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FOURIER AND SPECTRAL MULTIPLIERSIN RN AND IN THE HEISENBERG GROUP

prof. Fulvio Ricci

A.A. 2003-2004

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Contents

Chapter I Self-adjoint operators and spectral analysis

1. Review of notions from spectral theory2. The Heisenberg sub-Laplacian3. The spectral analysis of L1

4. The joint spectrum of L and i−1T

Chapter II Mihlin-Hormander multipliers for constant coefficient operators

1. Spectral and Fourier Lp-multipliers2. The Hardy-Littlewood maximal function3. Calderon-Zygmund operators on spaces of homogeneous type4. Integral Lipschitz conditions5. Non-isotropic dilations in Rn and Calderon-Zygmund kernels6. Mihlin-Hormander conditions on Fourier multipliers7. Applications

Chapter III Littlewood-Paley theory and Marcinkiewicz multipliers

1. Square functions2. Littlewood-Paley functions3. Marcinkiewicz multipliers4. Applications

Chapter IV Fourier analysis on the Heisenberg group

1. The Heisenberg group2. The group Fourier transform3. Fourier multipliers4. Radial functions and diagonal multipliers5. Radiality in Hn

6. Applications

Chapter V Spectral multipliers of the sub-Laplacian

1. The heat kernel on Hn

2. Smooth multipliers and Schwartz kernels3. Mihlin-Hormander multipliers of L

SELF-ADJOINT OPERATORS 3

CHAPTER I

SELF-ADJOINT OPERATORS

AND SPECTRAL ANALYSIS

1. Review of notions from spectral theory

Self-adjoint operators.

We sketch some basic facts about the spectral theory of (possibly unbounded)self-adjoint operators on a Hilbert space. For a complete treatment, the reader canconsult, e.g. M. Reed, B. Simon Methods of Modern Mathematical Physics, vol. I,Functional Analysis, vol. II, Fourier Analysis, Self-adjointness. We shall refer tothese books as [RS1, RS2] respectively.

Let H be a Hilbert space, and let T be a linear operator with domain D densein H and with values in H.

Let D′ consist of those elements u ∈ H for which an element w exists such that1

〈Tv|u〉 = 〈v|w〉

for all v ∈ D. This w is unique because D is dense, and it is denoted by T ∗u. Theoperator T ∗ defined on D′ is called the adjoint operator of T .T is called symmetric if D ⊆ D′ and T ∗

|D = T . It is called self-adjoint if it issymmetric and D′ = D. It is a known fact that a self-adjoint operator is closed(i.e. its graph in H ×H is closed). It follows from the closed graph theorem thata closed densely defined operator T is bounded if and only if D = H.

The most notable example of a self-adjoint operator is the following. Take H =L2(X,µ), with (X,µ) a measure space. Given an a.e. finite measurable real-valuedfunction ϕ on X, define

Tϕf(x) = ϕ(x)f(x) ,

on D = {f ∈ L2(X,µ) : ϕf ∈ L2(X,µ)}. That D is dense follows from the factthat, if we denote by Xn the set where ϕ(x) < n, then D contains all L2-functionssupported on Xn.

Given g ∈ L2(X,µ), we look for another L2-function h such that

∫

X

ϕ(x)f(x)g(x)dµ(x) =

∫

X

f(x)h(x)dµ(x) ,

1We adopt the following notation: 〈 | 〉 denotes the Hermitean product on a Hilbert space,

whereas 〈 , 〉 denotes a bilinear pairing, e.g. between a distribution and a test function.

Typeset by AMS-TEX

4 CHAPTER I

for every f ∈ D. This is possible if and only if h(x) = ϕ(x)g(x) a.e. Hence D′ = Dand then it is pretty obvious that T ∗

ϕ = Tϕ.Observe that Tϕ is bounded if and only if ϕ ∈ L∞(X,µ).

The following statement is easy to prove.

