ1 but does not depend at all on ξd whencealso does not decay in that direction. What we have shown is that the Fourier trans-form of a measure that lives on a plane does not decay in the direction normal to theplane. In contrast to this case, we shall demonstrate that nonvanishing Gaussiancurvature of a hypersurfaceM always guarantees decay at a universal rate.

To facilitate computations, fix someM and a measure on it with smooth com-pactly supported density. Working in local coordinates and fixing a direction inwhich we wish to describe the decay reduces matters to oscillatory integrals of theform

Ipλq »Rd

eiλφpξqapξq dξ , (6.15)

with a compactly supported function a P C80 and smooth, real-valued phase φ P

C8. The asymptotic behavior of the integral Ipλq as λ Ñ 8 is described by themethod of (non)stationary phase. This refers to the distinction between φ havinga critical point on supppaq or not. We begin with the easier case of non-stationaryphase. The reader should note that the standard Fourier transform is of this type.

Lemma 6.13. If ∇φ , 0 on supppaq then the integral (6.15) decays as »Rd

eiλφpξqapξq dξ ¤ CpN, a, φq λN , λÑ8

for arbitrary N ¥ 1.

Proof. Note that the exponential eiλφ is an eigenfunction for the operator

L : 1iλ∇φ

|∇φ|2∇ (6.16)

Therefore we can apply L to the exponential inside of (6.15) as often as we wish.The adjoint of L is

L iλ∇p ∇φ|∇φ|2 q

Then for any positive integer N, one has »Rd

eiλφpξqapξq dξ »

RdLNpeiλφpξqqapξq dξ

»Rd

eiλφpξqpLqNpapξqq dξ

¤»Rd|pLqNpapξqq| dξ

¤ CpN, a, φqλN


as claimed.

The situation changes when φ has a critical point inside supppaq. In that caseone no longer has arbitrary decay and it can be very difficult to determine the exactrate. However, if the critical point is nondegenerate, i.e., φ has a nondegenerateHessian at that point, then there is a precise answer as we now show.

Lemma 6.14. If ∇φpξ0q 0 for some ξ0 P supppaq, ∇φ , 0 away from ξ0 andthe Hessian of φ at the stationary point ξ0 is non-degenerate, i.e., det D2φpξ0q , 0,then for all λ ¥ 1, »

Rdeiλφpξqapξq dξ

¤ Cpd, a, φq λ d2 (6.17)

In fact, Bkλ

eiλφp0q

»Rd

eiλφpξqapξq dξ ¤ Cpd, a, φ, kq λ d

2k (6.18)

for any integer k ¥ 1.

Proof. Without loss of generality we assume ξ0 0. By Taylor expansion

φpξq φp0q 12xD2φp0qξ, ξy Op|ξ|3q

Since D2φp0q is non-degenerate we have |∇φpξq| & |ξ| on supppaq. We split theintegral into two parts, »

Rdeiλφpξqapξq dξ I II

where the first part localizes the contribution near ξ 0,

I :»Rd

eiλφpξqapξqχ0pλ 12 ξq dξ

and the second part restores the original integral, viz.

II »Rd

eiλφpξqapξqp1 χ0pλ 12 ξqq dξ

The first term has the desired decay

|I| ¤ C»Rd|χ0pλ 1

2 ξq| dξ ¤ C λd2

To bound the second term, we proceed as in the proof of the nonstationary case.Thus, let L be as in (6.16) and note thatDα

∇φ

|∇φ|2pξq

¤ Cα|ξ|1|α|


for any multi-index α. Thus, let N ¡ d be an integer. Then the Leibnitz rule yields

|II| ¤ C»

r|ξ|¡λ 12 s

|pLqNp1 χ0pλ 12 ξqqapξq| dξ

¤ C λN

8»λ

12

pλ N2 rN r2Nq rd1 dr ¤ C λ

d2

as claimed. The stronger bound (6.18) follows from the observation that

|φpξq φp0q| ¤ C |ξ|2and the same argument as before (with N ¡ d 2k) concludes the proof.

It is possible to obtain asymptotic expansions of Ipλq in inverse powers oflarge λ. In fact, the estimate provided by the preceding stationary phase lemma isoptimal.

Let us now return to the Fourier transform of dµ φ dσ where σ is the surfacemeasure of a hypersurface M in Rd and φ is compactly supported and smooth.Denote the unit normal vector toM at x PM by Npxq.

Corollary 6.15. Let ξ λν where ν P S d1 and λ ¥ 1. If ν is not parallel toNpxq for any x P supppφq, then µ Opλkq as λ Ñ 8 for any positive integer k.Assume thatM has nonvanishing Gaussian curvature on supppφq. Then

|µpξq| ¤ Cpµ,Mq |ξ| d12

for all λ ¥ 1 and all ν. This is optimal if ν does belong to the normal bundle ofMon supppφq.

Proof. One has

µpξq »M

eiλxν,xy φpxqσpdxqNote that the function x ÞÑ xν, xy as a function onM has a critical point at x0 PMif and only if ν is perpendicular to Tx0M, the tangent space at x0 toM. Lemma 6.13implies the first statement. For the second statement we note that the critical pointat x0 is nondegenerate if and only if the Gaussian curvature at x0 does not vanish,and thus the bound follows from stationary phase.

Lemma 6.14 in fact allows for a more precise description of µ. We illustratethis by means of zσS d1pxq for large x, where σS d1 is the surface measure of theunit sphere in Rd. This representation is very useful in certain applications wherethe oscillatory nature of the Fourier transform of the surface measure is needed,and not just its decay.

Note that the function zσS d1pxq is smooth and bounded, so we are interested inits asymptotic behavior for large |x|, both in terms of oscillations and size. We needto understand the oscillations in order to bound derivatives of zσS d1pxq. We remark


that the following result can also be obtained from the asymptotics of Bessel func-tions, but we make no use of special functions in this book.

Corollary 6.16. One has the representation

zσS d1pxq ei|x|ωp|x|q ei|x|ωp|x|q |x| ¥ 1 (6.19)

where ω are smooth and satisfy

|Bkrωprq| ¤ Ck r

d12 k @ r ¥ 1

and all k ¥ 0.

Proof. By definition,

zσS d1pxq »

S d1eixξ σS d1pdξq

which is rotationally invariant. Therefore we choose x p0, . . . , 0, |x|q and denotethe integral on the right-hand side by Ip|x|q. Thus,

Ip|x|q »

S d1ei|x|ξd σS d1pdξq

Note that ξd, viewed as a function on S d1, has exactly two critical points, namelythe north and south poles

p0, . . . , 0,1qwhich are, moreover, nondegenerate. We therefore partition the sphere into threeparts: a small neighborhood around each of these poles and the remainder. TheFourier integral Ip|x|q then splits into three integrals by means of a smooth partitionof unity adapted to these three open sets, and we write accordingly

Ip|x|q ¸

Ip|x|q Ieqp|x|q

The latter integral has non-stationary phase and decays like |x|N for any N, seeLemma 6.13. On the other hand, the integrals I exactly fit into the frame work ofLemma 6.14 which yields the desired result. Note that one can absorb the equato-rial piece Ieqp|x|q into either ei|x|ωp|x|q without changing the stated propertiesof ω, and (6.19) follows.

Notes

Many important topics are omitted here, such as spherical harmomics. A classicalas well as comprehensive discussion of the Euclidean Fourier transform is the book byStein and Weiss [49]. Folland [17] and especially Rudin [40] contain more information onthe topic of tempered distributions. Hormander [26] and Wolff [56] show that the Fouriertransform on Lp with p ¡ 2 take values in the distributions outside of the Lebesgue spaces(see Theorem 7.6.6 in [26]). For more on the topic of Sobolev embeddings and traceestimates see for example the comprehensive treatments in Gilbarg and Trudinger [22],as well as Evans [13], which do not invoke the Fourier transform. Stein [45] develops


Sobolev spaces in the context of the Fourier transform, but also uses the “fundamental the-orem of calculus” approach. A standard reference for asymptotic expansions in stationaryphase is the classical reference by Hormander [26]. For Exercise 6.2 see Theorem 6.25 inRudin [40].

Problems

Problem 6.1. Compute the Fourier transform of P.V. 1x . In other words, determine

limεÑ0

»|x|¡ε

e2πixξ dxx

for every ξ P R. Conclude that (up to an irrelevant normalization) the Hilbert transform isan isometry on L2pRq. This is the analogue for the line of Lemma 4.3.

Problem 6.2. Solve the boundary value problem in the upper half plane for the Laplaceequation via the Fourier transform. Compare your findings with Problem 4.5. Generalizeto higher dimensions.

Problem 6.3. Carry out the analogue Problem 1.11 with R instead of T and Rd insteadof Td (the case of the Schrodinger equation was already addressed in Exercise 6.8 above).Compute the explicit form of the heat kernel, i.e., the kernel Gt so that Gt u0 with t ¡ 0 isthe solution to the Cauchy problem for the heat equation ut ∆u 0, up0q u0. Discussthe limit t Ñ 0.

Problem 6.4. Prove the following generalization of Bernstein’s inequality, cf. Lemma 6.12:if suppp f q E Rd where E is measurable and f is Schwartz, say, prove that

f q ¤ |E| 1p

1q f p @ 1 ¤ p ¤ q ¤ 8 (6.20)

where |E| stands for the Lebesgue measure. Hint: First handle the case q 8, p 2, byPlancherel and Cauchy-Schwarz, and then dualize and interpolate.

Problem 6.5. Show that if f P L2pRq is supported in rR,Rs, then f extends to C as anentire function of exponential growth 2πR. The Paley-Wiener theorem states the converse:if F is an entire function of exponential growth 2πR and such that F P L2pRq, then F fwhere f P L2pRq is supported in rR,Rs. Prove this via a deformation of contour, andextend to higher dimensions.

Problem 6.6. Consider the kernel Kpx, yq 1xy on R2

and define the operator

pT f qpxq » 8

0

f pyqx y

dy @ x ¡ 0

Show that T is bounded on LppRq for every 1 p 8, but not at p 1 or p 8.

Problem 6.7. The Laplace transform is defined as

pL f qpsq :» 8

0esx f pxq dx, s ¡ 0 (6.21)

Show that L is bounded on L2pRq but not on any LppRq with p , 2. Verify thatL2Ñ2 ?

π. Use this to show that T2Ñ2 π where T is as in (6.21). Hint: Forthe L2 boundedness, write esx e

sx2 x

14 e

sx2 x

14 and apply Cauchy-Schwarz. For T

consider L2.


Problem 6.8. With L the Laplace transform on L2pRq as in the previous problem,find an inversion formula.

Problem 6.9. Establish a representation as in Corollary 6.16 for the ball instead of thesphere. I.e., find the asymptotics as in that corollary for xχB where B Rd is the unit-ballwith d ¥ 2. Generalize to the case of ellipsoids, i.e.,!

x px1, . . . , xdq P Rd |d

j1

λ2j x2

j ¤ 1)

where λ j ¡ 0 are constants. Contrast this to the Fourier transform of the indicator functionof the cube.

Problem 6.10. Let A be a positive dd-matrix. Compute the Fourier transform of theGaussian exAx,xy.

Problem 6.11. Compute the inverse Fourier transform of p4π2|ξ|2k2q1 in R3 wherek ¡ 0 is a constant. Denote this function by Gpx; kq. Show that the operator

pRpkq f qpxq »Rd

Gpx y; kq f pyq dy, f P SpR3q

is bounded on L2pRdq and that Rpkq f P C8pR3q as well as

p∆ k2qRpkq f f @ f P SpR3qConclude that Rpkq p∆ k2q1 for k ¡ 0, i.e., Rpkq is the resolvent of the Laplacian.

CHAPTER 7

Calderon-Zygmund Theory of Singular Integrals

In this chapter we take up the problem of developing a real-variable theory ofthe Hilbert transform. In fact, we will subsume the Hilbert transform in a classof operators defined in any dimension. We shall make no reference to harmonicfunctions as we did in Chapter 4, but rather rely only the properties of the kernelalone. For simplicity, we shall mostly restrict ourselves to the translation invariantsetting.

We begin with the following class of kernels which define operators by con-volution. As for the Hilbert transform, we include a cancellation condition, seepart iiiq in the following definition. For example, this condition guarantees thatthe principal value as in (7.1) exists (contrast this to the case Kpxq |x|d). Inaddition, this condition will be convenient in proving the L2-boundedness of theassociated operators.

This being said, it is often preferable to assume the L2-boundedness of theCalderon-Zygmund operator instead of the cancellation condition. The logic isthen to develop the L2-boundedness theory separately from the Lp-theory, with thegoal of finding necessary and sufficient conditions for L2-boundedness while theLp theory requires only condition iiq in the following definition. In many ways, iiq,which is called Hormander condition, is therefore the most important one.

Definition 7.1. Let K : Rdzt0u Ñ C satisfy, for some constant B,

i) |Kpxq| ¤ B|x|d for all x P Rd

ii)³|x|¡2|y| |Kpxq Kpx yq| dx ¤ B for all y , 0

iii)³

r |x| s Kpxq dx 0 for all 0 r s 8Then K is called a Calderon-Zygmund kernel.

Our first goal is With such a kernel we associate a translation invariant operatorby means of the principal value. Thus, the singular integral operator (or Calderon-Zygmund operator) with kernel K is defined as

T f pxq : limεÑ0

»|xy|¡ε

Kpx yq f pyq dy (7.1)

for all f P SpRdq.Exercise 7.1.

79

80 7. CALDERON-ZYGMUND THEORY OF SINGULAR INTEGRALS

a) Check that the limit exists in (7.1) for all f P SpRdq. How much reg-ularity on f suffices for this property (you may take f to be compactlysupported if you wish)?

b) Check that the Hilbert transform on the line with kernel 1x is a singular

integral operator.

There is the following simple condition that guarantees iiq.Lemma 7.2. Suppose |∇Kpxq| ¤ B|x|d1 for all x , 0 and some constant B.

Then »|x|¡2|y|

|Kpxq Kpx yq| dx ¤ CB (7.2)

with C Cpdq.Proof. Fix x, y P Rd with |x| ¡ 2|y|. Connect x and x y by the line segment

x ty, 0 ¤ t ¤ 1. This line segment lies entirely inside the ball Bpx, |x|2). Hence

|Kpxq Kpx yq| » 1

0∇Kpx tyqy dt

¤» 1

0|∇Kpx tyq||y| dt ¤ B2d1|x|d1|y|

Inserting this bound into the left-hand side of (7.2) yields the desired bound.

Exercise 7.2. Check that, for any fixed 0 α ¤ 1, and all x , 0,

sup|y| |x|

2

|Kpxq Kpx yq||y|α ¤ B|x|dα

also implies (7.2).

We now prove L2-boundedness of T , by computing K and applying Plancherel.This is an instance where the full force of Definition 7.1 comes into play. We shalluse all three conditions, in particular the cancellation condition.

Proposition 7.3. Let K be as in Definition 7.1. Then T2Ñ2 ¤ CB withC Cpdq.

Proof. Fix 0 r s 8 and consider

pTr,s f qpxq »Rd

Kpyqχrr |y| sspyq f px yq dy

Let

mr,spξq :»Rd

e2πixξχrr |x| ssKpxq dx

be the Fourier transform of the restricted kernel. By Plancherel’s theorem it sufficesto prove that

sup0 r s

mr,s8 ¤ CB (7.3)

7. CALDERON-ZYGMUND THEORY OF SINGULAR INTEGRALS 81

Indeed, if (7.3) holds, then

Tr,s2Ñ2 mr,s8 ¤ CB

uniformly in r, s. Moreover, for any f P SpRdq one has

T f pxq limrÑ0sÑ8

pTr,s f qpxq

pointwise in x P Rd. Fatou’s lemma therefore implies that T f 2 ¤ CB f 2 forany f P SpRdq. To verify (7.3) we split the integration in the Fourier transforminto the regions |x| |ξ|1 and |x| ¥ |ξ|1. In the former case »

r |x| |ξ|1e2πixξKpxq dx

»r |x| |ξ|1

pe2πixξ 1qKpxq dx

¤»|x| |ξ|1

2π|x||ξ||Kpxq| dx ¤

¤ 2π|ξ|»|x| |ξ|1

B|x|d1 dx

¤ CB|ξ||ξ|1 ¤ CB

as desired. Note that we used the cancellation condition iiiq in the first equalitysign. To deal with the case |x| ¡ |ξ|1 one uses the cancellation in e2πixξ whichin turn requires smoothness of K, i.e., condition ii) (but compare Lemma 7.2).Firstly, observe that»

s¡|x|¡|ξ|1Kpxqe2πixξ dx

»s¡|x|¡|ξ|1

Kpxqe2πi

x ξ

2|ξ|2

ξ

dx (7.4)

»

s¡|x ξ

2|ξ|2|¡|ξ|1

K

x ξ

2|ξ|2

e2πixξ dx (7.5)

Denoting the expression on the left-hand side of (7.4) by F one thus has

2F »

s¡|x|¡|ξ|1

Kpxq Kpx ξ

2|ξ|2 q

e2πixξ dx Op1q (7.6)

The Op1q term here stands for a term bounded by CB. Its origin is of course thedifference between the regions of integration in (7.4) and (7.5). We leave it to thereader to check that condition i) implies that this error term is really no larger thanCB. Estimating the integral in (7.6) by means of ii) now yields

|2F| ¤»|x|¡|ξ|1

Kpxq K

x ξ

2|ξ|2 dxCB ¤ CB

as claimed. We have shown (7.3) and the proposition follows.

Exercise 7.3. Supply the omitted details concerning the Op1q term in the pre-vious proof.


Our next goal is to show that singular integrals are bounded as operators fromL1pRdq into weak-L1. This requires the following basic decomposition lemma dueto Calderon-Zygmund for L1 functions. The proof is an example of a stopping timeargument.

Lemma 7.4. Let f P L1pRdq and λ ¡ 0. Then one can write f g b where|g| ¤ λ and b °

Q χQ f where the sum runs over a collectionB tQu of disjointcubes such that for each Q one has

λ 1|Q|

»Q| f | ¤ 2dλ (7.7)

Furthermore, ¤QPB

Q 1

λ f 1 (7.8)

Proof. For each ` P Z we define a collectionD` of dyadic cubes by means of

D` Πd

i1r2`mi, 2`pmi 1qq | m1, . . . ,md P Z(

Notice that if Q P D` and Q1 P D`1 , then either QXQ1 H or Q Q1 or Q1 Q.Now pick `0 so large that

1|Q|

»Q| f | dx ¤ λ

for every Q P D`0 . For each such cube consider its 2d “children” of size 2`01.Any such cube Q1 will then have the property that either

1|Q1|

»Q1

| f pxq| dx ¤ λ or1|Q1|

»Q1

| f pxq| dx ¡ λ (7.9)

In the latter case we stop, and include Q1 in the family B of “bad cubes”. Observethat in this case

1|Q1|

»Q1

| f pxq| dx ¤ 2d

|Q|»

Q| f pxq| dx ¤ 2dλ

where Q denotes the parent of Q1. Thus (7.7) holds. If, however, the first inequalityin (7.9) holds, then subdivide Q1 again into its children of half the size. Continuingin this fashion produces a collection of disjoint (dyadic) cubes B satisfying (7.7).Consequently, (7.8) also holds, since¤

B

Q ¤¸

B

|Q| ¸B

1λ

»Q| f pxq| dx ¤ 1

λ

»Rd| f pxq| dx

Now let x0 P RdzB Q. Then x0 is contained in a decreasing sequence tQ ju ofdyadic cubes each of which satisfies

1|Q j|

»Q j

| f pxq| dx ¤ λ


By Lebesgue’s theorem | f px0q| ¤ λ for a.e. such x0. Since moreover Rdz YB Qand Rdz YB Q differ only by a set of measure zero, we can set

g f ¸QPB

χQ f

so that |g| ¤ λ a.e. as desired.

We can now state the crucial weak-L1 bound for singular integrals. We shalldo this under more general hypotheses than those of Definition 7.1.

Proposition 7.5. Suppose T is a linear operator bounded on L2pRdq such that

pT f qpxq »Rd

Kpx yq f pyq dy @ f P L2comppRdq

and all x < suppp f q. Assume that K satisfies condition iiiq of Definition 7.1, butnot necessarily any of the other conditions. Then for every f P SpRdq there is theweak-L1 bound tx P Rd | |T f pxq| ¡ λu ¤ CB

λ f 1 @ λ ¡ 0

where C Cpdq.Proof. Dividing by B if necessary, we may assume that B 1. Now fix

f P SpRdq and let λ ¡ 0 be arbitrary. By Lemma 7.4 one can write f g b withthis value of λ. We now set

f1 g¸QPB

χQ1|Q|

»Q

f pxq dx

f2 b¸QPB

χQ1|Q|

»Q

f pxq dx ¸QPB

fQ

where we have set

fQ : χQ

f 1

|Q|»

Qf pxq dx

Notice that f f1 f2, f18 ¤ Cλ f 1, f21 ¤ 2 f 1, f11 ¤ 2 f 1, and»

QfQpxq dx 0

for all Q P B. We now proceed as follows:tx P Rd | |pT f qpxq| ¡ λu¤

x P Rd | |pT f1qpxq| ¡ λ

2

( x P Rd | |pT f2qpxq| ¡ λ

2

(¤ Cλ2 T f12

2 x P Rd | |pT f2qpxq| ¡ λ

2

( (7.10)

The first term in (7.10) is controlled by Proposition 7.3:Cλ2 T f12

2 ¤Cλ2 f12

2 ¤Cλ2 f18 f11 ¤ C

λ f 1


To estimate the second term in (7.10) we define, for any Q P B, the cube Q to bethe dilate of Q by the fixed factor 2

?n (i.e., Q has the same center as Q but side

length equal to 2?

n times that of Q). Thus x P Rd | |pT f2qpxq| ¡ λ

2

(¤

YB Q x P Rdz YB Q | |pT f2qpxq| ¡ λ

2

(¤ C

¸Q

|Q| 2λ

»RdzYQ

|pT f2qpxq| dx

¤ Cλ f 1 2

λ

¸QPB

»RdzQ

|pT fQqpxq| dx

Perhaps the most crucial point of this proof is the fact that fQ has mean zero whichallows one to exploit the smoothness of the kernel K. More precisely, for anyx P RdzQ,

pT fQqpxq »

QKpx yq fQpyq dy

»

QrKpx yq Kpx yQqs fQpyq dy

(7.11)

where yQ denotes the center of Q. Thus»RdzQ

|pT fQqpxq| dx ¤»RdzQ

»Q|Kpx yq Kpx yQq|| fQpyq| dy dx

¤»

Q| fQpyq| dy ¤ 2

»Q| f pyq| dy

To pass to the second inequality sign we used condition ii) in Definition 7.1 andour choice of Q. Hence the second term on the right bound side of (7.11) is nolarger than

Cλ

¸Q

»Q| f pyq|dy ¤ C

λ f 1

and we are done.

The assumption f P SpRdq in the previous proposition was for convenienceonly. It ensured that one could define T f by means of the principal value (7.1).However, observe that Proposition 7.3 allows one to extend T to a bounded opera-tor on L2. This in turn implies that the weak-L1 bound in Proposition 7.5 holds forall f P L1 X L2pRdq. Indeed, inspection of the proof reveals that apart from the L2

boundedness of T , see (7.10), the definition of T in terms of K was only used in(7.11) where x and y are assumed to be sufficiently separated so that the integralsare absolutely convergent.

Theorem 7.6 (Calderon-Zygmund). Let T be a singular integral operator as inDefinition 7.1. Then for every 1 p 8 one can extend T to a bounded operatoron LppRdq with the bound TpÑp ¤ CB with C Cpp, dq.


Proof. By Proposition 7.3 and 7.5 (and the previous remark) one obtains thisstatement for the range 1 p ¤ 2 from the Marcinkiewicz interpolation theorem.The range 2 ¤ p 8 now follows by duality. Indeed, for f , g P SpRdq one has

xT f , gy x f ,Tgy where Tgpxq limεÑ0

»|xy|¡ε

Kpx yqgpyq dy

and Kpxq : Kpxq. Since K clearly verifies conditions i)–iii) in Definition 77.1,we are done.

Remark. It is important to realize that the cancellation condition iii) was onlyused to prove L2 boundedness, but did not appear in the proof of Proposition 7.5explicitly. Therefore, T is bounded on LppRdq provided it is bounded for p 2and conditions i) and ii) of Definition 7.1 hold.