Proposition 1.1. Let A : H → H ′ be a unitary transformation between Hilbertspaces. Given a densely defined operator T on H with domain D, let T ′ be theoperator on H ′, with domain D′ = AD, given by T ′ = ATA−1. Then T is self-adjoint if and only if T ′ is.

In this case we say that T and T ′ are unitarily equivalent.As a direct consequence of self-adjointness of Tϕ for ϕ real-valued, we have the

following statement for constant coefficient differential operators.

Theorem 1.2. Let P be a polynomial in n variables with real coefficients, andconsider the differential operator L = P (i−1∂) = P (i−1∂x1

, . . . , i−1∂xn) on Rn.

With H = L2(Rn), take

(1.1) D = {f ∈ L2(Rn) : Lf ∈ L2(Rn)} ,

as the domain of L. Then L is self-adjoint.

Proof. Let A be the unitary transformation of L2(Rn) onto itself given by theFourier transform multiplied by (2π)−n/2. Then L is unitarily equivalent to TP .Since P has real coefficients, TP is self-adjoint. The conclusion follows from Propo-sition 1.1. �

The multiplication operator Tϕ described above is more than just an example.The following statement is proved in [RS1].

Theorem 1.3 (Spectral Theorem, version 1). Let H be a separable Hilbertspace and T be a self-adjoint operator on it with domain D. Then T is unitarilyequivalent to a multiplication operator Tϕ, with ϕ measurable, a.e. finite, and real-valued on some finite measure space (X,µ).

The resolvent set of a closed operator T on a HIlbert space H is defined as theset of those λ ∈ C for which λI −T has a bounded inverse. The complement of theresolvent set is the spectrum of T , denoted by σ(T ). The resolvent set is open andthe spectrum is closed.

If Tϕ is as above, it is simple to show that λ is in the resolvent set if and only if(λ− ϕ)−1 ∈ L∞(X,µ), i.e. if and only if there is δ > 0 such that

µ{x : |ϕ(x) − λ| < δ} = 0 .

The complement of this set, called the essential range of ϕ, is the spectrum ofTϕ. Since ϕ is real-valued, clearly σ(Tϕ) ⊆ R.

It is a general fact that the spectrum of a self-adjoint operator is contained inR. For separable H, one can appeal to Theorem 1.3, for general H see [RS1].

Spectral measure.

Definition. A regular projection-valued measure, also called a regular resolutionof the identity, on R is a map E assigning to each Borel subset ω of R an orthogonalprojection E(ω) on some fixed Hilbert space H, satisfying the following properties:

(1) E(∅) = 0 and E(R) = I;(2) E(ω ∩ ω′) = E(ω)E(ω′);(3) if {ωj} is a countable family of pairwise disjoint Borel sets, then

E( ⋃

j

ωj

)=

∑

j

E(ωj) ,

in the strong topology of L(H);(4) for every Borel set ω,

E(ω) = sup{E(ω′) : ω′ ⊂ ω , ω′ compact

}= inf

{E(ω′′) : ω′′ ⊃ ω , ω′′ open

},

with respect to the partial ordering in the space of bounded self-adjoint op-erators on H (for which T ≤ T ′ if and only if 〈Tu|u〉 ≤ 〈T ′u|u〉 for everyu ∈ H).

The support of the measure E is the smallest closed set F such that E(R\F ) = 0.

It follows from (2) that the projections E(ω) commute with each other.Observe that, if E is a regular resolution of the identity, then

v =

∫

R

dE(λ)v

for every v ∈ H. Also, given v, w ∈ H,

νv,w(ω) = 〈E(ω)v|w〉

is a (scalar-valued) finite Borel measure, and

〈v|w〉 =

∫

R

dνv,w =

∫

R

〈dE(λ)v|w〉 .

Clearly, νv,v is positive for every v, and ‖νv,v‖1 = ‖v‖2. Holder’s inequalityshows that, for general v, w ∈ H, ‖νv,w‖1 ≤ ‖u‖‖v‖.Proposition 1.4. Let m be an E-a.e. finite Borel function on R. Then

D ={v ∈ H :

∫

R

|m(λ)|2dνv,v <∞},

is dense in H and the operator

Smv =

∫

R

m(λ) dE(λ)v

is well-defined on D, it commutes with every E(ω), and

SmE(ω) = Smχω.

6 CHAPTER I

Moreover, Sm is bounded on H if and only if m is E-a.e. bounded, and it isself-adjoint if and only if m is E-a.e. real-valued.

If m1 and m2 are E-a.e. bounded, then Sm1m2= Sm1

Sm2.

Proof. Let Fn be the Borel set where |m(λ)| ≤ n, and let En = E(Fn). If v ∈ EnH,then νv,v(R \ Fn) = 0. It follows that v ∈ D.

Since the Fn form an increasing sequence, it follows from condition (3) that theEn converge in the strong topology to E

(⋃n Fn

). This is the identity operator,

because m is E-a.e. finite. This implies that

D ⊆⋃

n

EnH = H .

If v ∈ H and m is a simple function,

m(λ) =∑

j

mjχωj(λ) ,

with the ωj pairwise disjoint Borel sets, then

Smv =∑

j

mjE(ωj) ,

and

(1.2)

‖Smv‖2 =∑

j,k

mjmk〈E(ωj)v|E(ωk)v〉

=∑

j

|mj |2〈E(ωj)v|v〉

=

∫

R

|m(λ)|2 dνv,v(λ) .

A limiting argument shows that Smv is well defined for v ∈ D and that (1.2)holds for every v ∈ D. It also shows that Sm commutes with the E(ω) and thatSmE(ω) = Smχω

.In particular, this implies that each subspace EnH is mapped into itself by Sm.That the boundedness of Sm is equivalent to the E-a.e. boundedness of m follows

from (1.2). We show now that D is also the domain of the adjoint of Sm.If u, v ∈ D, then

〈Smv|u〉 =

∫

R

m(λ) dνv,u(λ)

=

∫

R

m(λ) dνu,v(λ)

= 〈Smu|v〉= 〈v|Smu〉 .

This shows that S∗m extends Sm, i.e. Sm is symmetric.


Take now u in the domain of S∗m. Then there is w ∈ H such that 〈Smv|u〉 = 〈v|w〉

for every v ∈ D. This is equivalent to saying that

(1.3)∣∣〈Smv|u〉

∣∣ ≤ Cu‖v‖

for some constant Cu and every v ∈ D.Let un = Enu. Then un ∈ D and un → u. Also, Smun is also in D. We can

then apply (1.3) with v = Smun. We find that

‖Smun‖2 = 〈S2mun|un〉

= 〈S2mun|Enu〉

= 〈S2mun|u〉

≤ Cu‖Smun‖ .

Hence ‖Smun‖ ≤ Cu for every n. From

‖Smun‖2 = ‖SmχFnu‖2 =

∫

Fn

m(λ)2 dνu,u(λ) ≤ Cu ,

we obtain that u ∈ D.Finally, the identity Sm1m2

= Sm1Sm2

is trivial if m1,m2 are simple functions,and the general case follows by approximation. �

The next theorem, the proof of which can be found in [RS1], says that resolutionsof the identity are in 1-1 correspondence with self-adjoint operators.

Theorem 1.5 (Spectral Theorem, version 2). Let T be a self-adjoint operatoron a Hilbert space H. Then there is one and only one regular resolution of theidentity of H, {E(ω)}, on R such that

T =

∫

R

λ dE(λ) .

The measure E is called the spectral measure of T . Its support coincides withσ(T ).

We describe the resolution of the identity of L2(X,µ) associated with a multi-plication operator Tϕ, ϕ being real-valued and a.e. finite.

Given a Borel set ω ⊆ R, let

E(ω)f = fχϕ−1(ω)

be the orthogonal projection onto the subspace of L2-functions on X supported onϕ−1(ω). This is clearly a regular resolution of the identity. For f ∈ L2(X,µ), wehave

νf,f (ω) =

∫

ϕ−1(ω)

|f(x)|2 dµ(x) .