We now present some of the most basic examples of singular integrals. Con-sider the Poisson equation 4u f in Rd where f P SpRdq, d ¥ 2. In the followingexercise we ask the reader to show that in dimensions d ¥ 3

upxq Cd

»Rd|x y|2d f pyq dy (7.12)

with some dimensional constant Cd is a solution to this equation. In fact, one canshow that (7.12) gives the unique bounded solution but we shall not need to knowthat here. In two dimensions, the formula reads

upxq C2

»R2

logp|x y|q f pyq dy (7.13)

The kernels |x|2d and log |x| are called Newton potentials. It turns out that thesecond derivatives B2u

BxiBx jwith u defined by these convolutions can be expressed

as Calderon-Zygmund operators acting on f (the so-called “double Riesz trans-forms”).

Exercise 7.4.

a) With f P S and a suitable constant Cd, show that (7.12) satisfies

∆u f

Hint: pass the Laplacian onto f and introduce a cut-off to the set r|xy| ¡εs in (7.12). Apply Green’s formula, and let εÑ 0 to recover f pxq.

b) With u as in (7.12), show that for f P SpRdq with d ¥ 3

B2uBxiBx j

pxq Cd pd 2qpd 1q»Rd

Ki jpx yq f pyq dy

in the principal value sense, where

Ki jpxq # xi x j

|x|d2 if i , jx2

i 1d |x|2

|x|d2 if i j

Find the analogue for d 2.


c) Verify that Ki j as above are singular integral kernels. Also show thatKipxq xi

|x|d1 is a singular integral kernel.

The operators Ri and Ri j defined in terms of the kernels Ki and Ki j, respectively,are called the Riesz transforms or double Riesz transforms. By Exercise 7.4

Ri jp4ϕq B2ϕ

BxiBx j(7.14)

for any ϕ P SpRdq. The following estimate is one of the most basic Lp estimate forelliptic equations.

Corollary 7.7. Let u P SpRdq. Then

sup1¤i, j¤d

B2uBxiBx j

LppRdq

¤ Cp,d4uLppRdq (7.15)

for any 1 p 8. Here Cp,d only depends on p and d.

Proof. This is an immediate consequence of the representation formula (7.14)and the Calderon-Zygmund Theorem 7.6.

Corollary 7.7 is remarkable since it states that the Hessian is controlled by itstrace. This cannot hold in the pointwise sense, which means that the corollary failsat p 8 and by duality therefore also at p 1.

Exercise 7.5. This exercise explains the aforementioned failure at p 1 orp 8 in more detail. The idea is simply to take u to be the Newton potential. Then4u δ0 which, at least heuristically, belongs to L1pRdq. However, one checks thatB2uBxiBx j

< L1pBp0, 1qq for any i, j, see the kernels Ki j form Exercise 7.4). In oder to

transfer δ0 to L1, we convolve the Newton potentials with an approximate identity(this is called mollifying); in addition, we truncate to a bounded set.

a) Let ϕ ¥ 0 be in C80 pRdqwith

³Rd ϕ dx 1. Set ϕεpxq : εdϕp x

εq for any0 ε 1. Clearly, tϕεu0 ε 1 form an approximate identity provided thelatter are defined analogously to Definition 1.3 on Rd. Moreover, let χ PC8

0 pRdq be arbitrary with χp0q , 0. Verify that, with Γdpxq |x|2d forn ¥ 3 and Γ2pxq log |x|, uεpxq : pϕε Γdqpxqχpxq has the followingproperties:

supε¡0

4uεL1 8

and

lim supεÑ0

B2uε

BxiBx jL1 8

for any 1 ¤ i, j ¤ d. Thus Corollary 7.7 fails on L1pRdq.b) Now show that Corollary 7.7 also fails on L8.


The convolution in (7.12) is an example of what one calls fractional integra-tion. More generally, for any 0 α d define

pIα f qpxq :»Rd|x y|αd f pyq dy, f P SpRdq (7.16)

We cannot use Young’s inequality since the kernel is not in any LrpRdq space, butit is clearly in weak-Lr with r d

dα . The following proposition shows that never-theless one still has (up to an endpoint) the same conclusion as Young’s inequalitywould imply.

Proposition 7.8. For any 1 p q 8 and any 0 α d there is thebound

Iα f q ¤ Cpp, q, dq f p provided1q 1

p α

dfor any f P SpRdq.

Proof. By Marcinkiewicz’s interpolation theorem it suffices for this bound tohold with weak Lq. Thus, we need to show that

|tx P Rd | |Iα f pxq| ¡ λu| ¤ Cλq f qp (7.17)

Fix some f with f p 1 and an arbitary λ ¡ 0. With r ¡ 0 to be determined,break up the kernel as follows

kαpxq : |x|αd kαpxqχr|x| rs kαpxqχr|x|¡rs : kp1qα pxq kp2qα pxqThen

kp2qα f 8 ¤ kp2qα p1 f p Crαdp

Determine r so that the right-hand side equals λ2 . Then

|tx P Rd | |Iα f pxq| ¡ λu| ¤ |tx P Rd | |pkp1qα f qpxq| ¡ λ2u|¤ Cλpkp1qα f p

p ¤ Cλpkp1qα p1

¤ Cλprαp Cλq

which is (7.17).

The condition on p and q is of course dictated by scaling. By Exercise 6.3, onehas

p∆q s2 f Cps, dq Is f @ f P SpRdq (7.18)

Thus, Proposition 7.8 implies the following Sobolev imbedding estimate.

Corollary 7.9. Let 0 ¤ s d2 . Then for all 1

2 1q s

d one has

f LqpRdq ¤ Cps, dq f 9HspRdq @ f P SpRdq (7.19)

Proof. Fix f P SpRdq and set g : p∆q s2 f . Then by (7.18) one has

f Cps, dqIsg

and the estimate (7.19) follows from the previous proposition.


We now address the question whether a singular integral operator can be de-fined by means of formula (7.1) even if f P LppRdq rather than f P SpRdq, say.This question is the analogue of Proposition 4.9 and should be understood as fol-lows: for 1 p 8we defined T “abstractly” as an operator on LppRdq by exten-sion from SpRdq via the apriori bounds T f p ¤ Cp,d f p for all f P SpRdq (thelatter space is dense in LppRdq, cf. Proposition 1.5). We now ask if the principalvalue (7.1) converges almost everywhere to this extension T for any f P LppRdq.Based on our previous experience with such questions, we introduce the associatedmaximal operator

T f pxq supε¡0

»|y|¡ε

Kpyq f px yq dy

and seek weak-L1 bounds for it. As in previous cases where we studied almosteverywhere convergence, this will suffice to conclude the a.e. convergence on Lp.In what follows we shall need the Hardy-Littlewood maximal function

M f pxq supxPB

1|B|

»B| f pyq| dy

where the supremum runs over all balls B containing x. M satisfies the same typeof bounds as those stated in Proposition 2.5.

We prove the desired bounds on T only for a subclass of kernels, namely thehomogeneous ones. This class includes the Riesz transforms from above. Moreprecisely, let

Kpxq :Ωp x

|x|q|x|d for x , 0 (7.20)

where Ω : S d1 Ñ C, Ω P C1pS d1q and³

S d1 Ωpxqσpdxq 0 where σ is thesurface measure on S d1. Observe the homogeneity Kptxq tdKpxq for all t ¡ 0.

Exercise 7.6. Check that any such K satisfies the conditions in Definition 7.1.

Proposition 7.10. Suppose K is of the form (7.20). Then T satisfies

pT f qpxq ¤ CrMpT f qpxq M f pxqswith some absolute constant C. In particular, T is bounded on LppRdq for 1 p 8. Furthermore, T is also weak-L1 bounded.

Proof. Let Kpxq : Kpxqχr|x|¥1s and more generally

Kεpxq εdKp xεq Kpxqχr|x|¥εs

Pick a smooth bump function ϕ P C80 pRdq, ϕ ¥ 0,

³Rd ϕ dx 1. Set

Φ : ϕ K K

and observe that ϕK is well-defined in the principal value sense. For any functionF on Rd let Fεpxq : εdF

xε

be its L1-normalized rescaling. Then Kε K,

Kε pKqε, and thus

Φε pϕ Kqε Kε ϕε Kε Kε ϕε K Kε


Hence, for any f P SpRdq, one has

Kε f ϕε pK f q Φε f

We now invoke the analogue of Lemma 2.7 for radially bounded approximate iden-tities in Rd. This of course requires that tΦεuε¡0 from such a radially boundedapproximate identity, the verification of which we leave to the reader (the case ofϕε is obvious). Indeed, one checks that

|Φpxq| ¤ C minp1, |x|d1qwhich implies the desired property. Therefore,

T f ¤ CpMpT f q M f qas claimed. The boundedness of T on LppRdq for 1 p 8 now follows fromthat of T and M. The proof of the weak-L1 boundedness of T is a variation of thesame property of T , of Proposition 7.5.

Exercise 7.7. Show that tΦεuε¡0 as it appears in the previous proof forms aradially bounded approximate identity.

Corollary 7.11. Let K be a homogeneous singular integral kernel as in (7.20).Then for any f P LppRdq, 1 ¤ p 8, the limit in (7.1) exists almost everywhere.

Proof. Let

Λp f qpxq | lim supεÑ0

pTε f qpxq lim infεÑ0

pTε f qpxq|

Observe that Λp f q ¤ 2T f . Fix f P Lp and let g P SpRdq so that

f gp δ

for a given small δ ¡ 0. Then Λp f q Λp f gq and therefore

Λ f p ¤ CΛp f gqp Cδ

if 1 p 8. Hence, Λ f 0. The case p 1 is similar.

Operators of the form (7.20) arise very naturally by considering multiplier op-erators of the form Tm f : pm f q_ where m is homogeneous of degree zero andbelongs to C8pRdzt0uq (for simplicity we assume infinite regularity). Indeed, withf P SpRdq one has over the tempered distributions

T f K f , K m

where m P S1pRdq is the distributional inverse Fourier transform. From the dilationlaw 6.2 one sees that K is homogeneous of degree d which means that for anyλ ¡ 0

xzmpλq, ϕy xm, λdϕpλ1qy xm, zϕpλqy xm, ϕpλqy xλdmpλ1q, ϕy


for all ϕ P SpRdq. Observer that constants are homogeneous of degree zero,whence 1 δ0 is homogeneous of degree d. This might seem less strange ifone remembers that as εÑ 0 one has

1|Bp0, εq|χBp0,εq Ñ δ0 in S1pRdq

since the left-hand scales like a function that is homogeneous of degree d. Wenow prove the following general statement which is very natural in view of the factthat the representation of m needs to be invariant under adding constants to m.

Lemma 7.12. Let m P C8pRdzt0uq be homogeneous of degree zero. Let xmyS d1

denote the mean of m on S d1. Then m P S1pRdq satisfies

m xmyS d1 δ0 P.V.KΩ

where KΩ is of the form (7.20) with xΩyS d1 0 and Ω P C8pS d1q.

Proof. We may assume that m has mean zero on the sphere, i.e.,»S d1

mpωqσpdωq 0

Observe that u j : Bdj m P S1pRdq is homogeneous of degree d and has mean

zero on S d1. This implies that P.V. u j is well-defined in the usual integral sense,and moreover, one has

u j P.V. u j ¸α

cα Bαδ0

by Exercise 6.2 where the sum is finite. In fact, by homogeneity, the sum is neces-sarily of the form c j δ0. Passing to the Fourier transforms in S1 one obtains

pu j P.V. u j c j

Next, one verifies that P.V. u jpxq is a smooth function on Rdzt0u which is homo-geneous of degree zero. Indeed, with χpξq a smooth radial, compactly supportedfunction which equals 1 on a neighborhood of the origin one has for any x , 0

P.V. u jpxq »RdBd

j mpξqχpξqpe2πixξ 1q dξ

d

j1

x j

2πi|x|2»RdB jBd

j mpξqp1 χpξqqe2πixξ dξ

where p j BBξ j

. Both integrals are absolutely convergent, and the first integralclearly defines a smooth function in x. We may repeat the integration by parts inthe second integral any number of times, which implies that the second integralalso defines a smooth function. The homogeneity of degree zero follows from thehomogeneity of P.V. u j. We may therefore write

mpxq Ω x|x|

|x|d x , 0


where Ω P C8pS d1q. Let ϕ P SpRdq be radial. Then

xm, ϕy xm, ϕy 0

since m has mean zero on spheres. Thus, Ω also has mean zero on S d1. Byconstruction, m P.V.KΩ is supported at t0u. By Exercise 6.2 we conclude thatthis difference is therefore a polynomial in derivatives of δ0. By homogeneity,it must be a constant. By the vanishing means, this constant is zero and we aredone.

As a consequence, we see that singular integral operators of the type Tm forman algebra, with Tm being invertible if and only if m , 0 on S d1. In Chapter 10 weshall see that one can still obtain Lp boundedness of Tm without the strong assump-tion of m being homogeneous of degree 0. Instead, one requires that |ξ|k|Bαmpξq|is bounded for all |α| k for finitely many k. While these conditions are satisfiedif m is homogeneous of degree zero, they are much weaker than that property.

We now wish to quantify the failure of the boundedness of singular integraloperators on L8. It turns out that the correct extension of L8 which containsT pL8q for any singular integral operator T is the space of functions of boundedmean oscillation on Rd, denoted by BMOpRdq. In what follows,?

Qf pyq dy |Q|1

»Q

f pyq dy

Definition 7.13. Let f P L1locpRdq. Define

f 7pxq : supQQx

?Q| f pyq fQ| dy (7.21)

where the supremum runs over cubes Q and fQ is the mean of f over Q. Thenf P BMOpRdq if and only if f 7 P L8pRdq. Then f BMO : f 78. If Q0 isa fixed cube, then BMOpQ0q is obtained by defining f 7 in terms of all subcubesof Q0.

Of course one may replace cubes here with balls without changing anything.Moreover, BMO is a norm only after factoring out the constants. Clearly,L8 BMO but BMO is strictly larger. In addition, BMO scales like L8, i.e., f pλqBMO f BMO. Suppose Q2 Q1 are two cubes with Q2 being half thesize of Q1. Then

| fQ1 fQ2 | ¤|Q1||Q2|?

Q1

| f pyq fQ1 | dy ¤ 2d f BMO (7.22)

Iterating this relation leads to the following: suppose Qn Qn1 . . .Q1 Q0is a sequence of nested cubes so that the diameters decrease by a factor of 2 at eachstep. Then

| fQn fQ0 | ¤ n 2d f BMO (7.23)


The important feature here is that while the size of Qn is by a factor of 2n smallerthan Q0, the averages increase only linearly in n. As an example, consider thefunction f defined on the interval r0, 1s as follows

g 8

n0

anχr0,2ns (7.24)

with 1 ¤ an for all n ¥ 0.

Exercise 7.8.

Show that g P BMOpr0, 1sq if and only if an Op1q. In that case, verifythat gpxq | log x| for 0 x 1

2 . Show that χr|x| 1s log |x| P BMOpr1, 1sq, but

χr|x| 1s signpxq log |x| < BMOpr1, 1sq Show that for any f as in Definition 7.13

f 77pxq : supxPQ

infc

?Q| f pyq c| dy

satisfies f 77 ¤ f 7 ¤ 2 f 77.

We shall not distinguish between f 7 and f 77. It is easy to show that singularintegrals take L8 Ñ BMO.

Theorem 7.14. Let T be a singular integral operator as in Definition 7.1. Then

T f BMO ¤ CB f 8 @ f P L2pRdq X L8pRdqWe are assuming here that f P L2pRdq so that T f is well-defined.

Proof. We may assume that B 1 in Definition 7.1. Fix some f as above, aball B0 : Bpx0,Rq Rd and define

cB0 :»r|yx0|¡2Rs

Kpx0 yq f pyq dy

Then, with B0 : Bpx0, 2Rq, one estimates»B0

|pT f qpxq cB0 | dx ¤»

B0

»r|yx0|¡2Rs

|Kpx yq Kpx0 yq|| f pyq| dydx

»

B0

TχB0

fpxq dx

¤ C|B0| f 8 |B0| 12 T2Ñ2χB0

f 2

¤ C|B0| f 8and we are done.


In fact, it turns out that BMO is the smallest space with contains T pL8pRdqq forevery singular integral operator T as in Definition 7.1, see the notes to this chapter.If BMO is supposed to be a useful substitute to L8pRdq, then we would like tobe able to use it in interpolation theory. This is indeed possible, as the followingtheorem shows.

Theorem 7.15. Let T be a linear operator bonded on Lp0pRdq for some 1 ¤p0 8 and bounded from L8 Ñ BMO. Then T is bounded on LppRdq for anyp0 p 8.

The proof of this result requires some machinery, more specifically a com-parison between the Hardy-Littlewood maximal function and the sharp maximalfunction as in (7.21). For this purpose it is convenient to use the dyadic maximalfunction, which we denote by Mdyad and which is defined as

pMdyad f qpxq : supxQQ

?Q| f pyq| dy

where the supremum now runs over dyadic cubes which are defined as the collec-tion

Qdyad : 2kr0, 1qd 2kZd | k P Z(

Note that any two dyadic cubes are either disjoint, or one contains the other. Sup-pose Q0 P Qdyad, and let f P L1pQ0q. Then for any

λ ¡?

Q0

| f pyq| dy (7.25)

we can perform a Calderon-Zygmund decomposition as in Lemma 7.4 to concludethat

tx P Q0 | pMdyad f qpxq ¡ λu ¤QPB

Q

whence in particular the measure of the left-hand side is λ1 f L1pQ0q. In otherwords, we have reestablished the weak-L1 bound for the maximal function. Thecondition (7.25) serves as a “starting condition” for the stopping-time constructionin Lemma 7.4. The other values of λ are of no interest, as in that case

λ1 f L1pQ0q ¥ |Q0|so the weak-L1 bound becomes trivial. The construction of course does not neces-sarily need to be localized to any Q0, but can also be carried out on Rd globally.

We now come to the aforementioned comparison between Mdyad f and f 7.While it is clear that f 7 ¤ 2M f , the usual Hardy-Littlewood maximal function,we require a lower bound. In essence, the following theorem says that on Lp withfinite p, the sharp function contains no new information.

Theorem 7.16. Let 1 ¤ p0 ¤ p 8 and suppose f P Lp0pRdq. Then»RdpMdyad f qppxq dx ¤ Cpp, dq

»Rd

f 7pxqp dx (7.26)


Proof. For simplicity, we write M instead of Mdyad. The proof is an immediateconsequence the following estimate, which is an example of a good λ inequality:for all γ ¡ 0 and λ ¡ 0 one has

|tx P Rd | M f pxq ¡ 2λ, f 7pxq ¤ γλu|¤ 2dγ|tx P Rd | M f pxq ¡ λu| (7.27)

To prove it, we write the set on the right-hand side as

QPB Q with dyadic Qvia Lemma 7.4. We use f P Lp0 here since this ensures that we may start thestopping time argument at any level λ ¡ 0 by taking the size of the first dyadiccube very large. By construction, these Q are the maximal dyadic cubes containedin tM f ¡ λu. Indeed, if Q Q denotes the unique dyadic cube of twice theside-length of Q (the parent cube) then by definition of B one has?

Q| f pyq| dy ¤ λ and λ

?Q| f pyq| dy ¤ 2dλ

Now fix one such Q and its parent Q. Then by the preceding, if x P Q and

pM f qpxq ¡ 2λ ùñ pMp f χQ fQqqpxq ¡ λ

Therefore,|tx P Q | pM f qpxq ¡ 2λu| ¤ |tx P Q | pMp f χQ fQqqpxq ¡ λu|

¤ λ1»

Q| f fQ|pyq dy ¤ γ|Q|

To pass to the final bound, we used that if there exists some x0 P Q so that f 7px0q ¤γλ (which we may assume), then»

Q| f fQ|pyq dy ¤

»Q| f fQ|pyq dy ¤ γλ|Q|

Summing over all Q P B establishes the claim (7.27). Now» λ0

0|tM f pxq ¡ 2λu| pλp1 dλ

¤» λ0

0|tM f pxq ¡ 2λ, f 7pxq ¤ γλu| pλp1 dλ

» λ0

0|t f 7pxq ¡ γλu| pλp1 dλ

¤ 2dpγ

» λ0

0|tM f pxq ¡ 2λu| pλp1 dλ γp f 7p

p

Choosing γ 2dp1 and letting λ0 Ñ8 concludes the proof.

It is now easy to prove the interpolation result.

Proof of Theorem 7.15. Consider S : f ÞÑ pT f q7. While it is not linear it issublinear which is sufficient for Marcinkiewicz interpolation. By definition, S isbounded on Lp0 and L8 whence it is also bounded on LppRdq for any p0 p 8.Now take f P Lp X Lp0 . Then T f P Lp0 and we conclude from Theorem 7.16 andT f ¤ MdyadpT f q a.e. that T is bounded on Lp.


We now take a closer look at the property (7.23). In fact, as suggested byExercise 7.8 we shall now show that any BMO function is exponentially integrable.The precise statement is given by the following John-Nirenberg inequality.

Theorem 7.17. Let f P BMOpRdq. Then for any cube Q one has

|tx P Q | | f pxq fQ| ¡ λu| ¤ C|Q| exp c

λ

f BMO

for any λ ¡ 0. The constants c,C are absolute.

Proof. We may take Q r0, 1qd. Assume f BMO ¤ 1 with fQ 0, andlet λ ¡ 10. Perform a Calderon-Zygmund decomposition of f on the cube Q at alevel µ ¥ 2 that we shall fix later. This yields f g b, |g| ¤ 2dµ and

b ¸

Q1PBχQ1 p f fQ1q,

¸Q1PB

|Q1| ¤ µ1

DefineEpλq : sup

gBMOpQq¤1|tx P Q | |gpxq gQ| ¡ λu|

ThenEpλq ¤ |tx P Q | |bpxq| ¡ λ 2dµu|

¤¸

Q1PB|tx P Q1 | | f pxq fQ1 | ¡ λ 2dµu|

¤¸

Q1PBEpλ 2dµq|Q1| ¤ µ1Epλ 2dµq

(7.28)

Iterating this relation implies

Epλq ¤ µnEpλ 2dµnqSetting for example µ 2 leads to the desired bound.

Exercise 7.9. Show that one can characterize BMO also by the following con-dition, where 1 ¤ r 8 is fixed but arbitrary:

supQ

?Q| f pyq fQ|r dy

1r 8 (7.29)

In fact, show that if the left-hand side is ¤ 1, then f BMO ¤ Cpd, rq. Conversely,by Holder’s f BMO ¤ 1 implies that the left-hand side of (7.29) is ¤ 1.

A remarkable characterization of BMO was found by Coifman, Rochberg andWeiss. The considered commutators between Calderon-Zygmund operators andoperators given by multiplication by a function, in other words, rT, bs Tb bT .While it is clear that rT, bs is bounded on Lp for 1 p 8 if b P L8, theseauthors discovered that this property remains true for b P BMOpRdq. What is more,if this property holds for a single Riesz-transform (or a single nonzero operator ofthe form (7.20), then necessarily b P BMO. We shall now prove one direction ofthis statement.


Theorem 7.18. Let T be a linear operator as in Proposition 7.5 with the addedproperty that for some fixed 0 δ ¤ 1 one has

|Kpx yq Kpx0 yq| ¤ A|x x0|δ|y x0|dδ

@ |y x0| ¡ 2|x x0| (7.30)

as well as T2Ñ2 ¤ A. Then for any b P BMOpRdq one has

rT, bspÑp ¤ Cpd, pqAbBMO

for all 1 p 8.

Proof. Since (7.30) implies the Hormander condition, we see that any operatorT as in the theorem is a Calderon-Zygmund operator. We can also assume thatbBMO ¤ 1 and A 1. Fix f P SpRdq, say. We shall show that for any 1 r 8

prT, bs f q7 ¤ Cpr, dqpMrpT f q Mr f q (7.31)

where

pMr f qpxq : supQQx

?Q| f pyq|r dy

1r

Letting 1 r p 8 and noting that Mr is Lp-bounded, we conclude from (7.31)that prT, bs f q7p ¤ Cpd, pq f p and Theorem 7.16 implies the desired estimate(the p0-condition in that theorem is clear by Theorem 7.17 and f P SpRdq).