If g(λ) =∑

j gjχωj(λ) is a simple function on R, with the ωj pairwise disjoint,

then∫

R

g(λ) dνf,f(λ) =∑

j

gj

∫

ϕ−1(ωj)

|f(x)|2 dµ(x) =

∫

X

g(ϕ(x)

)|f(x)|2 dµ(x) .

8 CHAPTER I

This identity can be easily extended to more general g.Consider

S =

∫

R

λ dE(λ) .

Hence the domain D of S consists of the functions f such that

∫

R

λ2 dνf,f (λ) =

∫

X

ϕ(x)2|f(x)|2 dµ(x) <∞ ,

i.e. those for which ϕf ∈ L2(X,µ).

This last example gives the spectral resolution for self-adjoint constant coefficientdifferential operators on Rn.

Proposition 1.6. Let L = P (i−1∂), where P a polynomial in n variables with realcoefficients, with the domain given in (1.1). Then

L =

∫

R

λ dE(λ) ,

where, denoting by F the Fourier transform,

E(ω)f = F−1(fχP−1(ω)) .

The notion of spectral measure associated to a self-adjoint operator T allows todevelop a functional calculus on T , i.e. to define other operators expressed in termsof the spectral measure, hence depending on T itself.

Let dE(λ) be the spectral measure of T . If m is a Borel measurable function onR, E-a.e. finite, define

m(T ) =

∫

R

m(λ) dE(λ) .

The function m is called a spectral multiplier.

Extensions of symmetric operators.

The general question if a symmetric operator admits a self-adjoint extension,and if this extension is unique, requires a detailed study, which is out of the scopeof these notes. The answer is that self-adjoint extensions do not always exist, andif they exist, they need not be unique. The only fact we want to mention concernspositive operators.

A symmetric operator T with domain D is called positive if

〈Tf |f〉 ≥ 0

for every f ∈ D.A well-known fact is that every positive symmetric operator admits at least one

self-adjoint extension (see [RS2] for the construcion of a canonical extension, calledthe Friedrichs extension).


For our purposes, it is important to mention the following application. LetX1, . . . , Xm be first-order operators on Rn, and denote by X ′

j the formal adjoint ofXj , i.e.

Xjf(x) =

n∑

k=1

ajk(x)∂xkf(x) + a0(x)f(x) ,

X ′jf(x) = −

n∑

k=1

∂xk

(aj,k(x)f(x)

)+ a0(x)f(x) .

We shall assume that the coefficients are defined and smooth on all of Rn.The operator L =

∑mj=1X

′jXj , initially defined on D0 = D(Rn) is symmetric,

because its adjoint is an extension of L itself, defined on

D ={f ∈ L2(Rn) : Lf ∈ L2(Rn)

},

(with Lf understood in the sense of distributions). It is also clear that L is positive.

Theorem 1.7. The operator L with domain D is self-adjoint, and it is the onlyself-adjoint operator, with domain containing D0, and equal to L on D0.

Proof. The first part of the statement follows from the Friedrichs construction. Forthe second part2, let (L,D′) be a self-adjoint extension of (L,D0). If f ∈ D′, byself-adjointness,

〈f |Lg〉 = 〈Lf |g〉for all g ∈ D′. If we restrict to g ∈ D0,this implies that h = Lf in the sense ofdistributions, so that f ∈ D.

Hence (L,D′) ⊆ (L,D). Passing to the adjoints, the inclusions are reversed; butboth operators are self-adjoint, hence D′ = D. �

Spectral analysis of commuting self-adjoint operators.

If T1, T2 are bounded self-adjoint operators on H, and T1T2 = T2T1, then anytwo projections E1(ω) and E2(ω

′) in the corresponding resolutions of the identityalso commute with each other (see [RS1]).

If we try to extend this statement to unbounded operators, we meet severaldifficulties. First of all, the composition T1T2 is well defined only on those elementsv in the domain of T2 such that T2v is in the domain of T1. These elements canform a very small space, and this space may not coincide with the one constructedby interchanging the role of T1 and T2. Worse than that, even though the equalityT1T2 = T2T1 holds on a dense subspace, the projections in the two resolutions ofthe identity need not commute.

We are so led to impose the following definition.

Definition. Let T1, T2 be self-adjoint operators on H. We say that T1 and T2 com-mute if the operators {E1(ω)} and {E2(ω)} forming the corresponding resolutionsof the identity all commute with each other.

We state without proof the following fact (see [RS1] for the proofs).

2Instead of using different symbols, like L′ or others for extensions of the “initial” L, we prefer

to keep the same symbol, specifying the domain whenever there is ambiguity. We shall also write

(L,D1) ⊆ (L,D2) to denote that the second operator is an extension of the first from the domainD1 to the domain D2.

10 CHAPTER I

Theorem 1.8. Let T1, T2 be self-adjoint operators. The following are equivalent:

(i) T1 and T2 commute;(ii) if λ, µ are non-real numbers, (λI − T1)

−1 and (µI − T2)−1 commute;

(iii) for every s, t ∈ R, eisT1 and eitT2 commute.

Observe that the operators in (ii) and (iii) are bounded, so that “commutation”is meant in the ordinary sense.

If T1 and T2 commute, we can define a joint spectral measure on R2, by defining

E(ω × ω′) = E1(ω)E2(ω′) ,

and extending E to every other Borel set in R2 so that conditions (3) and (4) aresatisfied.

Projecting E onto each component, we find E1 and E2 respectively. Hence

T1 =

∫

R2

λ dE(λ, µ) , T1 =

∫

R2

µ dE(λ, µ) .

A joint spectral multiplier of T1 and T2 is an E-a.e. finite Borel measurablefunction m(λ, µ), and one defines

m(T1, T2) =

∫

R2

m(λ, µ) dE(λ, µ) .

The extension of these notions to a larger number of mutually commuting self-adjoint operators is trivial. Proposition 1.4 remains true for multipliers of morethan one operator.

The support of E is called the joint spectrum of T1 and T2, denoted by σ(T1, T2).In contrast with what happens when taking tensor products of scalar-valued mea-sures, it may happen that the support of E is strictly contained in the product ofthe support of the Ej; in other words, the joint spectrum σ(T1, T2) can be strictlysmaller than σ(T1) × σ(T2).

What can happen is that E1(ω) and E2(ω′) are non-zero, but their product is

zero.Consider the case H = C3, and

T1 =

1 0 00 1 00 0 −1

, T2 =

2 0 00 3 00 0 3

.

Then σ(T1) = {1,−1} and σ(T2) = {2, 3}. The spectral projections have thefollowing ranges:

E1({1}) span (e1, e2)

E1({−1}) span (e3)

E2({2}) span (e1)

E2({3}) span (e2, e3) .


Hence the only non trivial products are E(1, 2), E(1, 3), E(−1, 3), so that

σ(T1, T2) = {(1, 2), (1, 3), (−1, 3)} .

A more interesting example is the following. On Rn, take T1 = i−1∂x1and

T2 = ∆, the Laplacian. The domains are those described before, making each ofthem self-adjoint.

By Proposition 1.6, if ω, ω′ are Borel subsets of R,

E1(ω)f = F−1(fχ{ξ:ξ1∈ω}) , E2(ω′)f = F−1(fχ{ξ:|ξ|2∈ω′}) .

This implies that E(ω × ω′) = 0 if ω × ω′ does not intersect the epi-parabolaµ ≥ λ2, and one can show easily that the joint spectrum is the full epi-parabola.

Fourier multipliers.

On Rn take Tj = i−1∂xj, for 1 ≤ j ≤ n. One can easily verify that the joint

spectrum is all of Rn, and that, if m(ξ) is an a.e. finite Borel function on Rn, then3

m(i−1∂)f(x) = F−1(fm)(x) .