To establish (7.31) fix any cube Q, which by scaling and translation invariancemay be taken to be the unit cube centered at 0. Denote by Q the cube of side-length2?

d, and write

rT, bs f Tpb bQq f1

Tpb bQq f2

pb bQqT f : g1 g2 g3

where f1 : χQ f , f2 f f1. First, by Exercise 7.9?Q|g3pxq| dx ¤

?Q|b bQ|r1 dx

1r1?

Q|T f |rpxq dx

1r ¤ C inf

QMrpT f q

Second, choose some 1 s r. Then by boundedness of T on Ls one obtains?Q|g1pxq| dx ¤

?Q|g1pxq|s dx

1s ¤

?Q|b bQ|s| f pxq|s dx

1s

¤ C infQ

Mr f

where we used (7.22) to compare bQ with bQ. Finally, set

cQ : Tpb bQq f2

p0q


Then by (7.30) one has the estimate?Q|g2pxq cQ| dx

¤ C»

Q

»Qc|Kpx yq Kp0 yq| |bpyq bQ| | f pyq| dydx

¤ Ç`¥0

»2`1Qz2`Q

2`pdδq|bpyq bQ| | f pyq| dy

¤ Ç`¥0

2`δ?

2`1Q|bpyq bQ|r1 dy

1r1?

2`1Q| f pyq|r dy

1r

¤ Ç`¥0

` 2`δ infQ

Mr f ¤ C infQ

Mr f

where we used (7.23) to pass to the final estimate.

We have merely scratched the surface of BMO theory. Of course one wouldlike to determine a duality relation analogous to the basic pL1pRdq L8pRdq.This turns out to be pH1pRdqq BMOpRdq where H1pRdq is the completionof SpRdq under the norm

f H1 : f 1 d

j1

R j f 1 (7.32)

Dually, one can then write

BMO L8 d

j1

R jL8 (7.33)

where the action of the Riesz transforms on L8 needs to be defined suitably. Inparticular, this shows that BMO is the smallest space into which singular integraloperators are bounded from L8. We shall establish some of these facts in a one-dimensional dyadic setting in a later chapter.

We close this introductory chapter by stating a number of natural questions.First, we may ask if a Calderon-Zygmund operator is bounded on Holder spacesCαpRdq where 0 α 1. As we shall see, this turns out to be the case. Theanalogue of Corollary 7.7 in this case is known as Schauder estimates which playa fundamental role in elliptic equations. There are a number of proofs known ofthis classical fact, and we shall develop one possible approach in Chapter 10.

Another question that one may ask, and which turned out to have far-reachingconsequences, is as follows: is there a proof of the L2-boundedness of the Hilberttransform which does not rely on the Fourier transform? In other words, we wishto find a proof of the L2-boundedness which relies exclusively on the propertiesof the kernel itself. We shall answer this question in the affirmative in the largercontext of Calderon-Zygmund operators, and introduce the important method ofalmost orthogonality in the process, see Chapter 8.


And finally, one should of course ask if there is a version of Calderon-Zygmundtheory for kernels which are not translation invariant. This question has far-reachingconsequences especially with respect to the L2 boundedness. While it is true thatthe Lp-theory is quite similar to the translation invariant case (see Problem 7.3),the L2 theory leads to the T p1q theorem. We shall develop the basic ideas for thattheorem by means of a dyadic model on the line.

Notes

For most of the basic material of this chapter, excluding BMO, see Stein’s 1970book [45]. For a systematic development of H1 and BMO see [46] which gives a systematicaccount of many aspects of Calderon-Zygmund theory that appeared from 1970 to roughly1992. A classic reference for H1 and the duality with BMO is the paper by Fefferman andStein [16], which contains (7.32). Uchiyama [55] gives a constructive proof of the decom-position of a BMO function as a sum of L8 and images thereof under the Riesz transforms,see (7.33). Coifman and Meyer [10] offer another perspective of Calderon-Zygmund the-ory with many applications and connections with wavelets. The proof of the Coifman,Rochberg, Weiss commutator estimate is from Janson [28] (credited to Stromberg), seealso Torchinsky [54]. One important aspect of Calderon-Zygmund theory that we do notdiscuss in this text is that of Ap-weights, which is closely related to BMO. For this materialsee Stein [46], Garcıa-Cuerva, Rubio de Francia [7], as well as Duoandikoetxea [12]. Theclassical “method or rotations” which can be used to reduce higher-dimensional results toone-dimensional ones, is discussed in [12].

Problems

Problem 7.1. Let tai jpxqudi, j1 be continuous in some domain Ω Rd, d ¥ 2, and

suppose that these matrices are elliptic in the sense that for all x P Ω

θ|ξ|2 ¤d

i, j1

ai jpxqξiξ j ¤ Θ|ξ|2 @ ξ P Rd

for some fixed 0 θ Θ. Assume that u P C2pΩq satisfiesd

i, j1

ai jpxq BBxi

BBx j

upxq f pxq

Then show that for any compact K Ω and any 1 p 8 one has the bound

D2uLppKq ¤ CpuLppΩq f LppΩqqwhere C only depends on p,K,Ω and the ai j.

Problem 7.2. This problem shows how symmetries limit the boundedness propertiesof operators which commute with these symmetries. We shall consider translation anddilation symmetries, respectively.

a) Show that for any translation invariant non-zero operator T : LppRdq Ñ LqpRdqone necessarily has q ¥ p.


b) Show that a homogeneous kernel as that in (7.20) can only be bounded from

LppRdq Ñ LqpRdqif p q.

Problem 7.3. Let T be a bounded linear operator on L2pRdq so that for some functionKpx, yq measurable and locally bounded on Rd Rdz∆ with ∆ tx yu being thediagonal, one has for all f P L2

comppRdq

pT f qpxq »Rd

Kpx, yq f pyq dy, x < suppp f q

Furthermore, assume the Hormander conditions»r|xy|¡2|yy1|s

|Kpx, yq Kpx, y1q| dx ¤ A @ y , y1»r|xy|¡2|xx1|s

|Kpx, yq Kpx1, yq| dy ¤ A @ x , x1(7.34)

Then T is bounded from L1 to weak-L1 and strongly on Lp for every 1 p 8.

Problem 7.4. This is the continuum analogue of Problem 4.3. Let m be a function ofbounded variation on R. Show that Tm f pm f q_ is bounded on LppRq for any 1 p 8. Find a suitable generalization to higher dimensions, starting with R2.

Problem 7.5. Let f P BMOpRdq. Show that for any σ ¡ d and 1 ¤ r 8 one has»Rd

| f pxq fQ0 |r1 |x|σ dx ¤ Cpd, σ, rq f BMO

where Q0 is any cube of side-length 1. What does one obtain by rescaling this inequality?

Problem 7.6. Let f P L1pRdq have compact support, say B : Bp0, 1q. Let M be theHardy-Littlewood maximal function. Show that M f P L1pBq if and only if f logp2| f |q PL1pBq. Now redo Problem 4.6 based on this fact and the F.&M. Riesz theorem. Hint:Reprove the weak-L1 bound on M by means of the Calderon-Zygmund decomposition. Inparticular, show that it in order to bound |tM f ¡ λu| it is enough to control the L1 normof f on t| f | ¡ c λu where c ¡ 0 is some constant. See Stein [45] page 23 for more details.

Problem 7.7. Show that the Hilbert transform H with kernel 1x is bounded on the

Holder spaces CαpRq, 0 α 1. In other words, prove that for any 0 α 1

rH f sα ¤ Cpαq r f sαfor all f P C1

0pRq, say. Here rsα is the Cα-seminorm. Use only properties of the kernel forthis problem.

Problem 7.8. Let A, B be reflexive Banach spaces (a reader not familiar with thismay take them to be Hilbert spaces). Let Kpx, yq be defined, measurable, and locallybounded on Rd Rdz∆ where ∆ tx yu is the diagonal, and take its values in LpA, Bq,the bounded linear operators from A to B. Define an operator T on C0

comppRd; Aq, thecontinuous compactly supported functions taking their values in A, such that

pT f qpxq »

Kpx, yq f pyq dy, x < suppp f q


for all f P C0comppRd; Aq. Assume that for some finite constant C0»

|xy|¡2|xx1|Kpx, yq Kpx1, yqLpA,Bq dy ¤ C0 @x , x1»

|xy|¡2|yy1|Kpx, yq Kpx, y1qLpA,Bq dx ¤ C0 @y , y1

Further, assume that T : LrpRd; Aq Ñ LrpRd; Bq for some 1 r 8 with norm boundedby C0. Show that then T : LppRd; Aq Ñ LppRd; Bq for all 1 p 8 with norm boundedby C0 C1 where C1 C1pd, rq.

Problem 7.9. Investigate the endpoint p dα

of Proposition 7.8. More precisely, findexamples which show that fractional integration in that case is not bounded into L8. Verifythat your examples get mapped into BMOzL8pRdq. For the positive result which provesboundedness Lp Ñ BMO see Adams [1].

Problem 7.10. Let f P L1pTq. Given λ ¡ 1, show that there exists E T (dependingon λ and f ) so that |E| λ1 and for all N P Z

1N

»TzE

N

n1

|S n f pθq|2 dθ ¤ Cλ f 21

C is a constant independent of f ,N, λ. Hint: First apply the Plancherel theorem in n, andthen use a Calderon-Zygmund decomposition.

CHAPTER 8

Almost Orthogonality

The proof of L2 boundedness of Calderon-Zygmund operators in the previouschapter was based on the Fourier transform. This is quite restrictive as it requiresthe operator to be translation invariant. We now present a device that allows one toavoid the Fourier transform in the context of L2-theory in many instances.

Let us start from the basic observation that the operator norm of an infinitediagonal matrix viewed as an operator on `2pZq is as large as its largest entry. Tobe specific, let

Ttξ ju jPZ

tλ jξ ju jPZ @ tξ ju jPZ P `2pZqwhere tλ ju jPZ is a fixed sequence of complex numbers. Then

T`2Ñ`2 supj|λ j|

More generally, suppose a Hilbert spaceH can be written as an infinite orthogonalsum

H àjH j, H Q f

¸j

f j, f j P H j

and suppose that T j are operators on H with T jHk t0u if k , j. Furthermore,assume that the range of T j is a subspace ofH j. Then

T f 2H

¸j,k

xT j f j,Tk fky ¸

j

T j f j2H j

¤ supjT j2

H jÑH j

¸j

f j2H j M2 f 2

H

where M sup j T jH jÑH j .Generalizing from the previous example, assume that T is a bounded operator

on the Hilbert space H which admits the representation T °j T j such that

RanpT jq K RanpTkq as well as RanpTj q K RanpT

k q for j , k. In other words,

we assume that Tk T j 0 and TkT

j 0 for j , k. Now recall that RanpTj q

kerpT jqK and denote by P j the orthogonal projection onto that subspace. Then forany f P H one has

T f ¸

j

T jP j f ¸

j

T j f j, f j P j f

101

102 8. ALMOST ORTHOGONALITY

which implies that

T f 2H

¸j

T j f j2H¤ sup

jT j2

¸j

f j2H M2 f 2

H

with M : sup j T j.As a final step, we shall now relax the orthogonality conditions T

k T j 0 andTkT

j 0 for j , k from above, which turn out to be too restrictive for mostapplications. The idea is simply to replace this strong vanishing requirement bya condition which ensures sufficient decay in | j k|. One can think of this asreplacing diagonal matrices with matrices whose entries decay in a controllablefashion away from the diagonal. The following lemma, known as Cotlar-Steinlemma, carries this idea out.

Lemma 8.1. Let tT juNj1 be finitely many operators on L2 such that for some

function γ : ZÑ R, one has

Tj Tk ¤ γ2p j kq, T jT

k ¤ γ2p j kqfor any 1 ¤ j, k ¤ N. Let

8

`8γp`q : A 8

Then °Nj1 T j ¤ A.

Proof. For any positive integer n,

pTT qn N

j1,..., jn1k1,...,kn1

Tj1Tk1T

j2Tk2 TjnTkn

We now take the operator norm of both sides of this identity, and apply the triangleinequality on the right-hand side. Furthermore, we bound the norm of the productsof operators by the products of the norms in two different ways, followed by takingthe square root of the product of the resulting estimate.Therefore, with sup1¤ j¤N T j : B

pTT qn

¤N

j1,..., jn1k1,...kn1

T j112 T

j1Tk112 Tk1T

j212 Tkn1T

jn12 T

jnTkn12 Tkn

12

¤N

j1,..., jn1k1,...,kn1

?B γp j1 k1qγpk1 j2qγp j2 k2q γpkn1 jnqγp jn knq

?B

¤ NBA2n1

8. ALMOST ORTHOGONALITY 103

Since TT is self-adjoint, the spectral theorem implies that pTT qn TTn T2n. Hence,

T ¤ pN BA1q 12n A

Letting n Ñ8 yields the desired bound.

Note that one needs both smallness of Tj Tk and T jT

k in the Cotlar-Steinlemma, as can be seen from the examples preceding the lemma.

In order to apply Lemma 8.1 one often invokes the following simple device.

Lemma 8.2. Define an integral operator on a measure space X Y with thepositive product measure µb ν via

pT f qpxq »

YKpx, yq f pyq νpdyq

where K is a measurable kernel. One has the following bounds (the first three itemsare known as Schur’s lemma):

T1Ñ1 ¤ supyPY³

X |Kpx, yq| µpdxq : A T8Ñ8 ¤ supxPX

³Y |Kpx, yq| νpdyq : B

TpÑp ¤ A1p B

1p1 where 1 ¤ p ¤ 8.

T1Ñ8 ¤ KL8pXYq

Proof. The first two items are immediate from the definitions, and the thirdthen follows by interpolation. Alternatively, one can use Holder’s inequality. Thefourth item is again evident from the definitions.

Henceforth, we shall simply refer to this lemma as “Schur’s test”. We will nowgive an alternative proof of L2 boundedness of singular integrals for kernels as inDefinition 7.1 which satisfy the stronger condition

|∇Kpxq| ¤ B|x|d1

cf. Lemma 7.2. This requires a certain standard partition of unity over a geometricscale which we now present.

Lemma 8.3. There exists ψ P C8pRdq with the property that supppψq Rdzt0uis compact and such that

8

j8ψp2 jxq 1 @ x , 0 (8.1)

For any given x , 0 at most two terms in this sum are nonzero. Moreover, ψ canbe chosen to be a radial nonnegative function.

Proof. Let χ P C8pRdq satisfy χpxq 1 for all |x| ¤ 1 and χpxq 0 for|x| ¥ 2. Set ψpxq : χpxq χp2xq. For any positive N one has

N

jN

ψp2 jxq χp2N xq χp2N1xq


If x , 0 is given, then we take N so large that χp2N xq 1 and χp2N1xq 0.This implies (8.1), and the other properties are immediate as well.

The point of the following corollary is the method of proof rather than thestatement (which is weaker than the one in the previous chapter).

Corollary 8.4. Let K be as in Definition 7.1 with the additional assumptionthat |∇Kpxq| ¤ B|x|d1. Then

T2Ñ2 ¤ CB

with C Cpdq.

Proof. We may take B 1. Let ψ be a radial function as in Lemma 8.3 andset K jpxq Kpxqψp2 jxq. One now easily verifies that these kernels have thefollowing properties: for all j P Z one has»

K jpxq dx 0,

∇K j8 ¤ C 2 j2 jd

In addition, one has the estimates»|K jpxq| dx C»

|x| |K jpxq| dx C 2 j

with some absolute constant C. Define

pT j f qpxq »Rd

K jpx yq f pyq dy

Observe that this integral is absolutely convergent for any f P L1`ocpRdq. We shall

now check the conditions in Lemma 8.1. Let rK jpxq : K jpxq. Then it is easy tosee that

pTj Tk f qpxq :

»RdprK j Kkqpyq f px yq dy

and

pT jTk f qpxq

»RdpK j rKkqpyq f px yq dy

Hence, by Young’s inequality,

Tj Tk2Ñ2 ¤ rK j Kk1

and

T jTk 2Ñ2 ¤ K j rKk1


It suffices to consider the case j ¥ k. Then, using the cancellation condition³Kkpyq dy 0, one obtainsprK j Kkqpxq

»Rd

K jpy xqKkpyq dy

»Rd

K jpy xq K jpxq

Kkpyq dy

¤»Rd∇K j8|y| |Kkpyq| dy

¤ C 2 j2 jd2k

Sincesupp prK j Kkq suppprK jq supppKkq Bp0,C 2 jq

we further conclude that

rK j Kk1 ¤ C 2k j C 2| jk|

Therefore, Lemma 8.1 applies with

γ2p`q C 2|`|

and the corollary follows.

Exercise 8.1.

a) In the previous proof it suffices to consider the case j ¡ k 0. Providethe details of this reduction.

b) Observe that Corollary 8.4 covers the Hilbert transform. In that case,draw the graph of Kpxq 1

x and also K jpxq 1xψp2 jxq and explain the

previous argument by means of pictures.

Another simple application of these almost orthogonality ideas is the Calderon-Vaillancourt theorem. This result concerns the L2-boundedness of pseudo-differentialoperators of the form

T f pxq »Rd

eixξ apx, ξq f pξq dξ, f P SpRdq (8.2)

where a P C8pRd Rdq is such that

supx,ξPRd

|Bαx apx, ξq| |Bαξ apx, ξq| ¤ B (8.3)

for all |α| ¤ d 1.

Proposition 8.5. Under the conditions (8.3) the operators in (8.2) are boundedon L2pRdq.

Proof. We may assume that B 1. Let χ be smooth and compactly supportedsuch that tχp kqukPZd forms a partition of unity in Rd:¸

kPZd

χpξ kq 1 @ ξ P Rd


To construct such a χ, start from a compactly supported smooth η ¥ 0 with theproperty that

ψpxq :¸

kPZd

ηpx kq ¡ 0 @ x P Rd

Note that ψ is periodic with respect to Zd and therefore uniformly lower bounded.Now define χ : χ

ψ .

Set

χk`px, ξq : χpx kqχpξ `qak,`px, ξq : apx, ξqχpx kqχpξ `q

and define

pTk,` f qpxq :»Rd

eixξ ak,`px, ξq f pξq dξ

for f P SpRdq. Note that we have dropped the Fourier transform of f , which isadmissible by Plancherel’s theorem. First, Lemma 8.2 implies that

supk,`Tk,`2Ñ2 ¤ C 1

Furthermore, we claim that

Tk1,`1Tk,`2Ñ2 ¤ Cxk1 kyd1 x`1 `yd1

Tk1,`1Tk,`2Ñ2 ¤ Cxk1 kyd1 x`1 `yd1

(8.4)

for all k, k1, `, `1 P Zd. Since the bounds on the right-hand side are summable overk, ` P Zd, Lemma 8.1 implies that ¸

|k| N

¸|`| N

Tk,` f

2 ¤ C f 2 @ f P SpRdq

for any positive integer N with some absolute constant C. Passing to the limitN Ñ8 concludes the proof. To prove (8.4) we start from

Tk,`gpξq

»Rd

eixξapx, ξq gpxq dx

which shows that Tk1,`1Tk,` , 0 requires that |kk1| ¤ C and similarly, Tk1,`1T

k,` , 0requires that |` `1| ¤ C. The kernel of T

k1,`1Tk,` is

Kk,`;k1,`1pξ, ηq :»Rd

eixpξηqak,`px, ξq ak1,`1px, ηq dx (8.5)

To prove the decay in |` `1| we may assume that this difference exceeds someconstant which is chosen such that |ξη| ¥ 1 on the support of ak,`px, ξq ak1,`1px, ηq.We now use the relation

iξ η

|ξ η|2∇xeixpξηq eixpξηq

in order to integrate by parts in (8.5) repeatedly, to precise d 1 times. This yields

|Kk,`;k1,`1pξ, ηq| ¤ C|` `1|d1


Since the support of the kernel with respect to either variable is of size ¤ C 1, weconclude from Schur’s test that

Tk1,`1Tk,`2Ñ2 ¤ C x`1 `yd1

where may assume that |k k1| ¤ C 1. The same type of argument now impliesthe second relation of (8.4) and we are done.

We now turn to Hardy’s inequality, which provides a natural transition betweenthis chapter and next one devoted to the uncertainty principle.

Theorem 8.6. For any 0 ¤ s d2 there is a constant Cps, dq with the property

that

|x|s f 2 ¤ Cps, dq f 9HspRdq (8.6)

for all f P 9HspRdq.

Proof. The stated bound is evident for s 0, so we may assume that 0 s d2 by interpolation (any reader uncomfortable with this interpolation may simplyassume that condition on s without missing anything from the following argument).By density, it suffices to prove this estimate for f P SpRdq. We use the partitionof unity from Lemma 8.3 and write Pk f pψp2kξq f q_ for any k P Z. Byconstruction, f °

k Pk f where the series converges in several ways, for examplein L2. Furthermore, we set χ`pxq ψp2`xq with ` P Z. Then

|x|s f 2 ¤¸

k

|x|sPk f 2

¤ C¸

k

¸`k¤0

22`sχ`Pk f 22 22skPk f 2

2

12

(8.7)

where the final term is the contribution from the region t|x| ¡ 2ku. The ideahere is that the main contribution should come from ` k 0 which immediatelyyields (8.6). The mechanism here derives from the uncertainty principle which wewill discuss in the following chapter; an indication of this is given by the trans-formation law (6.1) and in particular the special case of this, namely the dilationidentity (6.2).

To implement this idea, we let ψ be smooth and compactly supported in Rdzt0usuch that ψ 1 on supppψq. This implies that Pk f : pψp2kξq f q_ for any k P Zsatisfies PkPk PkPk Pk whence

χ`Pk f 2 χ`PkPk f 2 ¤ χ`Pk2Ñ2Pk f 2

We now claim that

χ`Pk22Ñ2 ¤ C 2dp`kq ` k ¤ 0 (8.8)

which is simple consequence of Schur’s test. We shall provide the details for thisestimate at the end of the proof. If this holds, then we can continue from (8.7) as


follows:¸k

¸`k¤0

22`sχ`Pk f 22

12 ¤ C

¸k

¸`k¤0

22`s22dp`kqPk f 22

12

¤ Ç

k

¸`k¤0

2pd2sqp`kq22skPk f 22

12

¤ Ç

k

2skPk f 2

In summary we arrive at (it is easy to see that we my drop the tilde on Pk)

|x|s f 2 ¤¸

k

22skPk f 2

But this is strictly weaker than what we would like which shows that one cannotgive away as much of the (almost) orthogonality of the Pk f as we did.

Hence, we should go about the first step (8.7) a bit differently:

|x|s f 22 ¤ C

¸`

22`sχ` f 22

¤ Ç`

22`s ¸

k`¤0

χ`Pk f 2

2¸`

22`sχ`P¡` f 22

(8.9)

where P` : °j¡` P j. For the sum involving both k, ` we invoke (8.8) to wit¸

`

22`s ¸

k`¤0

χ`Pk f 2

2¤ C

¸`

¸k`¤0

2pd2sqp`kq2skPk f 2

2

¤ Ç`

22ksPk f 22 ¤ C f 2

9Hs

since d2 s ¡ 0. To pass to the last line one can use Schur’s test for sums, for

example. The final sum in (8.9) uses that s ¡ 0:¸`

22`sχ`P¡` f 22 ¤

¸`

22`sP¡` f 22

¤ Ç`

22p`kqs ¸k`¡0

22skPk f 22

¤ Ç

k

22skPk f 22

¸`k¡0

22p`kqs ¤ C f 29Hs

which is (8.6).It remains to prove (8.8). For this write

pχ`Pk f qpxq »Rdψp2`xq2kdψp2kpx yqq f pyq dy


and observe that

supx

»Rd|ψp2`xq2kdψp2kpx yqq| dy ¤ C 1

supy

»Rd|ψp2`xq2kdψp2kpx yqq| dx ¤ C 2pk`qd

while implies the claim by Schur’s test.

Exercise 8.2. Show that one may “drop the tilde” as was done in the previousproof, i.e., prove that ¸

k

22ksPk f 22

¸k

22ksPk f 22

for any s.

Theorem 8.6 is sharp in the following sense: clearly, both sides scale the sameso it is impossible to use any other Sobolev space, for example. The conditions ¥ 0 is of course needed as we cannot allow for growing weights. And s ¥ d

2 isnot possible since in that case the estimate fails for Schwartz functions which donot vanish at the origin (the norm on the left-hand side is infinite in that case).

Several questions pose themselves naturally now: (i) is there a substitute fors ¡ d

2 in the previous theorem? (ii) Is there a version of Theorem 8.6 with Lp

instead of L2?In case of s 1, we shall answer these questions in the problems.

Notes

For more on the method of almost orthogonality, in particular the use of Cotlar’slemma, see Stein [46]. Hardy’s inequalities are basic tools that have been generalizedin many different directions, see for example Kufner and Opic [39]. For fractional s oneneeds more tools, namely Littlewood-Paley theory.

Problems

Problem 8.1. Let T be of the form (8.2) where a P C8pRd Rdq is such that

supxPRd

|Bαx apx, ξq| |Bαξ apx, ξq| ¤ Bxξys

for all |α| ¤ d1 and all ξ P Rd. Here s ¥ 0 is arbitrary but fixed. Show that T is boundedform HspRdq to L2pRdq.

Problem 8.2. Suppose Kpx, yq is defined on p0,8q2 and is homogeneous of degree1, i.e., Kpλx, λyq λ1Kpx, yq for all λ ¡ 0. Further, assume that for an arbitrary butfixed 1 ¤ p ¤ 8 one has » 8

0|Kpx, yq|y 1

p dy A

Show that pT f qpxq : ³80 Kpx, yq f pyq dy satisfies T f p ¤ A f p.


Problem 8.3. Prove Hardy’s original inequality, valid for every measurable f ¥ 0» 8

0

» x

0f ptq dt

pxr1 dx ¤

pr

p» 8

0px f pxqqp xr1 dx» 8

0

» 8

xf ptq dt

pxr1 dx ¤

pr

p» 8

0px f pxqqp xr1 dx

for every 1 ¥ p 8, r ¡ 0. Equality holds if and only if f 0 a.e. Hint: use theprevious problem.

Problem 8.4. Show the following Hardy’s inequality in Rd»Rd|∇upxq|p dx ¥

|d p|p

p»Rd

|upxq|p|x|p dx (8.10)

for all u P SpRdq if 1 ¤ p d and for all such u which also vanish on a neighborhood ofthe origin if 8 ¡ p ¡ d. The constants here are sharp. Hint: use the previous problem.

Problem 8.5. Show that if 1 ¤ p d and 1p 1

p 1d , then»

Rd

|upxq|p|x|p dx Cpp, dqup

p

provided u P SpRdq is radial and decreasing. Conclude that for such functions the Sobolevembedding inequality up ¤ Cpp, dq∇up. By means of the device of nonincreas-ing rearrangement this special case implies the general one. In other words, the Sobolevembedding holds without the radial decreasing assumption. The point here is that the re-arrangement does not change up and at most decreases ∇up, see [36]. This is usefulin order to find the optimal constants, see Frank, Seiringer [18].

CHAPTER 9

The Uncertainty Principle

This chapter deal with various manifestations of the heuristic principle thatit is not possible for both f and f the be localized on small sets. We alreadyencountered a rigorous, albeit qualitative and rather weak version of this principle:it is impossible for both f P L2pRdq and f to be compactly supported.

Here we are more interested in quantitative versions, and moreover, the tight-ness of these quantitative versions. What is meant by that can be seen from thetransformation law (6.1): let χ be a smooth bump function, and A be an invertibleaffine transformation that takes the unit ball onto an ellipsoid

E :!

x P Rd |d

j1

px j y jq2r2j ¤ 1

)where y j are arbitrary constants and r j ¡ 0. Then χA : χ A1 can be thought ofas a smoothed out indicator function of E. By (6.1), xχA then essentially “lives” onthe dual ellipsoid

E :!ξ P Rd |

d

j1

ξ2j r

2j ¤ 1

)Moreover, while χA is L8-normalized, xχA is L1-normalized. To “essentially liveon” E means that

|xχApξq| ¤ CN |E|1 |ξ|2

E

N

where

|ξ|2E

:d

j1

ξ2j r

2j @ ξ

is the Euclidean norm relative to E. What we can see from this is another, stillheuristic but more quantitative form of the uncertainty principle, namely the fol-lowing: if f P L2pRdq is supported in a ball of size R, then it is not possible for fto be “concentrated” on a scale much less than R1.

We begin the rigorous discussion with another form of Bernstein’s estimate.In what follows Bpx0, rq denotes the Euclidean ball in Rd which is of radius r andcentered at x0.

Lemma 9.1. Suppose f P L2pRdq satisfies suppp f q Bp0, rq. Then

Bα f 2 ¤ p2πrq|α| f 2 (9.1)

for all multi-indices α.

111

112 9. THE UNCERTAINTY PRINCIPLE

Proof. This follows from f P C8pRdq, the relation

Bα f pξq p2πiq|α|yxα f pξqand Plancherel’s theorem.

Heurstically speaking, this can be seen as a version of the non-concentrationstatement from before. Indeed, if f were to be concentrated on a scale of size! r1, then we may expect the left-hand side of (9.1) to be much larger than r|α|.Later in this chapter, we shall see a better reason why Bernstein’s estimate shouldbe viewed in the context of the uncertainty principle.

Next, we prove Heisenberg’s uncertainty principle.

Proposition 9.2. For any f P SpRq one has

f 22 ¤ 4πpx x0q f 2pξ ξ0q f 2 (9.2)

for all x0, ξ0 P R. This inequality is sharp, with extremizers being precisely givenby the modulated Gaussians

f pxq C e2πiξ0 x eπδpxx0q2

where C and δ ¡ 0 are arbitrary.

Proof. Without loss of generality x0 ξ0 0. Define D 12πi

ddx , and let

pX f qpxq x f pxq. Then the commutator

rD, Xs DX XD 12πi

whence for any f P SpRq, f 2

2 2πixrD, Xs f , f y 2πipxX f ,D f y xD f , X f yq 4πIm xD f , X f y ¤ 4πD f 2X f 2

(9.3)

as claimed.For the sharpness, note that if equality is achieved in (9.3) then from the con-

dition on equality in Cauchy-Schwarz one has D f λX f for some λ P C. ThenxD f , X f y λX f 2

2 which means that λ iδ with some δ ¡ 0. Finally, thisimplies that

f 1pxq 2πδ f pxq, f pxq Ceπδx2

The general case follows by translation in x and ξ (the latter being the same asmultiplication by eixξ0 .

There is an analogous statement in higher dimension which we leave to thereader. Once again, we can see Proposition 9.2 as a manifestation of the principlethat if f is concentrated on scale R ¡ 0, then f cannot be concentrated on a scale! R1.

9. THE UNCERTAINTY PRINCIPLE 113

Hardy’s inequality from the previous chapter can be seen as another instanceof this principle. Indeed, in R3 it implies that

|x x0|1 f 2 ¤ C∇ f 2 Cξ f 2 @ f P SpR3q (9.4)

for any x0 P R3. Thus, if f is supported on Bp0,Rq, then

supx0PR3

f L2pBpx0,ρq ¤ CρR f 2 0 ρ R1

We leave it to the reader to derive similar statements in all dimensions. In fact, theproof of Hardy’s inequality which we gave above clearly shows the dominant roleof the dual scales R in x, and R1 in ξ, respectively.

Let us reformulate (9.4) with x0 0 as follows:

x|x|2 f , f y ¤ Cx∆ f , f y @ f P SpR3qwhich can be rewritten as ∆ ¥ c

|x|2 for some c ¡ 0 as operators (in other words,∆ c

|x|2 is a nonnegative symmetric operator).

It is interesting to note that we can now give a proof of this special case ofHardy’s inequality using commutator arguments, similar to those in the proof ofProposition 9.2. In addition, this will yield the optimal value of the constant.

Proposition 9.3. One has ∆ ¥ pd2q2

4|x|2 in Rd with d ¥ 3, which means that

for any f P SpRdq

x∆ f , f y ¥ pd 2q24

x|x|2 f , f y ðñ |x|1 f 2 ¤ pd 2q24

∇ f 2

Proof. Let p i∇. The relevant commutator here is, with r |x|,irr1 pr1, Xs dr2

This implies that

dr1 f 22 2Im xr1 pr1 f , X f y @ f P SpRdq

Sincep|x|1 |x|1 p i

x|x|3

one obtains

pd 2qr1 f 22 2Im xp f , Xr2 f y @ f P SpRdq

Cauchy-Schwartz concludes the proof.

The bound in the previous proposition is optimal, but not attained in the largestadmissible class, i.e., 9H1pRdq. We shall not dwell on this issue here.

Next, we investigate the following question: if E, F Rd are of finite measure,can there be a nonzero f P L2pRdq with suppp f q E and suppp f q F ?

Note that if there were such an f , then T f : χEpχF f q_ satisfies T f f .In particular, this would imply T2Ñ2 ¥ 1. We might expect that T has small


norm if E and F are small, leading to a contradiction and negative answer to ourquestion. Now note that

pT f qpxq »RdχEpxqχFpx yq f pyq dy

In other words, T has kernel Kpx, yq χEpxqχFpx yq with Hilbert-Schmid norm»R2d|Kpx, yq|2 dxdy |E| |F| : σ2 8

We remark that σ is a scaling invariant quantity. Hence, T ¤ minpσ, 1q in oper-ator norm and T is a compact operator. So if σ 1, then we see that there can beno f , 0 as in the question. Interestingly, the answer is “no” in all cases and onehas the following quantitative expression of this property.

Theorem 9.4. Let E and F be sets of finite measure in Rd. Then

f L2pRdq ¤ Cp f L2pEcq f L2pFcqq (9.5)

for some constant CpA, B, dq.

The reader will easily prove the estimate in case σ 1. For the general case,we formulate the following equivalencies with the goal of showing that T 1for any E, F as in the theorem. In the lemma supp refers to the essential support.

Lemma 9.5. Let E, F be measurable subsets of Rd. Then the following areequivalent (C1 and C2 denote constants):

a) f L2pRdq ¤ C1p f L2pEcq f L2pFcqq for all f P L2pRdqb) there exists ε ¡ 0 so that f 2

L2pEq f 2L2pFq ¤ p2 εq f 2

2 for all

f P L2pRdqc) if suppp f q F, then f 2 ¤ C2 f L2pEcqd) if suppp f q E, then f 2 ¤ C2 f L2pFcqe) there exists 0 ρ 1 so that χEpχF f q_2 ¤ ρ f 2 for all f P L2pRdq.

Proof. aq ùñ bq: one has

f 2L2pEq f 2

L2pFq 2 f 22 f 2

L2pEcq f 2L2pFcq

¤ p2 p2C21q1q f 2

2

bq ùñ cq: for f as in cq, f 2L2pEq ¤ p1εq f 2

2 or f 22 ¤ ε1 f 2

L2pEq. cq ùñ aq:write f PF f PFc f where PF f pχF f q_ and set QEc f χEc f . Then

f 2 ¤ PF f 2 PFc f 2 ¤ C2QEc PF f 2 PFc f 2

¤ C2QEc f 2 C2QEc PFc f 2 PFc f 2

¤ C2QEc f 2 pC2 1qPFc f 2

Interchanging f with f one sees that property dq to the first three.


cq ùñ eq: with the projection P,Q as above one has

PF f 22 QEPF f 2

2 QEc PF f 22

¥ QEPF f 22 C2

2 PF22

which implies the desired bound with ρ p1C22 q 1

2 .eq ùñ cq: clear.

Proof of Theorem 9.4. We already observed that T QEPF is compact withHilbert-Schmid norm THS ¤ σ. In particular,

dimt f P L2pRdq | T f λ f u ¤ λ2σ2 (9.6)

To prove this bound, let t f jumj1 be an orthonormal sequence in the eigenspace on

the left. Then, with K being the kernel of K, one has

mλ2 ¸

j

»R2d

Kpx, yq f jpxq f jpyq dxdy2 ¤ K2

L2pR2d (9.7)

by Bessel’s inequality. Furthermore, as product of projections, T ¤ 1 in operatornorm. If T 1, then it follows from compactness of T that there exists f PL2pRdq with f , 0 and T f 2 f 2.

Thus, there exists f with suppp f q E and suppp f q F (as essential sup-ports). We shall now obtain a contradiction to (9.6) for λ 1. Inductively, define

S 0 : suppp f q, S 1 : S 0 Y pS 0 x0qS k1 : S k Y pS 0 xkq k ¥ 1

where the translations xk are chosen such that

|S k| |S k1| |S k| 2k

If follows that the collection fk : f p xkq with k ¥ 0 is linearly independentwith

suppp fkq S8 :8¤`0

S ` @ k ¥ 0

where |S8| 8, while maintaining suppp fkq F. This gives the desired contra-diction.

Next, we formulate some results that provide further evidence of the non-concentration property of functions with Fourier support in Bp0, 1q.

Proposition 9.6. Let α ¡ 0 and suppose that E Rd satisfies

|E X B| α|B| for all balls B or radius 1

Suppose f P L2pRdq satisfies suppp f q Bp0, 1q. Then

f L2pEq ¤ δpαq f 2 (9.8)

where δpαq Ñ 0 as αÑ 0.


Proof. Let ψ be a smooth compactly supported function with ψ 1 on Bp0, 1q.Define Tg : χEpψgq_. The kernel of this operator is

Kpx, yq χEpxqψpx yqand satisfies the bound, for any N ¥ 1,

|Kpx, yq| ¤ CN χEpxqp1 |x y|qN

We apply Schur’s lemma to this kernel. To this end let ε ¡ 0 be arbitrary but fixed.First, Kpx, qL1

y¤ C 1. Second, take N ¡ d and choose R so large that

supy

»|xy|¡R

CN p1 |x y|qN dx ¤ ε

Then we see by covering Bpx,Rq by finitely many balls of radius 1 that»|xy| R

CN χEpxqp1 |x y|qN dx ¤ ε

provided α ¡ 0 is small enough. Therefore, by Schur’s lemma 8.2 we concludethat T ¤ C

?ε. Hence, if f is as in the hypothesis, then χE f T f whence

f L2pEq ¤ C?ε f 2

as claimed.

We now establish a somewhat more refined version of the previous result,which goes by the name of Logvinenko-Sereda theorem.

Theorem 9.7. Suppose a measurable set E Rd satisfies the following “thick-ness” condition: there exists γ P p0, 1q such that

|E X B| ¡ γ|B| for all balls B of radius R1

where R ¡ 0 is arbitrary but fixed. Assume that suppp f q Bp0,Rq. Then

f L2pRdq ¤ C f L2pEq (9.9)

where the constant C only depends on d and γ.

By the same argument as in Proposition 9.6 (or directly from that proposition)one proves this result for γ near 1.

Exercise 9.1. Prove Theorem 9.7 if γ is close to 1.

We shall base the proof of Theorem 9.7 on arguments involving analytic func-tions of several complex variables. This is natural in view of the Paley-Wienertheorem, cf. 6.5 which we shall state below. However, we shall not obtain explicitconstants Cpd, γq from our argument.

In order to keep this discussion self-contained, we briefly review the few factsabout analytic functions of several complex variables that will be needed. LetΩ Cd be open and connected (a “domain”), and suppose G P CpΩq is complex-valued. We say that G is analytic in Ω if and only if

z j ÞÑ Gpz1, z2, . . . , z j1, z j, z j1, . . . , zdq


is analytic on the corresponding sections of Ω as all other z j are kept fixed. Wedenote this class by ApΩq. The Cauchy integral formula applies via integrationalong contours in the individual variables. Examples of such functions are powerseries

f pzq ¸α

aα pz z0qα, z P ∆ : tz | max1¤ j¤d

|z j z0j | Mu

provided |aα| ¤ M|α| for all multi-indices α and a constant M ¡ 0. Indeed, itis easy to see that the power series converges uniformly in ∆ and the sum f pzq isanalytic in ∆. In particular, if

|aα| ¤ MB|α|

α!@ α

then the power series represents an entire function in Cd.Conversely, if f P ApΩq, then

f pzq ¸α

Bαz f pz0qα!

pz z0qα |z z0| r

for some r ¡ 0. The uniqueness theorem therefore also holds in this setting: ifBαz f pz0q 0 for all α, then f 0 in Ω. Moreover, if Ω X Rd contains a setof positive measure on which f vanishes, then f 0 in Ω. Due to the Cauchyestimates, one has Montel’s theorem on normal families: if F ApΩq is a locallybounded family, then F is precompact, i.e., every sequence in F has a subsequencewhich converges locally uniformly on Ω to an element ofApΩq.

Finally, we recall the Paley-Wiener theorem: suppose f P L2pRdq satisfiessuppp f q Bp0,Rq. Then f extends to an entire function in Cd with exponentialgrowth:

| f pzq| ¤ C Rd2 e2π|R||z| f L2pRdq

which is proved by placing absolute values inside the integral defining f pzq andapplying Cauchy-Schwarz.

Proof of Theorem 9.7. We may assume that R 1. Let A ¥ 2 be a constantthat we shall fix later. Partition Rd into congruent cubes of sidelength 2, which weshall denote by Q. We say that one such Q is good if

Bα f L2pQq ¤ A|α| f L2pQq @ αWe now claim that

f L2pbad Qq ¤ CA1 f 2 (9.10)

To prove it, consider the sum¸α,0

¸Q bad

A2|α|Dα f 2L2pQq (9.11)


Since Q is bad, there is a lower bound of f 2L2pbad Qq. On the other hand, by

Lemma 9.1, one has an upper bound of the form¸α,0

¸Q bad

A2|α|p2πq|α| f 22

¤8

k1

pk 1qdp2πA2qk f 22 ¤ CA2 f 2

2

which proves the claim.Next, we claim that if Q is good, then there exists x0 P Q such that

|Bα f px0q| ¤ Cα A2|α|1 f L2pQq, @ α (9.12)

Indeed, if not, then

A2|Q| f 2L2pQq ¤

»Q

¸α

A3|α|A4|α|2 f 2L2pQq dx

¤»

Q

¸α

A3|α||Bα f pxq|2 dx

¤¸α

A3|α|A2|α| f 2L2pQq ¤

8

k0

pk 1qd2k f 2L2pQq

which is a contradiction for A ¥ 2 large.And finally, we claim that there exists η ¡ 0 which depends only on d, γ, A so

that

f L2pEXQq ¥ η f L2pQq for all good Q (9.13)

If not, there exist t fnu8n1 in L2pRdq with suppp fnq Bp0, 1q and measurable setsEn Rd with

|En X Bpx, 1q| ¥ γ|Bpx, 1q| @ x P Rd

and good cubes Qn such that

fnL2pQnq 1, fnL2pEnXQnq Ñ 0

as n Ñ8. After translation, Qn r1, 1sd : Q0. Denote the entire extension offn given by the Paley-Wiener theorem by Fn. We expand Fn around a good xn asin (9.12) to conclude that for any |z| ¤ R one has

|Fnpzq| ¤¸α

A2|α|1

α!|pz xnqα|

¤ 8

`0

A2`1

`!p2 |z|q`

d¤ CpA, d,Rq

So tFnu8n1 is a normal family in Cd and we may extract a locally uniform limitFn Ñ F8 (up to passing to a subsequence) whence F8L2pQ0q 1. On the other


hand, x P Q0 X En | | fnpxq| ¥ λn(

¤ λ2n fn2

L2pEnq λ2n η2

n

¤ γ

2|Bp0, 1q|

if ηn λn

bγ2 |Bp0, 1q|. In particular, λn Ñ 0. By the thickness condition,

Xn : tx P Q0 X En | | fnpxq| ¤ λnusatisfies |Xn| ¡ γ

2 . But then

X8 : lim supnÑ8

Xn

satisfies |X8| ¥ γ2 . By construction, λn Ñ 0 whence F8 0 on X8. By the

uniqueness theorem for analytic functions of several variables, F8 0 on Q0.This contradiction establishes our final claim. We can now finish the proof of

the theorem. Indeed, one has from (9.13) and (9.10)

f 2L2pEXQ good Qq

¸Q good

f 2L2pEXQq

¥¸

Q good

η2 f 2L2pQq η2 f 2

L2p good Qq

¥ η2 f 2L2 C2A2 f 2

2

¥ 1

2η2 f 2

2

if A was chosen large enough. This implies the desired estimate.

Exercise 9.2. Prove that Theorem 9.7 with R 1 can also be proved under thefollowing weaker hypothesis: there exists γ ¡ 0 and r ¡ 0 such that

|E X Bpx, rq| ¡ γ @ x P Rd

The constant in (9.9) then of course also depends on r.

We now formulate a second version of the Logvinenko-Sereda theorem. Webegin with a definition.

Definition 9.8. A set F Rd is called Bp0, 1q-negligible if and only if thereexists ε ¡ 0 so that for all f P L2pRdq one has either

f L2pRdzBp0,1qq ¥ ε f 2 or

f L2pRdzFq ¥ ε f 2(9.14)

By means of Theorem 9.7 we can characterize negligible sets.


Theorem 9.9. A set F is Bp0, 1q-negligible if and only if there exist r ¡ 0 andβ P p0, 1q such that for all x P Rd

|F X Bpx, rq| β|Bpx, rq| (9.15)

Proof. First, we note that the condition (9.15) is equivalent to the following:there exist r ¡ 0 and γ ¡ 0 so that

|Fc X Bpx, rq| ¡ γ @ x P Rd (9.16)

If (9.16) fails, then there exists x j P Rd and r j Ñ8 with

|Fc X Bpx j, r jq| Ñ 0 j Ñ8Now take ϕ Schwartz with supppϕq Bp0, 1q and set pϕ jpξq : e2πix jξϕ. Then pϕ jis still supported in Bp0, 1q and with B j : Bpx j, r jq, we have

ϕ jL2pRdzFq ¤ ϕ jL2pB jzFq ϕ jL2pRdzB jq

¤ ϕ8b|B jzF|

»r|xx j|¡r js

C|x x j|d1 dx 1

2 (9.17)

which vanishes in the limit j Ñ 8. But this contradicts (9.14) which finishes thenecessity.

For the sufficiency, let E : Fc. Applying Theorem 9.7, see also Exercise 9.2,we infer that for any f P L2pRdq with suppp f q Bp0, 1q the estimate

f 2 ¤ C f L2pEq

with C Cpγ, r, dq. Now split f f1 f2 with f1 pχBp0,1q f q_. If f22 ¤ε f 2, then

f L2pEq ¥ f1L2pEq f22

¥ C1 f 2 ε f 2 ¡ ε f 2

provided ε ¡ 0 is sufficiently small and we are done.

We shall now apply these uncertainty principle ideas to the local solvability ofpartial differential equations with constant coefficients. To be specific, suppose weare given a polynomial

ppξq ¸α

aα ξα

Then with D : 12πiBx one has

ppDq :¸α

aα p2πiqαDα

This definition is chosen so that for all f P SpRdqppDq f pξq ppξq f pξq @ ξ P Rd

The following theorem is known as Malgrange-Ehrenpreis theorem.


Theorem 9.10. Let Ω be a bounded domain in Rd and p , 0 be a polynomial.Then for all g P L2pΩq there exists f P L2pΩq such that ppDq f g in the sense ofdistributions.

The “distributional sense” means that for all ϕ P C8pΩq with compact supportsupppϕq Ω one has

pg, ϕq p f , ppDqϕqwhere ppξq : °

α aα ξα. The pairing p, q is the L2 pairing.We emphasize that we are not making any ellipticity assumptions in the theo-

rem. Even so, one can show that f can be taken to be C8pΩq, see the Problem 9.6below. This is very different from the assertion that every solution f is C8 whichis known as hypoellipticity. This is not only a large area of research onto itself, butalso false in the generality of the Malgrange-Ehrenpreis theorem.

The proof follows the common principle in functional analysis that uniqueness(for the adjoint operator) implies existence (for the operator). Recall that this is theessence of Fredholm theory which establishes this fact for compact perturbationsof the identity operator. Of course, we cannot expect uniqueness in our case with-out any boundary conditions, so we introduce a strong vanishing condition at theboundary. Henceforth we take Ω to be a ball, which we may do since Ω bounded.By dilation and translation, the ball may furthermore be taken to be Bp0, 1q.

Proposition 9.11. One has the bound

ppDqϕ2 ¥ C1ϕ2

for all ϕ P C8pBp0, 1qq which are compactly supported in Bp0, 1q. The constant Cdepends on p and the dimension.