Hence joint spectral multipliers for the system i−1∂ coincide with the Fouriermultipliers.

Similarly, if m is an a.e. finite Borel function on R, then

m(∆)f(x) = F−1(fm(| · |2)

)(x) ,

i.e. the spectral multipliers of ∆ coincide with the radial Fourier multipliers.

2. The Heisenberg sub-Laplacian

In this section we present what will be for us the main example of an operatorof the form described in Theorem 1.7. The group-theoretic notions connected withthe operators below are postponed to a future section.

We take R2n+1 = Rn × Rn × R with coordinates (x, y, t), and define 2n vectorfields X1, . . . , Xn, Y1, . . . , Yn as

(2.1) Xj = ∂xj− yj

2∂t , Yj = ∂yj

+xj

2∂t .

Then X ′j = tXj = −Xj , Y

′j = tYj = −Yj , so that the Heisenberg sub-Laplacian

(2.2) L =n∑

j=1

(X ′jXj + Y ′

jYj) = −n∑

j=1

(X2j + Y 2

j )

is positive on L2(R2n+1), and self-adjoint on the domain

D ={f ∈ L2(R2n+1) : Lf ∈ L2(R2n+1)

}.

3We use the symbol ∂ to denote the system (∂x1 , . . . , ∂xn) in short.

12 CHAPTER I

Notice that S(R2n+1) ⊂ D. The explicit expression of L,

L = ∆x + ∆y − |x|2 + |y|24

∂2t +

n∑

j=1

(xj∂yj− yj∂xj

)∂t ,

shows that L is not elliptic (e.g., L = ∆x + ∆y at the origin). It is howeverhypoelliptic, according to Hormander’s theorem4. This follows from the fact that,for every j,

[Xj , Yj] = ∂t ,

and the system of vector fields {X1, Y1, . . . , Xn, Yn, ∂t} gives a basis of the tangentspace at each point of R2n+1.

The constant vector field ∂t is usally denoted by T .

In order to work out its spectral decomposition, it is preferable to replace L byanother operator, unitarily equivalent to it.

Denote by Ft the partial Fourier transform in R2n+1 in the variable t. ThenA = (2π)−1/2Ft is a unitary operator on L2(R2n+1. We set

L = ALA−1 = FtLF−1t .

Then L is self-adjoint, with domain

D ={g ∈ L2(R2n+1) : Lg ∈ L2(R2n+1)

}.

If we perform the same conjugation by Ft on the Xj and Yj , we obtain

(2.3) Xj = FtXjF−1t = ∂xj

− iλ

2yj , Yj = FtYjF−1

t = ∂yj+ i

λ

2xj ,

and

(2.4) L =n∑

j=1

(X ′jXj + Y ′

j Yj) = −n∑

j=1

(X2j + Y 2

j ) .

It must be observed that derivatives are only taken in the variables xj , yj, and notin λ. We shall also regard the first-order operators in (2.3) as acting on functionsof (x, y) ∈ R2n, taking λ as a parameter.

When we do so, we call Xλ,j, Yλ,j the operators in (2.3), and Lλ the operator in(2.4). If we set gλ(x, y) = g(x, y, λ), this means that

Lg(x, y, λ) = Lλgλ(x, y) .

By Theorem 1.7, Lλ is self-adjoint on L2(R2n) for every λ, with domain

Dλ ={f ∈ L2(R2n) : Lλf ∈ L2(R2n)

}.

For λ = 0, Lλ = ∆; for λ 6= 0, Lλ is called the twisted Laplacian. The followingstatement is obvious.

4See the notes of the course “Sub-Laplacians on nilpotent Lie groups”.

Lemma 2.1. A function g(x, y, λ) ∈ L2(R2n+1) belong to D if and only if gλ ∈ Dλ

for a.e. λ and ∫

R

‖Lλgλ‖2

2 dλ <∞ .

We describe the spectral measure of L.