Proof of Theorem 9.10. We show how to deduce the theorem from the propo-sition. Let

X : ppDqϕ | ϕ P C8

comppBp0, 1qq(

viewed as a linear subspace of L2pBp0, 1qq. Define a linear functional ` on X by

`pppDqϕq pg, ϕqwhere the right-hand side is the L2-pairing. Proposition 9.11 implies that this iswell-defined and in fact one has a bound

|`pppDqϕq| ¤ g2ϕ2 ¤ Cg2ppDqϕ2

By the Hahn-Banach theorem we may extend ` as a bounded linear functional toL2pBp0, 1qqwith the same norm. So by the Riesz representation of such functionalsthere exists f P L2pBp0, 1qq with the property that

`pppDqϕq p f , ppDqϕq pg, ϕqfor all ϕ as above which proves the theorem.

The proof of Proposition 9.11 will be based on Theorem 9.7, which requiressome preparatory lemmas on polynomials.


uniformly in B. This follows form the previous inequality and the property thatthe highest order coefficients are invariant under translation. This concludes theproof.

Proof of Proposition 9.11. Lemma 9.13 guarantees that

F : tx P Rd | |ppxq| ¤ ε0uis Bp0, 1q-negligible. Thus, by Theorem 9.9,

ppDqϕ2 ppξqϕ2 ¥ pϕL2pFcq¥ ε0ϕL2pFcq ¥ ε1ϕL2pRdq

with some constant ε1 that depends on p and d.

There is no analogue of Theorem 9.10 for partial differential operators withnonconstant smooth coefficients. This is known as Lewy’s example, see for exam-ple [17].

Notes

A standard reference for the material in this chapter is Havin and Joricke’s book [24].Another basic resource in this area is Charles Fefferman’s survey [15]. An explicit constantin Theorem 9.4 of the form eC|E||F| in dimension 1 was obtained by Nazarov [38], and inhigher dimensions by Jaming [27]. For the solvability of linear constant coefficient PDEssee Jerison [29].

Problems

Problem 9.1. Does there exist a nonzero function f P L2pRdq so that both f 0 andf 0 on nonempty open sets?

Problem 9.2. Does there exist a nonzero f P L2pRq with f 0 on r1, 1s and f 0on a half-line? What about f 0 on a set of positive measure and f 0 on a half-line?

Problem 9.3. Suppose E, F Rd have finite measure. Show that for any g1, g2 PL2pRdq there exists f P L2pRdq with

f g1 on E, f g2 on F

This can be seen as the statement that f |E and f |F are independent.

Problem 9.4. Prove that for E, F Rd of finite measure one has

dimt f P L2pRdq | f 0 on E, f 0 on Fu 8Problem 9.5. There are several ways of proving Proposition 9.6. As an alternative to

the Schur lemma proof we gave, one can use Sobolev embedding on cubes together withBernstein’s theorem. This problem suggests yet another route.

Prove Poincare’s inequality: for any f P H1pRdq one has for every R ¡ 0»Bp0,Rq

| f pxq x f yR|2 dx ¤ CR2»

Bp0,Rq|∇ f pxq|2 dx

with an absolute constant. Here x f yR is the average of f on Bp0,Rq.


Combine this inequality with Bernstein’s estimate as in Lemma 9.1 to give anindependent proof of Proposition 9.6.

Problem 9.6. Show that f in Theorem 9.10 may be taken to be C8pΩq. Hint: UseSobolev spaces in the existence part in addition to L2.

Problem 9.7. Show Poincare’s inequality for general powers 1 ¤ p 8, i.e., verifythat for any f P SpRdq one has»

Bp0,Rq| f pxq x f yR|p dx ¤ Cpp, dqRp

»Bp0,Rq

|∇ f pxq|p dx

Deduce from this that 9W1,dpRdq ãÑ BMOpRdq, where f 9W1,dpRdq ∇ f LdpRdq.

CHAPTER 10

Littlewood-Paley theory

One of the central concerns of harmonic analysis is the study of multiplieroperators, i.e., operators T which are of the form

pTm f qpxq »Rd

e2πixξmpξq f pξq dξ (10.1)

where m : Rd Ñ C is bounded. By Plancherel’s theorem Tm2Ñ2 ¤ m8. Infact, it is easy to see that one has equality here. Over the distributions we may writeTm f K f for any f P SpRdq with K : m P S1. Then Tm is bounded on L1 ifand only if the associated kernel is in L1, in which case

Tm1Ñ1 K1

There are many cases, though, where K < L1pRdq but T is bounded on Lp forsome (or all) 1 p 8. The Hilbert transform is one such example. We shallnow discuss one of the basic results in the field, which describes a large class ofmultipliers m that give rise to Lp bounded operators for 1 p 8.

In Chapter 7 we previously studied the case where m is homogeneous of de-gree 0 and we showed that the Tm is a Calderon-Zygmund operator with homo-geneous kernel, and thus Lp-bounded for 1 p 8. In the following theorem,which goes by the name of Mikhlin multiplier theorem, we shall allow for moregeneral functions m which satisfy finitely many derivative conditions of the zero-degree type..

Theorem 10.1. Let m : Rdzt0u Ñ C satisfy, for any multi-index γ of length|γ| ¤ d 2

|Bγmpξq| ¤ B|ξ||γ|for all ξ , 0. Then for any 1 p 8 there is a constant C Cpd, pq so that

pm f q_p ¤ CB f p

for all f P S.

Proof. Let ψ give rise to a dyadic partition of unity as in Lemma 8.3. Definefor any j P Z

m jpξq ψp2 jξqmpξqand set K j m j. Now fix some large positive integer N and set

Kpxq N

jN

K jpxq

125

126 10. LITTLEWOOD-PALEY THEORY

We claim that under our smoothness assumption on m one has the pointwise esti-mates

|Kpxq| ¤ CB|x|d, |∇Kpxq| ¤ CB|x|d1 (10.2)where C Cpdq. One then applies the Calderon-Zygmund theorem and lets N Ñ8.

We will verify the first inequality in (10.2). The second one is similar, and willbe only sketched. By assumption, Dγm j8 ¤ CB2 j|γ| and thus

Dγm j1 ¤ CB2 j|γ|2 jd

for any multi-index |γ| ¤ d 2. Similarly,

Dγpξim jq1 ¤ CB2 jp|γ|1q2 jd

for the same γ. Hence,

xγ m jpxq8 ¤ CB2 jpd|γ|q

andxγDm jpxq8 ¤ CB2 jpd1|γ|q .

Since |x|k ¤ Cpk, dq°|γ|k |xγ| one concludes that

|m jpxq| ¤ CB2 jpdkq|x|k (10.3)

and|Dm jpxq| ¤ CB2 jpd1kq|x|k (10.4)

for any 0 ¤ k ¤ d 2 and all j P Z, x P Rdzt0u. We shall use this with k 0 andk d 2. Indeed,

|Kpxq| ¤¸

j

|m jpxq| ¤¸

2 j¤|x|1

|m jpxq| ¸

2 j¡|x|1

|m jpxq|

¤ CB¸

2 j¤|x|1

2 jd CB¸

2 j¡|x|1

2 jdp2 j|x|qpd2q

¤ CB|x|d CB|x|2|x|d2 CB|x|d ,

as claimed. To obtain the second inequality in (10.2) one uses (10.4) instead of(10.3). Otherwise the argument is unchanged. Thus we have verified that K sat-isfies the conditions i) and iii) of Definition 7.1, see Lemma 7.2. Furthermore,m8 ¤ B so that pm f q_2 ¤ B f 2. By Theorem 7.6, and the remark followingit, one concludes that

pm f q_p ¤ Cpp, dq f p

for all f P S and 1 p 8, as claimed.

The number of derivatives entering into the hypothesis can be decreased, seethe problem section. Theorem 10.1 allows one to give proofs of Corollaries 7.7without going through Exercise 7.4. Indeed, one hasB2u

BxiBx jpξq ξiξ j

|ξ|2 x4upξq

10. LITTLEWOOD-PALEY THEORY 127

Since mpξq ξiξ j

|ξ|2 satisfies the conditions of Theorem 10.1 (this is obvious, as mis homogeneous of degree 0 and smooth away from ξ 0q, we are done. Observethat this example also shows that Theorem 10.1 fails if p 1 or p 8.

Theorem 10.1 is formulated as an apriori inequality for f P SpRdq. In thatcase m f P S1 so that pm f q_ P S1. The question arises whether or not pm f q_ ismeaningful for f P LppRdq. If 1 p ¤ 2, then f P L2 L8pRdq (in fact f P Lp1

by Hausdorff-Young), so that m f P S1 and therefore pm f q_ P S1 is well-defined. Ifp ¡ 2, however, it is known that there are f P LppRdq such that f P S1 has positiveorder (see the Notes to Chapter 6). For such f it is in general not possible to definepm f q_ in S1.

We now present an important application of Mikhlin’s theorem to Littlewood-Paley theory. With ψ as in Lemma 8.3 we define P j f pψ j f q_ where ψ jpξq ψp2 jξq (it is customary to assume ψ ¥ 0). Then by Plancherel,

C1 f 22 ¤

¸jPZP j f 2

2 ¤ f 22 (10.5)

for any f P L2pRdq. Observe that the middle expression is equal to S f 22 with

S f ¸

j

|P j f |2 1

2

This is called the Littlewood-Paley square-function. It is a basic result of harmonicanalysis that (10.5) generalizes to

C1 f p ¤ S f p ¤ C f p (10.6)

for any f P LppRdq provided 1 p 8 and with C Cpp, dq. It is of course im-portant to understand why such a result might hold, and so we offer some heuristicexplanations.

Consider a lacunary trigonometric polynomial of the form

T pθq M

n1

cn ep2nπθq

where θ P T. Let p ¥ 1 be an integer and estimate by Plancherel’s theorem» 1

0

M

n1

cn ep2nθq2p

dθ » 1

0

M

n1,...,np1

cn1 cnp ep2nθq2 dθ

¤ CppqM

n1,...,np1

|cn1 cnp |2 ¸

n

|cn|2p

We used here that for every positive integer N the number of ways in which it canbe written in the form

N 2n1 2np , n j ¥ 0


is bounded by a constant that depends only on p. In other words, given a sequencetcku P `2pZq so that ck 0 unless |k| is of the form 2n with n ¥ 0, we have theproperty ¸

k

|ck|2 1

2 ¤¸

k

ck epkπθq

p¤

¸k

|ck|2 1

2(10.7)

for every p ¥ 2 (and by Holder’s inequality one can lower this to p ¥ 1, see theproof of Lemma 10.3 below). Clearly, (10.7) is remarkable as it says that any L2pTqfunction f so that f pkq 0 unless |k| is a power of 2 has finite Lp-norm for everyp 8 (but of course not for p 8). We remark in passing that this applies toany lacunary or gap Fourier series, but this is harder to see (the standard argumentinvolves Riesz products).

The rough idea behind (10.6) can be construed as follows: if we consider an ar-bitrary trigonometric polynomial

°n anepnθq, then (10.7) is of course false. How-

ever, we may try to salvage something of (10.7) by setting

ckpθq :¸

2k1 |n|¤2k

anepnθq, @ k ¥ 1

We might then expect that the square function

S f pθq :¸

k

|ckpθq|2 1

2

satisfies S f p f p.It is insightful to view lacunary characters such as tep2nθqun¥1 from a proba-

bilistic angle. Indeed, we can view these functions as being close to independentrandom variables on the interval r0, 1s. For example, on any interval rk2n, pk 1q2nq r0, 1q with integer k, on which sinp2nπθq has a fixed sign, the func-tion sinp2n1πθq is equally likely to be positive or negative. The reader shouldalso compare the graphs of Re ep2nθq with those of the Rademacher functions rndefined below.

Taking another leap of faith, we are thus lead to viewing lacunary series fromthe point of view of the central limit theorem in probability theory, which at leastheuristically implies that a lacunary Fourier series f as above with f 2 1 satis-fies tθ P T | | f pθq| ¡ λu ¤ Ce

λ22

with a suitable constant C ¡ 0. In particular, this ensures that f satisfies an estimateof the type (10.7).

It should therefore come as no surprise to the reader that some probabilisticelements enter into the proof of (10.6). In fact, we shall now derive (10.6) by meansof a standard randomization technique. Let tr ju be a sequence of independentrandom variables with Prr j 1s Prr j 1s 1

2 , for all j (in other words,the r j are a coin tossing sequence). Readers familiar with the central limit theoremwill not be surprised by the following sub-Gaussian bound.


Lemma 10.2. For any positive integer N and ta juNj1 C one has

P N

j1

r ja j ¡ λ

N

j1

|a j|2 1

2¤ 4e

λ22 (10.8)

for all λ ¡ 0.

Proof. Assume first that a j P R. Then

E etS N ΠNjiE

et r ja j

ΠNj1 coshpta jq

where we have set S N °N

j1 r ja j. Now invoke the calculus fact

cosh x ¤ ex22 @ x P Rto conclude that

E etS N ¤ ΠNj1et2a2

j2 expt2

N

j1

a2j2

Hence, with σ2 °N

j1 a2j ,

PS N ¡ λσ

¤ et2σ22eλtσ ¤ eλ22

where the final inequality follows by minimizing in t, i.e., by setting t λσ . Simi-

larly,

PrS N λσs ¤ eλ22

so thatPr|S N | ¡ λσs ¤ 2eλ

22

The case of a j P C follows by means of a decomposition into real and imaginaryparts.

This sub-Gaussian estimate on the so-called “tails” of the distribution of therandom sum S N : °N

j1 r ja j imply that all expressions pE |S N |pq1p are compara-

ble. This fact is known as Khinchin’s inequality.

Lemma 10.3. For any 1 ¤ p 8 there exists a constant C Cppq so that

C1 N

j1

|a j|2 p

2 ¤ E N

j1

r ja j

p ¤ C N

j1

|a j|2 p

2(10.9)

for any choice of positive integer N and ta juNj1 P C.

Proof. We start with the upper bound in (10.9). It suffices to consider the case°Nj1 |a j|2 1. Setting

°Nj1 a jr j S N one has

E |S N |p » 8

0Pr|S N | ¡ λspλp1 dλ ¤

» 8

04eλ

22 pλp1 dλ : Cppq 8


For the lower bound it suffices to assume that 1 ¤ p ¤ 2, in fact, p 1. ByHolder’s inequality,

E |S N |2 E |S n|23|S N |43 ¤ pE |S N |q23pE |S n|4q13

¤ CpE |S N |q23pE |S N |2q43where the final inequality follows from the case 2 ¤ p 8 just considered. Thisimplies that

E |S N |2 ¤ CpE |S N |q2and we are done.

Khinchin’s inequality is often formulated for the Rademacher functions, whichare a concrete realization of the sequence tr ju on the interval r0, 1s given byr jpxq signpsinpπ2 jxqq for j ¥ 0. The explicit form of the Rademacher functionsallows for a different proof of Khinchin’s inequality, which are based on expand-ing and integrating out E |S N |p when p is an even integer. We invite the reader todetermine the growth (at least an upper bound on the growth) of the constant Cppqin Khinchin’s inequality as p Ñ8.

Exercise 10.1.

Draw the graphs of the first few Rademacher functions and explain wheythey are a realization of independent identically distributed random vari-ables as those appearing above (coin tossing sequence).

By explicit expansion and integration for even integer p, prove that» 1

0

N

j1

a j r jptqp dt ¤ Cppq

N

j1

|a j|2 p

2

Then recover the general case of 1 ¤ p 8.

We are now ready to prove the Littlewood-Paley theorem (10.6).

Theorem 10.4 (Littlewood-Paley). For any 1 p 8 there is a constantC Cpp, dq such that

C1 f p ¤ S f p ¤ C f p

for any f P S.

Proof. Let tr ju be as above. The proof rests on the fact that

mpξq :N

jN

r jψ jpξq

satisfies the conditions of the Mikhlin multiplier theorem uniformly in N and uni-formly in the realization of the random variables tr ju. Indeed, for any γ,

|Dγmpξq| ¤N

jN

|Dγψ jpξq| ¤ CN

jN

|ξ|γ|pDγψqp2 jξq| ¤ C|ξ|γ


To pass to the final inequality one uses that only an absolutely bounded numberof terms is non-zero in the sum preceding it for any ξ , 0. Hence, in view ofLemma 10.3,»

Rd

pS f qpxqp dx ¤ C lim sup

NÑ8E

»Rd

N

jN

r jpP j f qpxqp dx ¤ C f p

p

as desired.To prove the lower bound we use duality: choose a function ψ so that ψ 1 on

supp pψq and ψ is compactly supported with supp ψ Rdzt0u. Defining P j like P jwith ψ instead of ψ yields tP ju satisfying P jP j P j. Therefore, for any f , g P S,and any 1 p 8,

|x f , gy| |¸

j

xP j f , P jgy| ¤»Rd

¸j

|pP j f qpxq|2 1

2¸

k

|pPkgqpxq|2 1

2dx

¤ S f pS gp1 ¤ CS f pgp1

For the final bound we use that the argument for the upper bound equally wellapplies to S instead of S . Thus, f p ¤ CS f p, as claimed.

It is desirable to formulate Theorem 10.4 on LppRdq rather than on SpRdq.This is done in the following corollary. Observe that S f is defined pointwise iff P S1pRdq since P j f is a smooth function.

Corollary 10.5.

a) Let 1 p 8. Then for any f P LppRdq one has S f P Lp and

C1p,d f p ¤ S f p ¤ Cp,d f p

b) Suppose that f P S1 and that S f P LppRdq with some 1 p 8. Thenf g P where P is a polynomial and g P LppRdq. Moreover, S f S gand

C1p,dS f p ¤ gp ¤ Cp,dS f p

Proof. To prove part aq, let fk P S so that fk f p Ñ 0 as k Ñ8. We claimthat

limkÑ8

S fk S f p 0 (10.10)

If (10.10) holds, then passing to the limit k Ñ8 in

C1p,d fkp ¤ S p fkqp ¤ Cp,d fkp

implies part aq. To prove (10.10) one applies Fatou repeatedly. Fix x P Rd. Then

|S fkpxq S f pxq| |tP j fkpxqu8k8`2 tP j f pxqu8j8`2 |¤ tP j fkpxq P j f pxqu8j8`2

S p fk f qpxq ¤ lim infmÑ8 S p fk fmqpxq


Therefore,

S fk S f p ¤ lim infmÑ8 S p fk fmqp ¤ Cp,d lim inf

mÑ8 fk fmp

The claim (10.10) now follows by letting k Ñ8.For the second part we argue as in the proof of the lower bound in Theo-

rem 10.4: Let f P S1 and h P S with suppphq Rdzt0u. Then

|x f , hy| ¤ CpS f pS hp1 ¤ CpS f php1

where S is the modified square function from the proof of Theorem 10.4. By theHahn-Banach theorem there exists g P Lp so that x f , hy xg, hy for all h as abovesatisfying gp ¤ CpS f p. By Exercise (6.2) one has f g P. Moreover,S f S g and

S f p S gp ¤ Cgp

by part i).

The following exercise shows that the Littlewood-Paley theorem fails at theendpoints p 1 and p 8.

Exercise 10.2.

a) Show that Theorem 10.4 fails at p 1. The intuition is of course to takef δ0. For that case verify that that

pS f qpxq |x|d

so that S f < L1pRdq. Next, transfer this logic to L1pRdq by means ofapproximate identities.

b) Show that Theorem 10.4 fails for p 8.

It is natural to ask at this point whether the Littlewood-Paley theorem holdsfor square functions defined in terms of sharp cut-offs rather than smooth ones asabove. More precisely, suppose we set

pS new f q2 ¸jPZ|pχr2 j1¤|ξ| 2 js f pξqq_|2

Is it true thatC1

p f p ¤ S new f p ¤ Cp f p (10.11)

for 1 p 8? Due the boundedness of the Hilbert transform the answer is “yes”in dimension 1, but it is rather deep fact that it is “no” in dimensions d ¥ 2, see thenotes and problems to this chapter.

As a first application of (10.6) we now finish the proof of (6.8).

Proposition 10.6. For any 0 ¤ s d and with 12 1

p sd one has

f LppRdq ¤ Cps, dq f 9HspRdq

for all f P SpRdq.


Proof. In Chapter 6 we already settled the case where f is localized to a dyadicshell t|ξ| 2 ju. Thus, it remains to prove (6.11). In fact, we shall now prove thatfor any finite p ¥ 2

f p ¤ C¸

jPZP j f 2

p

12

with a constant that only depends on p and d. By (10.6)

f p ¤ Cpp, dq S f p Cpp, dq tP j f u jPZ`2pZq p

¤ Cpp, dqtP j f pu jPZ`2pZq

where we used the triangle inequality in the final step (see the following exercise).

Exercise 10.3.

Show that for any 8 ¥ p ¥ q ¥ 1 and on any measure space pX, µq, t f ju jPZ`qpZq

Lppµq ¤ t f jLppµqu jPZ

`qpZq

Also show that for 8 ¥ q ¥ p ¥ 1 one has the reverse inequality. Show that (6.11) fails for every 1 ¤ p 2.

As evidenced by the proof of Proposition 10.6, Exercise 10.3 is a useful (andsharp) tool when combined with (10.6). In particular, it is used in comparingSobolev with Besov spaces, see the problem section below.

As final general comment about Littlewood-Paley theory, we ask the reader toverify that there can be no version of 10.6 in which the square function is replacedby some `q-based norm.

Exercise 10.4. Suppose that for some 1 ¤ q ¤ 8 and 1 ¤ p ¤ 8 one has tP j f u jPZ`qpZq

LppRdq f p @ f P SpRdqwith constants that do not depend on f . Show that q 2.

We now present a connection between BMO and Littlewood-Paley theory. Thiswill give us the opportunity to introduce some notions that historically played animportant role in the development of the theory, such as Carleson measure and theCalderon reproducing formula, which can be seen as a continuum version of thediscrete Littlewood-Paley decomposition.

Definition 10.7. Let µ be a (complex) measure on the upper half-space Rd1 .

Denote a general dyadic cube in Rd by I and the cube in Rd1 with base I and

of height `pIq where ` is the side-length of I, by QpIq. In other words, QpIq :tpx, tq | x P I, 0 t `pIqu. One says that µ is a Carleson measure if

µC : supI

|µ|pQpIqq|I| 8

The following result may seem unmotivated for now, but it is a standard deviceby which a BMO function can be broken up into basic constituents.


Proposition 10.8. Let f P BMOpRdq with f BMO ¤ 1 have compact supportand mean zero. There exist functions tbIuI indexed by the dyadic cubes in Rd andscalars λI such that

f °I λI bI

supppbIq 3I ³Rd bIpxq dx 0

∇bI8 ¤ `pIq1

°IJ |λI|2|I| ¤ |J| for all dyadic cubes J

The final property means that°

I |λI|2|I|δpxI ,`pIqq is a Carleson measure where xIis the center of I.

Proof. Let ϕ P SpRdq be real-valued, radial, and such that supppϕq Bp0, 1qand with »

Rdϕptξq2 dt

t 1 @ ξ , 0 (10.12)

Note that the left-hand side is a radial function and by scaling does not even dependon |ξ|. In other words, it does not depend on the choice of ξ , 0 at all. The onlyrequirement is that ϕp0q 0 or equivalently,

³ϕ 0. Moreover, any ϕ 0 which

satisfies this condition can be normalized so as to fulfil (10.12).Clearly, (10.12) is a continuum analogue of the discrete partition of unity in

Lemma 8.3. The Calderon reproducing formula is now the following statement,which is implied by (10.12) and the vanishing

³f 0:

f pxq » 8

0pϕt ϕt f qpxq dt

t

¸

I

»TpIq

ϕtpx yqpϕt f qpyq dtt

dy ¸

I

bIpxq(10.13)

where T pIq : tpx, tq | x P I, t P p`pIq2, `pIqqu and ϕtpxq tdϕpxtq. Byconstruction,

³bI 0 and supppbIq 3I. Moreover, if we define with a suitable

constant C

λI : C|I|1

»TpIq

|ϕt f pyq|2t1 dtdy 1

2

then one checks by means of Cauchy-Schwarz that

|∇bIpxq| »

TpIq∇ϕtpx yqpϕt f qpyq dt

tdy ¤ λI `pIq1

Define bI : λ1I bI . We have verified all properties up to the Carleson measure. To

obtain this final claim, we first show that for any f P BMOpRdq and any (dyadic)cube I Rd »

QpIq|pϕt f qpyq|2 t1 dtdy ¤ C f 2

BMO|I| (10.14)


By the scaling and translation symmetries, we may assume that I is the cuber1, 1sd. Define I : r2, 2sd and split

f fI p f fIqχI p f fIqχRdzI : f1 f2 f3

The constant f1 does not contribute to (10.14) and f2 is estimated by»Rd1

|pϕt f2qpyq|2 t1 dtdy f222 ¤ C f 2

BMO

Finally, »QpIq

|pϕt f3qpxq|2 t1 dtdx

¤» 2

0

dtt

»p1,1qd

dx »Rdzp2,2qd

p f pyq fIq tdϕ x y

t

dy2

¤ C» 2

0

dtt

»p1,1qd

dx»Rdzp2,2qd

| f pyq fI |2 td|ϕ x yt

| dy

¤ C»Rd

| f pyq fI |21 |y|d1 dy ¤ C f 2

BMO

as desired. For the final step, see Problem 7.5. Hence (10.14) holds and therefore¸JI

|λJ|2|J| ¸JI

»TpJq

|pϕt f qpxq|2 t1 dtdx

C»

QpIq|pϕt f qpxq|2 t1 dtdx ¤ C f 2

BMO|I|

and we are done.

Exercise 10.5. Give a detailed proof of (10.13).

To conclude this chapter, we shall show that singular integrals as in Defini-tion 7.1 are bounded on CαpRdq, 0 α 1. Problem 7.7 of the previous chapterdeals with the Cα-boundedness of the Hilbert transform, which can be proved onthe level of the kernel alone. Here we shall follow a different route based on theprojections P j. We begin with a characterization of CαpRdq for 0 α 1 in termsof the projections P j. The reader should realize that the proof of the followinglemma is reminiscent of the proof of Bernstein’s Theorem 1.12.

Lemma 10.9. Let | f | ¤ 1. Then f P CαpRdq for 0 α 1 if and only if

supjPZ

2 jαP j f 8 ¤ A (10.15)

Moreover, the smallest A for which (10.15) holds is comparable to r f sα.

Proof. Set ψ jpxq 2 jnψp2 jxq : ϕ jpxq. Hence ϕ j1 ψ1 for all j P Z and»Rdϕ jpxqxγ dx 0


for all multi-indices γ. First assume f P Cα. Then

|P j f pxq| ¤»Rd| f px yq f pxq| |ϕ jpyq| dy

¤»Rdr f sα|y|α|ϕ jpyq| dy 2 jαr f sα

»Rd|y|α|ψpyq| dy

Hence A ¤ Cr f sα, as claimed. Conversely, define

g`pxq ¸

`¤ j¤`pP j f qpxq

for any integer ` ¥ 0. We need to show that, for all y P Rd,

sup`|g`px yq g`pxq| ¤ CA|y|α

with some constant C Cpdq. Fix y , 0 and estimate ¸|y|1 2 j¤2`

pP j f qpxq ¤ ¸

2 j¡|y|1

A2 jα ¤ CA|y|α (10.16)

Secondly, observe that

|P j f px yq P j f pxq| ¤ ∇P j f 8|y| ¤ C2 jP j f 8|y|¤ C2 jp1αqA|y|

(10.17)

where we invoked Bernstein’s inequality to pass to the second inequality sign.Combining (10.16) and (10.17) yields

|g`px yq g`pxq|¤

¸2`¤2 j¤|y|1

|P j f px yq P j f pxq| ¸

|y|1 2 j¤2`2P j f 8

¤¸

2 j¤|y|1

CA2 jp1αq|y| ¸

|y|1 2 j

2A2 jα ¤ CA|y|α(10.18)

uniformly in ` ¥ 1.So tg` g`p0qu8`1 are uniformly bounded on CαpKq for any compact K. By

the Arzela-Ascoli theorem one concludes that

g` g`p0q Ñ g

uniformly on any compact set and therefore rgsα ¤ CA by (10.18) (up to passingto a subsequence t`iu). It remains to show that f has the same property. This willfollow from the following claim: f g const. To verify this property, note thatg` g`p0q Ñ g in S1. Thus, also g` δ0g`p0q Ñ g in S1 which is the same as¸

`¤ j¤`ψp2 jξq f pξq δ0g`p0q Ñ g in S1


So if h P Swith suppphq Rdzt0u, then x fg, hy 0, i.e., we have suppp fgq t0u. By Exercise 6.2 therefore

p f gqpxq ¸

|γ|¤M

Cγxγ (10.19)

On the other hand, since gp0q 0,

|p f gqpxq| ¤ f 8 |gpxq gp0q| ¤ 1CA|x|αSince α 1, comparing this bound with (10.19) shows that the polynomial in(10.19) is of degree zero. Thus, f g const, as claimed.

Next, we need the following result which gives an example of a situation whereT maps into L1pRdq. We postpone the proof.

Proposition 10.10. Let K be a singular integral kernel as in Definition 7.1. Forany η P SpRdq with

³Rd ηpxq dx 0 one has

Tη1 ¤ CpηqBwith a constant depending on η.

We can now formulate and prove the Cα bound for singular integrals.

Theorem 10.11. Let K be as in Definition 7.1 and 0 α 1. Then for anyf P L2XCαpRdq one has T f P CαpRdq and

T f

α¤ Cα Br f sα with Cα Cpα, dq.

Proof. We use Lemma 10.9. In order to do so, notice first that

T f 8 ¤ CBpr f sα f 2qwhich we leave to the reader to check. Therefore, it suffices to show that

supj

2 jαP jT f 8 ¤ CBr f sα (10.20)

Let P j be defined asP ju

ψp2 jqupξq_

where ψ P C80 pRdzt0uq and ψψ ψ. Thus, P jP j P j. Hence,

P jT f 8 P jP jT f 8 P jT P j f 8¤ P jT8Ñ8P j f 8¤ P jT8Ñ8Cr f sα2 jα

It remains to show that sup j P jT8Ñ8 ¤ CB, see (10.20). Clearly, the kernel of

P jT is 2 jd rψp2 jq K so that

P jT8Ñ8 ¤ 2 jd rψp2 jq K1 rψ 2 jdKp2 jq1 ¤ CB (10.21)

by Proposition 10.10. Indeed, set η rψ in that proposition so that»Rdηpxq dx rψp0q 0


as required. Furthermore, we apply Proposition 10.10 with the rescaled kernel2 jdKp2 jq. As this kernel satisfies the conditions in Definition 7.1 uniformly inj, one obtains (10.21) and the theorem is proved.

It remains to show Proposition 10.10. We will build up from the case of an“atom” as defined by the following lemma.

Lemma 10.12. Let f P L8pRdq with³

f pxq dx 0, suppp f q Bp0,Rq and f 8 ¤ Rd. Then

T f 1 ¤ CB

Proof. By Cauchy-Schwarz,»|x|¤2R

pT f qpxq dx ¤ T f 2CRd2 ¤ CB f 2R

d2

¤ CBRdRd2 R

d2 CB

Furthermore,»|x|¡2R

|pT f qpxq| dx ¤»Rd

»|x|¡2|y|

|Kpx yq Kpxq| dx| f pyq| dy ¤ B f 1 ¤ CB

as desired.

In passing, we point out a simple relation between these atoms and BMO. Infact, let ϕ P BMO and a f as defined in Lemma 10.12. Then

|xa, ϕy| »

Bapxqpϕpxq ϕBq dx

¤ a8»

B|ϕpxq ϕB| dx

¤?

B|ϕpxq ϕB| dx ¤ ϕBMO

Moreover, it is easy to see from this that supa |xa, ϕy| ϕBMO. It is there-fore natural to define the atomic space H1

atompRdq as all sums f °j c j a j with°

j |c j| 8 and to define the norm f H1 as the smallest possible value of the sum°j |c j| for a given f .

It is a remarkable fact that this atomic space coincides with H1 as definedin (7.32). We will establish this in a later chapter in a one-dimensional dyadicsetting. In the following lemma we show that one can write a general Schwartzfunction with mean zero as a superposition of infinitely many atoms.

Lemma 10.13. Let η P SpRnq, ³Rd ηpxq dx 0. Then one can write

η 8

`1

c`a`

with³Rn a`pxq dx 0, a`8 ¤ `d, supppa`q Bp0, `q for all ` ¥ 1 and

8

`1

|c`| ¤ Cpηq


with a constant Cpηq depending on η.

Proof. In this proof, we let

xgyS : 1|S |

»S

gpxq dx

for any g P L1pRdq and S Rd with 0 |S | 8. Moreover, B` : Bp0, `q for` ¥ 1, and χ` χB` (indicator of B`q. Define

f1 : pη xηyB1qχ1 , η1 : η f1

and set inductively "f`1 : pη` xη`yB`1q χ`1 andη`1 : η` f`1

(10.22)

for ` ¥ 1 (one can take this also with ` 0 and η0 : η). Observe that

η η1 f1 f1 f2 η2 . . . M

`1

f` ηM1 (10.23)

We need to show that we can pass to the limit M Ñ8 and that

a` : f` `d

f`8 for ` ¥ 1 (10.24)

have the desired properties. By construction, for all ` ¥ 1,»Rd

a`pxq dx 0 , supppa`q B`

and a`8 ¤ `d. It remains to show that

c` : `d f`8 (10.25)

satisfies°8`1 c` 8 and that ηM18 Ñ 0, see (10.23). Clearly,

η1 " xηyB1 on B1η on RdzB1

Hence,

η2 " xη1yB2 on B2η1 η on RdzB2

By induction, for ` ¥ 1, one checks that

η`1 " xη`yB`1 on B`1η on RdzB`1.

(10.26)

Moreover, induction shows that »Rdη`pxq dx 0 (10.27)

for all ` ¥ 0. Indeed, this is assumed for η0 η. Since³

f`pxq dx 0 for ` ¥ 1by construction, one now proceeds inductively via the formula

η`1 η` f`1


Property (10.27) implies thatxη`yB`1

¤ 1|B`1|

»RdzB`1

|η`pxq| dx

¤ 1|B`1|

»RdzB`1

|ηpxq| dx ¤ C`20d(10.28)

since η` η on RdzB`, see (10.26), and since η has rapid decay. (10.28) impliesthat

η`18 ¤ C`20d,

see (10.26), and also f`18 ¤ C`20d

see (10.22). We now conclude from (10.23)–(10.25) that

η 8

`1

f` 8

`1

cà`

with, see (10.25) and (10.24),8

`1

|c`| 8

`1

`d f`8 ¤8

`1

C`19d 8

and the lemma follows.

Proof of Proposition 8. Let η be as in the statement of the proposition. ByLemma 10.13,

η 8

`1

cà`

with c` and a` as stated there. By Lemma 10.12,

T pcà`q1 ¤ |c`|Ta`1 ¤ CB|c`|for ` ¥ 1. Hence,

8

`1

T pcà`q1 ¤ CηB

and this easily implies that Tη1 ¤ CηB, is claimed.

A typical application of Theorem 10.11 is to the so-called “Schauder estimate”.

Corollary 10.14. Let f P C2,α0 pRdq. Then

sup1¤i, j¤d

B2 fBxiBx j

α

¤ Cpα, dqr4 f sα (10.29)

for any 0 α 1.


Proof. As in the Lp case, see Corollary 7.7, this follows from the fact that

B2 fBxiBx j

Ri jp4 f q for 1 ¤ i, j ¤ d

where Ri j are the double Riesz transforms. Now apply Theorem 10.11 to the sin-gular integral operators Ri j.

For an extension of this corollary to variable coefficients, see Problem 10.6below.

Exercise 10.6. Show that Theorem 10.11 fails at α 0 and α 1.

As a final application of Lemma 10.9 we remark that it easily implies Sobolev’simbedding 9W1,ppRdq ãÑ CαpRdq in the range d p 8 with α 1 d

p . SeeProblem 10.12.

Notes

The Littlewood-Paley theorem can be proved in a number of different ways, see for ex-ample Stein [47], which also contains a martingale difference version of this theorem. Formany applications of Littlewood-Paley theory (such as to Besov spaces) and connectionswith wavelet theory, see Frazier, Jawerth and Weiss [19].

Concerning (10.11), it is a relatively easy consequence of the Lp-boundedness of theHilbert transform that the answer is “yes” in one dimension. On the other hand, it is a veryremarkable result of Charles Fefferman that the answer is “no” for dimensions d ¥ 2. Infact, Fefferman [14] showed that the ball multiplier χB where B is any ball in Rd is boundedon LppRdq only for p 2. This latter result is based on the existence of Kakeya sets andwill shall return to it in a later chapter. The positive answer for d 1 is presented asProblem 10.4 below.

The “atoms” of Lemma 10.12 are exactly what one calls an H1-atom where H1pRdqis the real-variable Hardy space. Lemma 10.13 is an instance of an atomic decompositionwhich exists for any f P H1, see Stein [46] and Koosis [34].

Proposition 10.8 is from [55].

Problems

Problem 10.1. The conditions in Theorem 10.1 can be relaxed. Indeed, it suffices toassume the following:

supR¡0

R2|α|?

R |ξ| 2R|Bαξ mpξq|2 dξ ¤ A2 8

for all |α| ¤ k where k is the smallest integer ¡ d2 .

Hint: Verify the Hormander condition iiq of Definition 7.1 directly rather than goingthrough Lemma 7.2.

Problem 10.2. Let T by a bounded linear operator on LppX, µq where pX, µq is somemeasure space and 1 ¤ p 8. Show the following vector-valued Lp estimate:tT f ju`2

p ¤ CppqTpÑp

t f ju`2

p


for any sequence t f ju of measurable functions for which the right-hand side is finite. Suchvector-valued extensions are very useful. A challenging question is how to extend thisproperty to sub-linear operators such as the Hardy-Littlewood maximal function, see [46],page 51.

Problem 10.3. Let 1 ¤ p 2. By means of Khintchin’s inequality show that forevery ε ¡ 0 there exists f P SpRdq such that f p1 ¤ ε f p, cf. Corollary 6.8.

Problem 10.4. In this problem the dimension is d 1.

a) Deduce (10.11) from Theorem 10.4 by means of the following “vector-valued”inequality: Let tI ju jPZ be an arbitrary collection of intervals. Then for any 1 p 8

pχI j f jq_(

j

Lpp`2q

:¸

j

pχI j f jq_2 1

2

p

¤ Cpt f juLpp`2q

(10.30)

where t f ju jPZ are an arbitrary collection of functions in SpR1q, say.b) Now deduce (10.30) from Theorem 4.8 by means of Khinchin’s inequality Hint:

express the operator f ÞÑ pχI f q_ by means of the Hilbert transform via the sameprocedure which was used in the proof of Theorem 4.11.

c) By similar means prove the following Littlewood-Paley theorem for functions inLppTq: For any f P L1pTq let

S f 8

j0

|P j f |2 1

2

wherepP j f qpθq

¸2 j1¤|n| 2 j

f pnqepnθq

for j ¥ 1 and 40 f pθq f p0q. Show that, for any 1 p 8C1

p f LppTq ¤ S f LppTq ¤ Cp f LppTq (10.31)

for all f P LppTq.d) As a consequence of (10.11) with d 1 show the following multiplier theorem:

Let m : Rzt0u Ñ C have the property that, for each j P Zmpξq m j constant

for all 2 j1 ¤ |ξ| 2 j. Then, for any 1 p 8pm f q_LppRq ¤ Cp sup

jPZ|m j| f p

for all f P SpRq. Prove a similar theorem for LppTq using (10.31).

Problem 10.5. Prove the following stronger version of Problem 7.4. By a dyadicinterval we mean an interval of the form r2k, 2k1q where k P Z. Suppose m is a boundedfunction on R which is locally of bounded variation. In fact, assume that for some finite Bone has m8 ¤ B and »

I|dm| ¤ B


for all dyadic intervals I. Show that m is bounded on LppRq as a Fourier multiplier forany 1 p 8 with pm f q_p ¤ Cppq B f p. For an extension to higher dimensionssee [45].

Problem 10.6. We extend Corollary 10.14 to estimates for elliptic equations on a re-gion Ω Rd with variable coefficients, i.e.,

n

i, j1

ai jpxq B2uBxiBx j

pxq f pxq in Ω

where°

i, j ai jξiξ j ¥ λ|ξ|2 and ai j P CαpΩq. By “freezing x”, prove the following apriori

estimate from (10.29) for any f P C2,αpΩq:

sup1¤i, j¤d

B2 fBxiBx j

CαpKq

¤ Cpα, d,Kqr f sCαpΩq f L8pΩq

for any compact K Ω. Hint: See Gilbarg-Trudinger [22] for this estimate and muchmore.

Problem 10.7. Under the same assumptions as in Theorem 10.1 one has

rpm f q_sα ¤ CαB r f sαfor any 0 α 1 and f P CαpRdq X L2pRdq. Hint: This is a corollary of the proof ofTheorem 10.1. Analyze which properties of K it establishes and what is needed in orderfor the proof of Theorem 10.11 to go through.

Problem 10.8. For this problem we need to recall the definition of a martingale differ-ence sequence: let tFnu8n1 be an increasing sequence of σ-algebras on some probabilityspace pΩ,Pq (this is called a filtration). Assume that the function Xn : Ω Ñ R is mea-surable with respect to Fn and E rXn|Fn1s 0. Then tXnu8n1 is called a martingaledifference sequence. Suppose tXnu8n1 is such a martingale difference sequence adopted tosome filtration tFnu8n1. Show that

P N

n1

Xn ¡ λ

N

n1

Xn28

12

¤ Cecλ2

for any N 1, 2, . . ., and λ ¡ 0. Here C, c ¡ 0 are absolute constants.

Problem 10.9. Let tr ju81 be a coin tossing sequence as before. Show that for anyta ju P C, and N P Z

P

sup0¤θ¤1

N

j1

r j a je2πi jθ ¡ C0

N

j1

|a j|2 1

2a

log N¤ C0N2

provided C0 is a sufficiently large absolute constant.

Problem 10.10. Using the ideas of this chapter as well as the proof strategy of The-orem 8.6, try to prove (8.10) (but with an arbitrary multiplicative constant). For which pdoes this approach succeed?

Problem 10.11. This problem introduces Sobolev and Besov spaces and studies someembeddings. We remark that the spaces with dots are called “homogeneous”, whereasthose without are called “inhomogeneous”.


Let 1 ¤ p 8 and s ¥ 0. Define the Sobolev spaces 9W s,ppRdq and W s,ppRdqas completions of SpRdq under the norms

p∆q s2 f p respectively x∆y s

2 f p

where xay ?1 a2. The operators here are interpreted as Fourier multipliers.

Show that for any 1 p 8

f 9W s,p

¸jPZ

22 js|P j f |2 1

2

p

with implicit constants that only depend on s, p, d. Formulate and prove ananalogous statement for W s,ppRdq. Does anything change for s 0? Note thats 0 naturally arises by the duality relation p 9W s,pq 9Ws,p. For which s isthis valid? What is the analogue for the inhomogeneous spaces?

Define the Besov spaces 9Bsp,qpRdq and Bs

p,qpRdq by means of the norms (for qfinite) ¸

jPZ

2q jsP j f qp

1q

respectively¸

j¥0

2q jsP j f qp

1q

where in the second sum P0 is interpreted as the projection onto all frequencies¤ C 1. By means of the results of this chapter obtain embedding theoremsbetween Besov and Sobolev spaces.

Prove the embedding 9Bσq,2pRdq ãÑ LppRdq for 2 ¤ q ¤ p 8, σ ¥ 0, and1q 1

p σd . Does the same hold for the inhomogeneous spaces? What if

anything can one change in that case? By means of the previous item, deduceembedding theorems for the Sobolev spaces from this.

Problem 10.12. Prove that r f sα ¤ Cpα, dq f 9W1,ppRdq with d p 8, α 1 d

p

and f P SpRdq. This is called Morrey’s estimate. For the case p d see Problem 9.7.

Problem 10.13. Show that 9Bd22,1pRdq ãÑ L8pRdq but 9B

d22,qpRdq 6ãÑ L8pRdq for any 1

q ¤ 8. Show the embedding 9Bd22,8pRdq ãÑ BMOpRdq. Conclude that if f

9Bd2

2,qpRdq¤ A for

some 1 ¤ q 8, then w : e f satisfies?Q

wpxq dx?

Qw1pxq dx ¤ CpA, d, qq

uniformly for all cubes Q Rd.

CHAPTER 11

Fourier Restriction and Applications

In the late 1960’s, Elias M. Stein posed the following question: is it possible torestrict the Fourier transform f of a function f P pRdqwith 1 ¤ p ¤ 2 to the sphereS d1 as a function in LqpS d1q for some 1 ¤ q ¤ 8? By the uniform boundednessprinciple this is the same as asking about the validity of the inequality

f S d1LqpS d1q ¤ C f LppRdq (11.1)

for all f P SpRdq, with a constant C Cpd, p, qq? As an example, take p 1,q 8 and C 1. On the other hand, p 2 is impossible, as f is no betterthan a general L2-function by Plancherel. Stein asked whether it is possible to find1 p 2 so that for some finite q one has the estimate (11.1).

We have encountered an estimate of this type in Chapter 6, under the name oftrace estimate. In fact, by Lemma 6.11 one has

f S d18 ¤ C xxyσ f 2

as long as σ ¡ 12 (some work is required to pass from the planar case to the curved

one). However, (11.1) is translation invariant and therefore much more useful forapplications to nonlinear partial differential equations. In addition, as we shall see,the answers to (11.1) crucially involve curvature, whereas the trace lemmas do notdistinguish between flat and curved surfaces.

Exercise 11.1. Suppose S is a bounded subset of a hyper-plane in Rd. Provethat if f S 1 ¤ C f LppRdq for all f P SpRdq, then necessarily p 1. In otherwords, there can be no nontrivial restriction theorem for flat surfaces.

The following Tomas-Stein theorem settles the important case q 2 of Stein’squestion. For simplicity, we state it for the sphere. But it equally well applies toany bounded subset of any smooth hyper-surface inRd with nonvanishing Gaussiancurvature.

Theorem 11.1. For every dimension d ¥ 2 there is a constant Cpdq such thatfor all f P LppRdq

f S d1L2pS d1q ¤ Cpdq f LppRdq (11.2)

with p ¤ pd : 2d2d3 . Moreover, this bound fails for p ¡ pd.

The left-hand side in (11.2) is »S d1

| f pwq|2 σpdwq 1

2

145

146 11. FOURIER RESTRICTION AND APPLICATIONS

where σ is the surface measure on S d1.We shall prove this theorem below, after making some initial remarks. Firstly,

there is nothing special about the sphere. In fact, if S 0 is a compact subset of ahypersurface S with nonvanishing Gaussian curvature, then

f S 0L2pS 0q ¤ Cpd, S 0q f L

2d2d3 pRdq

(11.3)

for any f P SpRdq. For example, take the truncated paraboloid

S 0 : tpξ1, |ξ1|2q | ξ1 P Rd1, |ξ1| ¤ 1uwhich is important for the Schrodinger equation. On the other hand, (11.3) failsfor

S 0 : tpξ1, |ξ1|q | ξ1 P Rd1 , 1 ¤ |ξ1| ¤ 2usince this piece of the cone has exactly one vanishing principal curvature, namelythe one along a generator of the one. This latter example is relevant for the waveequation as the characteristic surface of the wave equation is a cone, and so weshall need to find a substitute of (11.3) for the wave equation. A much simplerremark concerns the range 1 ¤ p ¤ pd in Theorem 11.1: for p 1, one has

f S d1L2pS d1q ¤ f S d18|S d1| 12 ¤ f L1pRdq|S d1| 1

2 . (11.4)

Hence, it suffices to prove Theorem 11.1 for p pd since the cases 1 ¤ p pdfollows by interpolation with (11.4). The Stein-Tomas theorem is more accessiblethan the general restriction problem with q ¡ 2; in fact, we shall see that theappearance of L2pS d1q allows one to use duality in the proof. To do so, we needto identify the adjoint of the restriction operator

R : f ÞÑ f S d1

Lemma 11.2. For any finite measure µ in Rd, and any f , g P SpRdq one has theidentity »

Rdf pξqgpξq µpdξq

»Rd

f pxqpg µqpxq dx

Proof. We use the following elementary identity for tempered distributions: Ifµ is a finite measure, and φ P S, thenxφµ φ µTherefore (and occasionally using F p f q synonymously with f for notational rea-sons), »

Rdf pξqgpξq µpdξq

»Rd

f pxqF g µpxq dx

»

Rdf pxqppg µqpxq dx

»Rd

f pxqpg µqpxq dx

since pg pg g.

Lemma 11.3. Let µ be a finite measure on Rd, and d ¥ 2. Then the followingare equivalent:

11. FOURIER RESTRICTION AND APPLICATIONS 147

a) xfµLqpRdq ¤ C f L2pµq for all f P SpRdqb) gL2pµq ¤ CgLq1 pRdq for all g P SpRdqc) µ f LqpRdq ¤ C2 f Lq1 pRdq for all f P SpRdq.

Proof. By the previous lemma, for any g P SpRdq,

gL2pµq supfPS, f L2pµq1

»Rd

gpξq f pξqµpdξq

supfPS, f L2pµq1

»Rd

gpxqxfµpxq dx (11.5)

Hence, if a) holds, then the right-hand side of (11.5) is no larger than gLq1 pRdqand b) follows. Conversely, if b) holds, then the entire expression in (11.5) is nolarger than CgLq1 pRdq, which implies a). Thus a) and b) are equivalent with thesame choice of C. Clearly, applying first b) and then a) with f g yields

F pgµqLqpRdq ¤ CgLq1 pRdq

for all g P SpRdq. Since F pgµq gpq µ, part c) follows. Finally, we note therelation »

gpxqpµ f qpxq dx »

gpxqF pµ f qpxq dx »

gpξq f pξq µpdξq

for any f , g P SpRdq. Hence, if c) holds then »Rd

gpξq f pξq µpdξq ¤ C2gLq1 pRdq f Lq1 pRdq

Now set f pxq gpxq. Then»Rd|gpξq|2 µpdξq ¤ C2g2

Lq1 pRdq

which is b).

Setting µ σ σS d1 , the surface measure of the unit sphere in Rd, one nowobtains the following:

Corollary 11.4. The following assertions are equivalent

a) The Stein-Tomas theorem in the “restriction form”:

f S d1L2pσq ¤ C f Lq1 pRdq

for q1 2d2d3 and all f P SpRdq

b) the “extension form” of the Stein-Tomas theorem

gσS d1LqpRdq ¤ CgL2pσq

for q 2d2d1 and all g P SpRdq.


c) The composition of a) and b): for all f P SpRdq f zσS d1LqpRdq ¤ C2 f Lq1 pRdq

with q 2d2d1 .

Proof. Set µ σS d1 σ in Lemma 10.2.

Exercise 11.2. In general, a) and b) above remain true, whereas c) requiresL2pσq. More precisely, show the following: The restriction estimate

f S d1Lppσq ¤ C f Lq1 pRdq @ g P Sis equivalent to the extension estimate

gσS d1LqpRdq ¤ CgLp1 pσq @ g P S

We shall now show via part b) of Corollary 11.4 that the Stein-Tomas theoremis optimal. This is the well-known Knapp example.

Lemma 11.5. The exponent pd 2d2d3 in Theorem 11.1 is optimal.

Proof. This is equivalent to saying that the exponent q 2d2d1 in part b) of

the previous corollary is optimal. Fix a small δ ¡ 0 and let g P S such that g 1on Bped;

?δq, g ¥ 0, and supppgq Bpen; 2

?δq where ed p0, . . . , 0, 1q is a unit

vector.Then

|xgσpξq| » e2πirx1ξ1ξdp?

1|x1|21qs gpx1,a

1 |x1|2qa1 |x1|2

dx1

¥ » cosp2πpx1 ξ1 ξdp

b1 |x1|2 1qqqgpx1,

a1 |x1|2qa

1 |x1|2dx1

¥ cos

π

4

»g dσ ¥ C1δ

d12

(11.6)

provided |ξ1| ¤ p?δq1

100 |ξd| ¤ δ1

100 . Indeed, under these assumptions, and for δ ¡ 0small,

|x1 ξ1 ξdb

1 |x1|2 1 ¤ ?

δ p?δq1

100 δ1

100

?δ2¤ 1

50,

so that the argument of the cosine in (11.6) is smaller than 2π50 ¤ π

4 in absolutevalue, as claimed.

Hence,

gσS d1LqpRdq ¥ C1δpd1q

2 δ

pd1q2 δ1

1q

C1δd1

2 δpd1q

2q


whereas gL2pσq ¤ Cδd1

4 . It is therefore necessary that

d 14

¤ d 12

d 12q

or q ¥ 2d2d1 , as claimed.

For the proof of Theorem 11.1 we need the following decay estimate for theFourier transform of the surface measure σS d1 , see Corollary 6.16:zσS d1pξq

¤ Cp1 |ξ|qp d12 q (11.7)

It is easy to see that (11.7) imposes a restriction on the possible exponents for anextension theorem of the form

fσS d1LqpRdq ¤ C f LppS d1q (11.8)

Indeed, setting f 1 implies that one needs

q ¡ 2dd 1

by (11.7). On the other hand, one has

Exercise 11.3. Check by means of Knapp’s example from Lemma 10.3 that(11.8) can only hold for

q ¥ d 1d 1

p1 (11.9)

The still unproved (in dimensions d ¥ 3) restriction conjecture states that (11.8)holds under these conditions, i.e., provided

8 ¥ q ¡ 2dd 1

and q ¥ d 1d 1

p1

Observe that the Stein-Tomas theorem with q 2d2d1 and p 2 is a partial result

in this direction. In dimension d 2 the conjecture is true and can be proved byelementary means, see the following chapter.

It is known, see Wolff’s notes, that the full restriction conjecture implies theKakeya conjecture which states that Kakeya sets in dimension d ¥ 3 have (Haus-dorff) dimension d. This latter conjecture appears to be also very difficult.

Proof of the Stein-Tomas theorem for p 2d2d3 . Let¸

jPZψp2 jxq 1

for all x , 0 be the usual Littlewood-Paley partition of unity. By Lemma 10.2 andCorollary 11.4 it is necessary and sufficient to prove

f zσS d1Lp1 pRdq ¤ C f LppRdq


for all f P SpRdq. Firstly, let

ϕpxq 1¸j¥0

ψp2 jxq

Clearly, ϕ P C80 pRdq, and

1 ϕpxq ¸j¥0

ψp2 jxq for all x P Rd

Now observe that ϕ zσS d1 P C80 so that

f ϕ zσS d1Lp1 pRdq ¤ C f LppRdq (11.10)

with C ϕ zσS d1Lr where 1 1p1 1

r 1p , i.e., 2

p1 1r . It therefore remains to

controlK j : ψp2 jxq zσS d1pxq

in the sense that we need to prove an estimate of the form

f K jLp1 pRdq ¤ C2 jε f LppRdq @ f P SpRdq (11.11)

for all j ¥ 0 and some small ε ¡ 0. It is clear that the desired bound follows bysumming (11.10) and (11.11) over j ¥ 0. To prove (11.11) we interpolate a 2 Ñ 2and 1 Ñ8 bound as follows:

f K jL2 f L2K jL8 f L22n jψp2 jq σS d1L8

¤ C f L22d j 2 jpd1q C2 j f L2

(11.12)

To pass to the estimate (11.12) one uses that

supxσS d1pBpx, rqq ¤ Crd1

as well as the fact that ψ has rapidly decaying tails, which implies that the estimateis the same as for compactly supported ψ.

On the other hand,

f K j8 ¤ K j8 f 1 ¤ C2 j pd1q2 f 1 (11.13)

since the size of K j is controlled by (11.7). Interpolating (11.12) with (11.13)yields

f K jp1 ¤ C2 j d12 θ2 jp1θq f p

where 1p1 θ

8 1θ2 1θ

2 . We thus obtain (11.11) provided

0 d 12

θ p1 θq d 12

θ 1

pd 1q1

2 1

p1 1 d 1

2 d 1

p1

This is the same as p1 ¡ 2d2d1 or p 2d2

d3 , as claimed.


Exercise 11.4. Provide the details concerning the “rapidly decaying tails” inthe previous proof.

The previous proof strategy of the Tomas-Stein theorem does not achieve thesharp exponent p 2d2

d3 because (11.11) leads to a divergent series with ε 0. Inorder to achieve this endpoint exponent, one therefore has to avoid interpolating theoperator bounds on each dyadic piece separately. The idea is basically to sum firstand then to interpolate, rather than interpolating first and then summing. Of course,one needs to explain what it means to sum first: recall that the proof of the Riesz-Thorin interpolation theorem is based on the three lines theorem from complexanalysis. The key idea in our context is to sum the dyadic pieces T j : f ÞÑ f K jtogether with complex weights w jpzq in such a way that

Tz :¸j¥0

w jpzqT j

converges on the strip 0 ¤ Re z ¤ 1 to an analytic operator-valued function withthe property that

Tz : L1 Ñ L8 for Re z 1 and

Tz : L2 Ñ L2 for Re z 0

It then follows that Tθ LppRdq Ñ Lp1pRdq for 1p1 1θ

2 . The “art” is then tochoose the weights w jpzq so that Tθ at a specific θ is a prescribed operator such asconvolution by zσS d1 in our case.

While it is conceptually correct and helpful to describe complex interpolationas a method of summing divergent series, it is rarely implemented in this fashion.Rather, one tries to embed the desired operator under consideration into an analyticfamily from the start without ever breaking it up into dyadic pieces. There arecertain standard ways of doing this which involve a little complex analysis anddistribution theory (on the level of integration by parts and analytic continuation).We shall present such a standard approach now.

Proof of the endpoint for Tomas-Stein. We consider a surface of non-zero cur-vature which can be written locally as a graph: ξd hpξ1q, ξ1 P Rd1. Define

Mzpξq 1Γpzq

ξd hpξ1qz1

χ1pξ1qχ2ξd hpξ1q (11.14)

where χ1 P C80 pRd1q, χ2 P C8

0 pRq are smooth cut-off-functions, Γ is the Gammafunction, and Re z ¡ 0. Moreover, pq refers to the positive part. We will showthat

Tz f : pMz f q_ (11.15)can be defined by means of analytic continuation to Re z ¤ 0. The main estimatesare now

Tz2Ñ2 ¤ Bpzq for Re z 1 (11.16)

Tz1Ñ8 ¤ Apzq for Re z d 12

(11.17)


where Apzq, Bpzq grow now faster than eC|z|2 as |Im z| Ñ 8. Moreover, we willsee that the singularity of

ξd hpξ1qz1

at z 0 “cancels” out against the simplezero of 1

Γpzq at z 0 so as to produce

M0pξq χ1pξ1qδ0pξd hpξ1qq dξ1 (11.18)

see (11.21); this means that M0pξq is proportional to surface measure on the graph.It then follows from Stein’s complex interpolation theorem that

f ÞÑ xM0 f

is bounded from Lp Ñ Lp1 where1p1 θ

8 1 θ

2, 0 θd 1

2 1 θ

which implies that1p1 d 1

2d 2

as desired. It remains to check (11.16)–(11.18). To do so, recall first that 1Γpzq is

an entire function with simple zeros at z 0, 1, 2 , . . .. It has the productrepresentation

1Γpzq zeγz

8¹ν1

1 z

ν

ezν, z x iy

which converges everywhere in C. Thus, 1Γpzq

2 ¤ |z|2e2γx8¹ν1

1 x

ν

2 y2

ν2

e

2ν x

¤ |z|2e2γx8¹ν1

e

2xν

|z|2

ν2 e2xν

|z|2e2γxe|z|

2 π26

(11.19)

In particular, if Re z 1, then for all ξ P Rd,

|Mzpξq| ¤ p1 y2qe2γep1y2q π26 χ1pξ1qχ2

ξd hpξ1q ¤ Cecy2

(11.16) holds with the stated bound on Bpzq. We remark that the bound we justobtained is very wasteful. The true growth, as given by the Stirling formula, is ofthe form |y| 1

2 eπ2 |y| as |y| Ñ 8 but this makes no difference in our particular case.

Now let ϕ P SpRdq. Thus, for Re z ¡ 0,»Rd

Mzpξqϕpξq dξ

1Γpzq

»Rd1

» 8

0χ2ptqϕpξ1, t hpξ1qqtz1 dt χ1pξ1q dξ1

1zΓpzq

»Rd1

» 8

0

ddt

χ2ptqϕ

ξ1, t hpξ1q tz dt χ1pξ1q dξ1

(11.20)


Observe that the right-hand side is well-defined for Re z ¡ 1. Furthermore, atz 0, using zΓpzqz0 1»

RdM0pξqϕpξq dξ

»Rd1

χ2p0qϕpξ1, hpξ1qqχ1pξ1q dξ1

which shows that the analytic continuation of Mz to z 0 is equal to (settingχ2p0q 1)

M0pξq χ1pξ1q dξ1δ0pξd hpξ1qq (11.21)

Clearly, M0 is proportional to surface measure on a piece of the surface

S pξ1, hpξ1qq | ξ1 P Rd1(This is exactly what we want, since we need to bound xσS f .

Observe that (11.20) defines the analytic continuation to Re z ¡ 1. Integrat-ing by parts again extends this to Re z ¡ 2 and so forth. Indeed, the right-handside of»

RdMzpξqϕpξq dξ p1qk

zpz 1q . . . pz k 1qΓpzq»Rd1

χ1pξ1q» 8

0tzk1

dk

dtk

χ2ptqϕpξ1, ξd kpξ1qq dt dξ1

is well-defined for all Re z ¡ k. Next we prove (11.17) by means of an estimateon xMz8. This requires the following preliminary calculation: let N be a positiveinteger such that N ¡ Re z 1 ¡ 0. Then we claim that » 8

0e2πitτtzχ2ptq dt

¤ CNp1 |z|qN

1 Re zp1 |τ|qRe z1 (11.22)

To prove (11.22) we will distinguish large and small tτ. Let ψ P C80 pRq be

such that ψptq 1 for |t| ¤ 1 and ψptq 0 for |t| ¥ 2. Then, since 0 ¤ χ2 ¤ 1,we have » 8

0e2πitτtzψptτqχ2ptq dt

¤ » 8

0tRe zψpt|τ|q dt

¤ |τ|Re z1» 2

0tRe z dt ¤ C

Re z 1|τ|Re z1

(11.23)

If |τ| ¤ 1, then (11.23) is no larger than» 8

0tRe zχ2ptq dt ¤ C

Re z 1

Hence

(11.23) ¤ C1 Re z

p1 |τ|qRe z1 (11.24)


in all cases. To treat the case tτ large, which implies that τ is large, we exploitcancellation in the phase. More precisely, » 8

0e2πitτtzp1 ψptτqqχ2ptq dt

¤ 12π|τ|

N» 8

0

dN

dτN

tzp1 ψptτqqχ2ptq

dt

¤ CN

12π|τ|

N» 8

0dt|zpz 1q . . . pz N 1q|tRe zNp1 ψptτqqχ2ptq

tRe z|ψpNqptτq|τNχ2ptq tRe zp1 ψptτqq|χpNq2 ptq|

¤ CN

12π|τ|

N|zpz 1q . . . pz N 1q|

» 8

1τ

tRe zN dt

Cp2π|τ|qN |τ|N |τ|Re z1 C|τ|N

» 1

0tRe z|χpNq2 ptq| dt

¤ CN|zpz 1q . . . pz N 1q| 1

|τ|Re z1 CN |τ|N (11.25)Observe that the indefinite integrals here converge because of Re z N 1.Moreover, the second term in (11.25) is ¤ |τ|Re z1 by the same condition (recallthat we are taking τ to be large). Hence (11.22) follows from(11.24) and (11.25).We now compute xMzpxq. Let k be a positive integer with Re z ¡ k. Then xMzpxqequals»

e2πixξ 1Γpzqpξd hpξ1qqz1

χ1pξ1qχ2pξd hpξ1qq dξ1 dξd

1Γpzq

» 8

0e2πixdt tz1 χ2ptq dt

»Rd1

e2πirx1ξ1xdhpξ1qs χ1pξ1q dξ1

p1qkΓpzqz pz 1q . . . pz k 1q

» 8

0

e2πixdtχ2ptq

pkqtzk1 dt

»Rd1

e2πirx1ξ1xdhpξ1qsχ1pξ1q dξ1

(11.26)

where the final expression is well-defined for Re z ¡ k. Now suppose that Re z d1

2 and pick k P Z such that 1 k ¤ Re z k 2, i.e., d12 ¤ k d1

2 1.Apply (11.22) with z k 1 instead of z and with N 2. Then the first integral(11.26) is bounded by » 8

0

e2πixdtχ2ptq

pkqtzk1 dt

¤ Cp1 |z|q2Re z k

p1 |xd|qRe zk Ckp1 |xd|qk

¤ Ck p1 |z|q2p1 |xd|qRe z

(11.27)

On the other hand, the second integral in (11.26) is controlled by the stationaryphase estimate, cf. (11.7), »

Rd1e2πirx1ξ1xdhpξ1qsχ1pξ1q dξ1

¤ Cp1 |x|q d12 (11.28)


Observe that the growth in |xd| for Re z d12 is exactly balanced by the decay

in (11.28). One concludes that for Re z pd1q2 and with k as in (11.27)

xMz8 ¤ Cd

1Γpzqz pz 1q . . . pz k 1q

p1 |z|q2see (11.26)–(11.28). Thus (11.17) follows from the growth estimate (11.19) orStirling’s formula, and we are done.

The endpoint of the Tomas-Stein theorem is so important because of its scalinginvariance. To see this, let upt, xq be a smooth solution of the Schrodinger equation"

1i Btu 1

2π 4Rd u Fu|t0 f

(11.29)

where f P SpRdq, say. The constant 12π is rather arbitrary and chosen for cosmetic

reasons having to do with our choice of normalization of the Fourier transform. Itcould be replaced by any other positive constant by rescaling; one can even changethe sign of the constant by passing to complex conjugates. We first treat the caseF 0. Then

upt, xq »Rd

e2πipxξt|ξ|2q f pξq dξ p fµq_pt, xq (11.30)

where µ is the measure in Rd1 defined by the integration»Rd1

Fpξ, τq µpdpξ, τqq »Rd

Fpξ, |ξ|2q dξ

for all F P C0pRd1q. Now let ϕ P C80 pRd1q, ϕpξ, τq 1 if |ξ| |τ| ¤ 1. Then

the endpoint of Stein-Tomas applies and one concludes that

p fϕµq_LqpRd1q ¤ C f L2pϕµq

where q 2d4d 2 4

d . In other words, if suppp f q Bp0, 1q, then

p fµq_L2 4

d pRd1q ¤ C f L2pRdq C f L2pRdq (11.31)

To remove the condition suppp f q Bp0, 1q we rescale. Indeed, the rescaled func-tions

fλpxq f pxλquλpx, tq upxλ, tλ2q (11.32)

satisfy the equation "1i Btuλ 1

2π 4 uλ 0uλ|t0 fλ

If suppp f q is compact, then suppp fλq Bp0, 1q if λ is large. Hence, in view of(11.30) and (11.31),

uλLqpRd1q ¤ C fλL2pRdq (11.33)However,

fλL2pRdq λd2 f L2pRdq


and

uλLqpRd1q λd2

q uLq λd2 uLq

Thus, (11.33) is the same as

uLqpRd1q ¤ C f L2pRdq (11.34)

for all f P S will suppp f q compact. These functions are dense in L2pRdq. There-fore, (11.34), which is the original Strichartz bound for the Schrodinger equation ind spatial dimensions, follows for all f P L2pRdq. Note the main difference betweenthe Strichartz estimate (11.33) and the Stein-Tomas endpoint estimate: while thelatter is confined to compact surfaces, the former applies to functions which live onthe non-compact characteristic variety of the Schrodinger equation, i.e., the parab-oloid tpξ, |ξ|2q | ξ P Rdu. Therefore, the Strichartz estimate necessarily obeys thescaling symmetry of that variety which is also all we needed to pass from compactsurfaces to this specific non-compact surface.

While the approach we followed to derive the estimate (11.34) for solutionsof (11.29) with F 0 is the original one used by Strichartz and instructive dueto its reliance on the restriction property of the Fourier transform, it is technicallyadvantageous for many reasons to use a shorter and somewhat different argumentwhich was introduced and developed by Ginibre and Velo in a series of papers.We will present this argument now to derive the following theorem. We call a pairpp, qq Strichartz admissible if and only if

2p d

q d

2(11.35)

and 2 ¤ p ¤ 8 with pp, qq , p2,8q.Theorem 11.6. Let F be a space-time Schwartz function in d 1 dimensions,

and f a spatial Schwartz function. Let upt, xq solve (11.29). Then

uLpt Lq

xpRd1q ¤ C f L2pRdq FLa1

t Lb1x pRd1q

(11.36)

where pp, qq and pa, bq are Strichartz admissible with a ¡ 2 and p ¡ 2. Finally,these estimates localize in time: if u is restricted to some time-interval I on theleft-hand side, then F can be restricted to I on the right-hand side.

Some remarks are in order: first, the estimate (11.36) is scaling invariant underthe law (11.32) if and only if pp, qq and pa, bq obey the relation (11.35). Hence thelatter is necessary for the Strichartz estimates (11.36) to hold. Note that the pairp q 2 4

d which we derived above for F 0 is of this type. On the otherhand, the restriction p ¡ 2 is technical and the important endpoint estimate forp 2 and d ¥ 3 was proved by Keel and Tao. Finally, under the strong conditionson F and f it is easy to actually solve (11.29) by means of an explicit expression:

uptq ei

2π4t f » t

0e

i2π4ptsq Fpsq ds (11.37)


with the understanding that the operator needs to be understood as a Fourier multi-plier:

ei

2π4t fpxq :

»e2πipxξ|ξ|2tq f pξq dξ

which is well-defined for f P SpRdq, say.

Proof of Theorem 11.6. We again start with F 0 and let Uptq denote thepropagator, i.e., Uptq f is the solution of (11.29) as give by (11.37): Uptq e

i2π4t.

By (6.14) we have the bound

Uptq f LqpRdq ¤ C|t| d2 p 1

q1 1

q q f Lq1 pRdq (11.38)

for 1 ¤ p ¤ 2. Indeed, for p 1 this follows by placing absolute values in-side (6.14), whereas p 2 is Plancherel’s theorem. The intermediate p valuesthen follow by interpolation. Our goal for now is to prove that for any pp, qq as inthe theorem

U f Lpt Lq

x¤ C f 2

where pU f qpt, xq pUptq f qpxq, slightly abusing notation. In analogy with the“duality equivalence” Lemma 11.3 one now has that each of the following estimateimplies the other two:

U f Lpt Lq

x¤ C f L2

x

UFL2x¤ CF

Lp1t Lq1

x

U ULpt Lq

x¤ C2 F

Lp1t Lq1

x

As in the case of the Tomas-Stein theorem, the key is to prove the third property.Note that one has

UF » 8

8UpsqFpsq ds

whence by the group property of Uptq

pU UFqptq » 8

8Upt sqFpsq ds

By (11.38),

pU UFqptqLqx¤ C

» 8

8|t s| d

2 p 1q1 1

q qFpsqLq1

xds

whence by fractional integration in time with

1 1p 1

p1 d

2p 1q1 1

qq ðñ (11.35)

provided

0 d2p 1q1 1

qq 1 ðñ p ¡ 2 (11.39)

one hasU UFLp

t Lqx¤ CF

Lp1t Lq1

x


which is the desired estimate for the case F 0. Now suppose f 0, and writethe solution of (11.29) as

uptq » t

0Upt sqFpsq ds

» 8

8χr0 s ts Upt sqFpsq ds

We now claim two estimates on u as a space-time function:

uLpt Lq

x¤ CF

Lp1t Lq1

x

uLpt Lq

x¤ CFL1

t L2x

The first one is proved by the same argument as before, whereas the second one isproved as follows:

uptqLqx¤» 8

8χr0 s ts Upt sqFpsqLq

xds

¤» 8

8Upt sqFpsqLq

xds

whence

uLpt Lq

x¤ C

» 8

8Upt sqFpsqLp

t Lqx

ds

¤ C» 8

8FpsqL2

xds FL1

s L2x

By duality and interpolation one now concludes that

uLpt Lq

x¤ CFLa1

s Lb1x

for any admissible pair pa, bq and pp, qq with a ¡ 2 and p ¡ 2. This concludesthe argument. The statement about time-localizations follows from the solutionformula (11.37).

Exercise 11.5. The previous argument of course offers an alternative to thecomplex interpolation approach that we used to obtain the Stein-Tomas endpoint.Provide the details of this alternative proof to Theorem 11.1 for general boundedsubsets of surfaces of nonvanishing Gaussian curvature.

As an application of Theorem 11.6 we now solve the nonlinear equation

iBtψ ∆ψ λ|ψ| 4dψ (11.40)

in d spatial dimensions and with an arbitrary real-valued constant λ for small L2-data ψ|t0 ψ0. First, it is not apriori clear what we mean by “solve” since thedata are L2, nor is it immediately clear why we do not consider smoother data.Second, we remark that the choice of power in (11.40) is not accidental. In fact, itis the unique power for which the rescaled functions

λd2ψpλx, λ2tq (11.41)


are again solutions. The relevance of this scaling law is the fact that the L2-invariantscaling of the data is λ

d2ψ0pλxq, which therefore leaves the smallness condition un-

changed. Moreover, the induced rescaling of the solution is then given by (11.41).Hence, the only setting in which we can hope for a meaningful global small L2-datatheory is (11.40).

As in the existence and uniqueness theory of ordinary differential equations,one reformulates (11.40) as an integral equation as follows (this is called Duhamelformula):

ψptq eit∆ψ0 iλ» t

0eiptsq∆ |ψpsq| 4

dψpsq ds (11.42)

and then seeks a solution of this integral equation by contraction or Picard iterationbelonging to the space Ctpr0,8q; L2

xpRdqq. This is very natural, as (11.40) con-serves the L2-norm for real λ (see the problem section). However, this space is stilltoo large to formulate a meaningful theory in, and so we restrict it further in a waythat allows us to invoke Theorem 11.6.

Definition 11.7. By a global weak solution of (11.40) with data ψ0 P L2pRdqwe mean a solution to the integral equation (11.42) which lies in the space

X : pCt X L8t qpr0,8q; L2xpRdqq X Lp0

t pr0,8q; L2p0x pRdqq (11.43)

with p0 : 1 4d .

We remark that the power p0 in the definition is very natural. In fact, passingL2-norms onto (11.42) yields

ψptq2 ¤ Cψ02 |λ|

» t

0ψpsqp0

2p0ds

which is at least finite for ψ P X. Now we will show much more.

Corollary 11.8. The nonlinear equation (11.40) in R1dt,x admits a unique

weak solution in the sense of Definition 11.7 for any data ψ0 P L2pRdq providedψ02 ¤ ε0pλ, dq.

Proof. For the sake of simplicity, we set d 2 which implies that the nonlin-earity takes the form λ|ψ|2ψ and p0 3. We leave the case of general dimensionsto the reader. Let us first prove uniqueness. Thus, let ψ and ϕ be two weak solutionsto (11.42). Taking differences yields

pψ ϕqptq iλ» t

0eiptsq∆ p|ψpsq|2ψpsq |ϕpsq|2ϕpsqq ds

Therefore, applying the Strichartz estimates of the previous theorem locally in timeon I r0,T q yields, with XpIq being as in (11.43) localized to I,

ψ ϕXpIq ¤ C|λ||ψpsq|2ψpsq |ϕpsq|2ϕpsqL1pr0,Tq;L2pR2q¤ C|λ|pψXpIq ϕXpIqq2 ψ ϕXpIq


However, if I is small enough then C|λ|pψXpIq ϕXpIqq2 1 whence ψ ϕXpIq 0. Thus, ψ ϕ on I. This argument shows that the set where ψ ϕ isboth open and closed and since it is nonempty (it contains t 0) therefore equalsthe whole time-line.

To prove existence, we set up a contraction argument in the space X for smalldata. Thus, we define and operator A on X by the formula

pAψqptq : eit∆ψ0 iλ» t

0eiptsq∆ |ψpsq|2ψpsq ds

and note that by Theorem 11.6

AψX ¤ Cpψ02 |λ|ψ3Xq

Therefore, if ψX ¤ R, and if

Cpψ02 |λ|R3q R (11.44)

we conclude that A takes and R-ball in X into itself. To satisfy (11.44) we simplychoose ε so small that R Cε will verify (11.44) for any ψ02 ε. Takingdifferences as in the uniqueness proof then shows that A is in fact a contraction.Therefore, there exists a unique fixed-point ψ in the R-ball in X. But Aψ ψ isprecisely the definition of a weak solution and we are done.

Corollary 11.8 is only one of many results which can be obtained in this area.It is also clear that many interesting questions pose themselves now, such as thequestion of global existence for large data (which depends on the sign of λ) whichis relatively easy for data in H1 but hard for data in L2, persistence of regularity,spatial decay of solutions etc. We present some of the easier results in this directionin the problem section. Showing that an equation with a derivative nonlinearitysuch as

iBtψ ∆ψ |ψ|2 B1ψ (11.45)

has global solutions for small data in S, say, cannot be accomplished by means ofStrichartz estimates alone. Indeed, the Strichartz estimates are not able to make upfor the loss of a derivative on the right-hand side.

We conclude this chapter with a sketch of Strichartz estimates for the waveequation in R1d

t,x :

u Bttu ∆u F

ut0 f , Btu

t0 g

(11.46)

The solution to this Cauchy problem is

uptq cospt|∇|q f sinpt|∇|q|∇| g

» t

0

sinppt sq|∇|q|∇| Fpsq ds


at least for Schwartz data, say. The meaning of the operators appearing here is ofcourse based on the Fourier transform. In other words,

rcospt|∇|q f spxq ¸

12

»Rd

e2πipxξt|ξ|q f pξq dξ

Proceeding analogously to the Schrodinger equation were are therefore lead tointerpreting these integrals as Fourier transforms of measures living on the doublelight-cone

Γ : pξ, τq P Rd1 | |τ| |ξ|(This is different from the paraboloid in the Schrodinger equation in two importantways: first, Γ has one vanishing principal curvature namely along the generatorsof the light cone. Second, Γ is singular at the vertex of the cone. To overcome thelatter difficulty, we define

Γ0 : tpτ, ξq P R1d | 1 ¤ |ξ| τ ¤ 2uto be a section of the light cone in R1d. Since the cone has one vanishing principalcurvature, the Fourier transform of the surface measure on Γ0 exhibits less decayas for the paraboloid which reflects itself in a different Stein-Tomas exponent (infact, the numerology here is such that the dimension d is reduced to d1 reflectingthe loss of one principal curvature).

Exercise 11.6.

By means of the method of stationary phase from Chapter 6 show thatzϕσΓ0pt, xq ¤ Cp1 |pt, xq|q pd1q

2 (11.47)

for all pt, xq P R1d. Moreover, this is optimal for all directions pt, xqbelonging to the dual cone Γ (which is equal to Γ if the opening angle is90).

Check that the complex interpolation method from above therefore im-plies that there is the following restriction estimate for Γ0:

f Γ0L2pσΓq ¤ C f LppRdq

where p 2d2d3 and d ¥ 2 (for d 1 this amounts to a trivial bound).

Via a rescaling and Littlewood-Paley theory we now arrive at the followinganalogue of the argument leading to (11.34) for the Schrodinger equation.

Theorem 11.9. Let Γ be the double light-cone in R1dτ,ξ with d ¥ 2, equipped

with the measure µpdpτ, ξqq dξ|ξ| . Then »

Γ

| f pτ, ξq|2 µpdpτ, ξqq 1

2 ¤ C f LppRdq (11.48)

with p 2d2d3 .


Proof. Let Γ0 be the cone restricted to 1 ¤ |τ| ¤ 2 as above. Then »λ¤|pτ,ξq|¤2λ

| f ppτ, ξqq|2 µpdpτ, ξqq 1

2

¤ »

Γ0

| f pλpτ, ξqq|2µpdpλpτ, ξqqq 1

2

¤ Cλd1

2

1λd1 f

1λ

LppR1dq

Cλd1

2 λd1 λ d1p f LppRd1q C f LppRd1q

(11.49)

by choice of p. To sum up (11.49) for λ 2 j we use the Littlewood-Paley theoremtogether with Exercise 10.3. In fact, since p 2, we conclude that

f L2pµq ¤ C¸

jPZP j f 2

Lp

12 ¤ C

¸jPZ

P j f2 1

2

Lp¤ C f LppRd1q

which is (11.48).

For the wave equation this means the following.

Corollary 11.10. Let upt, xq be a solution of (11.46) with f , g P SpRdq, d ¥ 2.Then

uL

2d2d1 pRd1q

¤ Cp f 9H

12 pRdq g 9H 1

2 pRdqqwith a constant C Cpdq.

Proof. For simplicity, set f 0. Then

upt, xq »Rd

e2πipt|ξ|xξqgpξq dξ4πi|ξ|

»Rd

e2πipt|ξ|xξqgpξq dξ4πi|ξ|

pFµq_px, tqwhere Fpξ,|ξ|q f pξq and dµpξ,|ξ|q dξ

|ξ| . By the dual to Theorem 11.9,

pF µq_Lp1 pRd1q ¤ CFF2pµq (11.50)

where p1 2d2d1 . Clearly,

FL2pµq »Rd|gpξq|2 dξ

|ξ| 1

2 g9H 1

2

so that (11.50) implies that

uL

2d2d1 pRd1q

¤ Cg9H 1

2 pRdq

as claimed.

Exercise 11.7. Derive estimates for the wave equation as in the previous corol-lary but with f

9H1 gL2 on the right-hand side.


Notes

A standard introductory reference in this area are Stein’s Beijing lectures [48] andWolff’s course notes [56]. The original reference by Strichartz is [50]. For an accountsummarizing more recent developments see Tao [53]. Strichartz estimates are a vast area,and continue to be developed, especially in the variable coefficient setting, see Bahouri,Chemin, and Danchin [3] for that aspect. A standard references on dispersive evolutionequations is Tao [52]. Keel, Tao [31] prove the endpoint for the Strichartz estimates, i.e.,L2

t -type estimates. For more on the Schrodinger equation see Cazenave [8], as well asSulem and Sulem [51]. For the wave equation, see Shatah and Struwe [42]. Strichartzesimates for the Klein-Gordon equation, which behaves like the wave equation for largefrequencies and the Schrodinger equation for small frequencies, respectively, are derivedin [37]. Some of the original work by Ginibre and Velo on Strichartz estimates for variousdispersive equations can be found here [23]. Finally, for the solution theory of equationsof the type (11.45) which require smoothing estimates for the Schrodinger equation, seeKenig, Ponce and Vega [32].

Problems

Problem 11.1. Suppose that φ is a smooth function on p1, 1q with φ1ptq 0 if andonly if t 0. Assume further that φ2p0q 0 and φ3p0q , 0. Let a P C8p1, 1q withsupppaq p1, 1q. Determine the sharp decay of» 1

1eiλφptq aptq dt

as λÑ8.

Problem 11.2. Investigate the restriction of the Fourier transform of a function in R3

to a (section of a) smooth curve in R3. Assume that the curve has nonvanishing curvatureand torsion. Try to generalize to other dimensions and codimensions.

Problem 11.3. This problem expands on Corollary 11.8.

Show that if ψ0 P HkpRdq (one can again take d 1 or d 2 for simplicity)where k is a positive integer, then ψ P Ctpr0,8q; HkpRdqq. Conclude via Sobolevimbedding that if ψ0 P SpRdq, say, then ψpt, xq is smooth in all variables andsatisfies the nonlinear equation (11.40) in a pointwise sense.

Show that if ψpt, xq is any solution (not necessarily small) of (11.40) with Btψ PCpr0,T q; L2pRdqq and ψ P Cpr0,T q; H2pRdqq, then one has conservation of mass(recall λ is real)

Mpψq : 12ψptq2

2 12ψ02

2 @ 0 ¤ t T

and energy

Epψq :»Rd

12|∇ψpt, xq|2 λ

2 4d

|ψpt, xq|2 4d

dx

Hint: differentiate M and E with respect to time, and integrate by parts.


Prove the relationeit∆x px 2i∇qeit∆

as operators acting on Schwartz functions and use this to show that if ψ0 PSpRdq, then the solution constructed in Corollary 11.8 has the property that forany time t one has ψptq P SpRdq.

Show that the solution of Corollary 11.8 with ψ0 P L2pRdq preserves the massMpψq and the energy provided the data satisfy ψ0 P H1pRdq.

Problem 11.4.

For any power p ¥ 2 show that the equation

iBtψ Bxxψ λ|ψ|p1ψ (11.51)

on the line R, has local in-time unique weak solutions for any data ψ0 P H1pRq.These solutions again need to be interpreted in the Duhamel integral equationsense, and the space in which to contract is Cpr0,T q; H1pRqq where T ¡ 0 canbe chosen so that it is bounded below by a function of ψ0H1pRq alone. Showthat these solutions conserve mass and energy.

Show that if T is the maximal time so that a weak solution exists on r0,Tq,and if T 8, then ψptqH1pRq Ñ 8 as t Ñ T. Conclude that for λ ¡ 0 onehas global solutions, i.e., T 8. Note: for λ 0 this is false, for examplesolutions of negative energy blow up in finite time in the sense that T 8,see [8] or [51].

For small data in H1pRq and powers p ¥ 5 show that one has global solutionsirrespective of the sign of λ.

Problem 11.5. For the one-dimensional wave equation utt uxx 0 with smooth dataut0 f and Btu

t0 g show that the solution is given by

upt, xq 12p f px tq f px tqq 1

2

» xt

xtgpyq dy

for all times. Conclude that such waves do not decay, precluding any possibility of aStrichartz estimate other than one involving L8t .

Problem 11.6. Carry out the Ginibre-Velo approach to Strichartz estimates for thewave equation, in analogy with the Schrodinger equation as in Theorem 11.6. This needs tobe done for a fixed Littlewood-Paley piece, say P0, and then rescaled and summed up usingthe results of the previous chapter. This approach leads to Besov-space estimates whichare stronger than the Sobolev ones (one can switch to the latter by means of Exercise 10.3as we did in the proof of Corollary 11.10). Hint: See [42] for details.

Bibliography

[1] Adams, D. R. A note on Riesz potentials. Duke Math. J. 42 (1975), no. 4, 765–778.[2] Antonov, N. Yu. Convergence of Fourier series. Proceedings of the XX Workshop on Function

Theory (Moscow, 1995). East J. Approx. 2 (1996), no. 2, 187–196.[3] Bahouri, H., Chemin, J.-Y., Danchin, R. Fourier analysis and nonlinear partial differential

equations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Math-ematical Sciences], 343. Springer, Heidelberg, 2011.

[4] Bergh, J., Lofstrom, J. Interpolation spaces. An introduction. Grundlehren der MathematischenWissenschaften, No. 223. Springer-Verlag, Berlin-New York, 1976.

[5] Bourgain, J. Bounded orthogonal systems and the Λppq?(p)-set problem. Acta Math. 162(1989), no. 3-4, 227–245.

[6] Carleson, L. On convergence and growth of partial sums of Fourier series. Acta Math. 116,135–157 (1966).

[7] Garca-Cuerva, J., Rubio de Francia, J., Weighted norm inequalities and related topics. North-Holland Mathematics Studies, 116. Notas de Matematica [Mathematical Notes], 104. North-Holland Publishing Co., Amsterdam, 1985.

[8] Cazenave, T. Semilinear Schrodinger equations. Courant Lecture Notes in Mathematics, 10.New York University, Courant Institute of Mathematical Sciences, New York; American Math-ematical Society, Providence, RI, 2003.

[9] Christ, M. Lectures on singular integral operators. CBMS Regional Conference Series in Math-ematics, 77. Published for the Conference Board of the Mathematical Sciences, Washington,DC; by the American Mathematical Society, Providence, RI, 1990.

[10] Coifman, R., Meyer, Y. Wavelets. Calderon-Zygmund and multilinear operators. Translatedfrom the 1990 and 1991 French originals by David Salinger. Cambridge Studies in AdvancedMathematics, 48. Cambridge University Press, Cambridge, 1997.

[11] Davis, K. M., Chang Lectures on Bochner Riesz means. London Mathematical Society, LectureNote Series 114, 1987.

[12] Duoandikoetxea, J. Fourier analysis. (English summary) Translated and revised from the 1995Spanish original by David Cruz-Uribe. Graduate Studies in Mathematics, 29. American Math-ematical Society, Providence, RI, 2001.

[13] Evans, L. C. Partial differential equations. Second edition. Graduate Studies in Mathematics,19. American Mathematical Society, Providence, RI, 2010.

[14] Fefferman, C. The multiplier problem for the ball. Ann. of Math. (2) 94 (1971), 330–336.[15] Fefferman, CThe uncertainty principle. Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 2, 129–

206.[16] Fefferman, C., Stein, E. M. HpHp spaces of several variables. Acta Math. 129 (1972), no. 3-4,

137–193.[17] Folland, Gerald B. Introduction to partial differential equations. Second edition. Princeton Uni-

versity Press, Princeton, NJ, 1995.[18] Frank, R., Seiringer, R. Non-linear ground state representations and sharp Hardy inequalities.

J. Funct. Anal. 255 (2008), no. 12, 3407–3430.[19] Frazier, M., Jawerth, B., Weiss, G. Littlewood-Paley theory and the study of function spaces.

CBMS Regional Conference Series in Mathematics, 79. Published for the Conference Board

165

166 BIBLIOGRAPHY

of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Prov-idence, RI, 1991.

[20] Garnett, J. Bounded analytic functions. Revised first edition. Graduate Texts in Mathematics,236. Springer, New York, 2007.

[21] Garnett, J. B., Marshall, D. E. Harmonic measure. Reprint of the 2005 original. New Mathe-matical Monographs, 2. Cambridge University Press, Cambridge, 2008.

[22] Gilbarg, D., Trudinger, N. Elliptic partial differential equations of second order. Second edition.Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of MathematicalSciences], 224. Springer-Verlag, Berlin, 1983.

[23] Ginibre, J., Velo, G. On a class of nonlinear Schrodinger equation. I. The Cauchy problems; II.Scattering theory, general case, J. Func. Anal. 32 (1979), 1-32, pp. 33–71; Scattering theoryin the energy space for a class of nonlinear Schrodinger equations, J. Math. Pures Appl. (9)64 (1985), no. 4, pp. 363–401; The global Cauchy problem for the nonlinear Klein-Gordonequation. Math. Z. 189 (1985), no. 4, 487–505.; Time decay of finite energy solutions of thenonlinear Klein-Gordon and Schrodinger equations. Ann. Inst. H. Poincare Phys. Theor. 43(1985), no. 4, 399–442.

[24] Havin, V., Joricke, B. The uncertainty principle in harmonic analysis. Ergebnisse der Mathe-matik und ihrer Grenzgebiete (3), 28. Springer-Verlag, Berlin, 1994.

[25] Hoffman, K. Banach spaces of analytic functions. Reprint of the 1962 original. Dover Publica-tions, Inc., New York, 1988.

[26] Hormander, L. The analysis of linear partial differential operators. I. Distribution theory andFourier analysis. Second edition. Springer Study Edition. Springer-Verlag, Berlin, 1990.

[27] Jaming, P. Nazarov’s uncertainty principles in higher dimension. J. Approx. Theory 149 (2007),no. 1, 3041.

[28] Janson, S. Mean oscillation and commutators of singular integral operators. Ark. Mat. 16(1978), no. 2, 263–270.

[29] Jerison, D. An elementary approach to local solvability for constant coefficient partial differen-tial equations. Forum Math. 2 (1990), no. 1, 45–50.

[30] Katznelson, I. An introduction to harmonic analysis. Dover, New York, 1976.[31] Keel, M., Tao, T. Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), pp. 955–980.[32] Kenig, C. E., Ponce, G., Vega, L. The initial value problem for the general quasi-linear

Schrodinger equation. Recent developments in nonlinear partial differential equations, 101–115, Contemp. Math., 439, Amer. Math. Soc., Providence, RI, 2007.

[33] Kolmogorov, A. N. Une serie de Fourier-Lebesgue divergente presque partout. FundamentaMathematicae (4), 1923, 324–328.

[34] Koosis, P. Introduction to Hp spaces. Second edition. With two appendices by V. P. Havin [Vik-tor Petrovich Khavin]. Cambridge Tracts in Mathematics, 115. Cambridge University Press,Cambridge, 1998.

[35] Konyagin, S. V. On the divergence everywhere of trigonometric Fourier series. (Russian) Mat.Sb. 191 (2000), no. 1, 103–126; translation in Sb. Math. 191 (2000), no. 1-2, 97–120.

[36] Lieb, E. H., Loss, M. Analysis. Second edition. Graduate Studies in Mathematics, 14. AmericanMathematical Society, Providence, RI, 2001.

[37] Nakanishi, K., Schlag, W. Invariant Manifolds and Dispersive Hamiltonian Evolution Equa-tions., EMS, 2011.

[38] Nazarov, F. L. Local estimates for exponential polynomials and their applications to inequali-ties of the uncertainty principle type. Algebra i Analiz 5 (1993), no. 4, 3–66; translation in St.Petersburg Math. J. 5 (1994), no. 4, 663–717.

[39] Opic, B., Kufner, A. Hardy-type inequalities. Pitman Research Notes in Mathematics Series,219. Longman Scientific & Technical,

[40] Rudin, W. Functional analysis. Second edition. International Series in Pure and Applied Math-ematics. McGraw-Hill, Inc., New York, 1991.

[41] Rudin, W. Trigonometric series with gaps. J. Math. Mech. 9 1960, 203–227.

BIBLIOGRAPHY 167

[42] Shatah, J., Struwe, M. Geometric wave equations. Courant Lecture Notes in Mathematics, 2.American Mathematical Society, Providence, RI, 1998.

[43] Sjolin, P. An inequality of Paley and convergence a.e. of Walsh-Fourier series. Ark. Mat. 7,551–570 (1969).

[44] Stein, E. M. On limits of seqences of operators. Ann. of Math. (2) 74 (1961), 140–170.[45] Stein, E. M. Singular integrals and differentiability properties of functions. Princeton Mathe-

matical Series, No. 30 Princeton University Press, Princeton, N.J., 1970.[46] Stein, E. M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory inte-

grals. With the assistance of Timothy S. Murphy. Princeton Mathematical Series, 43. Mono-graphs in Harmonic Analysis, III. Princeton University Press, Princeton, N.J., 1993.

[47] Stein, E. M. Topics in harmonic analysis related to the Littlewood-Paley theory. Annals ofMathematics Studies, No. 63 Princeton University Press, Princeton, N.J.; University of TokyoPress, Tokyo 1970.

[48] Stein, E. M. Oscillatory integrals in Fourier analysis. Beijing lectures in harmonic analysis(Beijing, 1984), 307355, Ann. of Math. Stud., 112, Princeton Univ. Press, Princeton, NJ, 1986.

[49] Stein, E. M., Weiss, G. Introduction to Fourier analysis on Euclidean spaces. Princeton Math-ematical Series, No. 32. Princeton University Press, Princeton, N.J., 1971.

[50] Strichartz, R. S. Restrictions of Fourier transforms to quadratic surfaces and decay of solutionsof wave equations. Duke Math. J. 44 (1977), no. 3, 705–714.

[51] Sulem, C., Sulem, P-L. The nonlinear Schrodinger equation. Self-focusing and wave collapse,Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999.

[52] Tao, T. Nonlinear dispersive equations. Local and global analysis. CBMS Regional ConferenceSeries in Mathematics, 106. AMS Providence, RI, 2006.

[53] Tao, T. Some recent progress on the restriction conjecture. Fourier analysis and convexity, 217–243, Appl. Numer. Harmon. Anal., Birkhauser Boston, Boston, MA, 2004.

[54] Torchinsky, A. Real-variable methods in harmonic analysis. Pure and Applied Mathematics,123. Academic Press, Inc., Orlando, FL, 1986.

[55] Uchiyama, A. A constructive proof of the Fefferman-Stein decomposition of BMOpRnq. ActaMath. 148 (1982), 215–241.

[56] Wolff, T. H. Lectures on harmonic analysis. With a foreword by Charles Fefferman and prefaceby Izabella Łaba. Edited by Łaba and Carol Shubin. University Lecture Series, 29. AmericanMathematical Society, Providence, RI, 2003.

[57] Yafaev, D. Sharp constants in the Hardy-Rellich inequalities. J. Funct. Anal. 168 (1999), no. 1,121–144.

[58] Zygmund, A. Trigonometric series. Vol. I, II. Third edition. With a foreword by Robert A.Fefferman. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2002.

harmonic analysis w. schlag - department of mathematicsschlag/harmonicnotes.pdf · chapter 1...

Documents