Proposition 2.2. Denote by E(ω) the L-spectral measure of a Borel subset ω ofR, and by Eλ(ω) its Lλ-spectral measure. Then

E(ω)g(x, y, λ) = Eλ(ω)gλ(x, y) .

Proof. One easily checks that E is a regular projection-valued measure. Then wejust need to identify the operator

A =

∫

R

µ dE(µ)

together with its domain.Setting

νg,h(ω) = 〈E(ω)g|h〉 , νgλ,hλ(ω) = 〈Eλ(ω)gλ|hλ〉 ,

we have

νg,g(ω) =

∫

R

〈Eλ(ω)gλ|gλ〉 dλ =

∫

R

∫

ω

dνgλ,gλ(µ) dλ .

Hence the domain of A consists of the functions g such that∫

R

µ2 dνg,g(µ) =

∫

R2

µ2dνgλ,gλ(µ) dλ <∞ .

But this means that gλ ∈ Dλ for a.e. λ and that∫

R

‖Lλgλ‖2

2 dλ <∞ ,

i.e. g is in the domain of L. Moreover,

〈Ag|h〉 =

∫

R

µ dνg,h(µ)

=

∫

R2

µ dνgλ,hλ(µ) dλ

=

∫

R

〈Lλgλ, hλ〉 dλ

= 〈Lg|h〉 . �

In view of Proposition 2.2, it will be interesting to obtain an explicit descriptionof the spectral measures Eλ. The case λ = 0 is very simple, because L0 is theLaplacian, but of course we are more interested in λ 6= 0.

The following lemma shows that the operators Lλ with λ 6= 0 can be conjugatedamong themselves, up to a constant factor, by unitary operators.

14 CHAPTER I

Lemma 2.3. For s > 0, let As be the unitary operator Asf(x, y) = snf(sx, sy) onL2(R2n). Then

L±s2 = s2AsL±1A−1s .

If Bf(x, y) = f(x,−y), then

L−λ = BLλB−1 .

If Fλ is the spectrum of Lλ, and λ 6= 0,

Fλ = |λ|F1 ={|λ|µ : µ ∈ F1

}.

Finally, if m(µ) is a spectral multiplier and λ 6= 0, and ε = 0, 1 depending onwhether λ is negative or positive,

m(Lλ) = BεA|λ|

12m

(|λ|L1

)A−1

|λ|12B−ε .

Proof. Given a Schwartz function f on R2n, it follows from (2.3) that

Xλ,j(Asf)(x, y) = sn+1∂xjf(sx, sy)− i

λ

2snyjf(sx, sy)

= sn+1Xλ/s2,jf(sx, sy)

= sAs(Xλ/s2,jf)(x, y) .

It follows that AsDλ/s2 = Dλ and

LλAs = s2AsLλ/s2 .

This gives the first assertion, and the second can be proved in a similar andsimpler way.

By uniqueness of the spectral measure associated with a self-adjoint operator,setting C = BεA

|λ|12, from the identity∫

R

µ dEλ(µ) = Lλ

= |λ|CL1C−1

= |λ|C( ∫

R

µ dE1(µ)

)C−1

= C

( ∫

R

µ dE1

(|λ|−1µ

))C−1

we derive thatEλ(ω) = CE1

(|λ|−1ω

)C−1 .

Hence the support Fλ of Eλ and the support F1 of E1 are in the stated relation.To conclude,

m(Lλ) =

∫

R

m(µ) dEλ(µ)

= C

( ∫

R

m(µ) dE1

(|λ|−1µ

))C−1

= C

( ∫

R

m(|λ|µ) dE1

(µ))C−1

= Cm(|λ|L1

)C−1 . �


3. The spectral analysis of L1

In order to complete the analysis, we then have to determine the spectral measureE1 of L1. The first remark is that we can reduce our analysis to n = 1, i.e. to theoperator

(3.1) −(∂x − i

2y

)2

−(∂y +

i

2x

)2

on R2.This reduction is based on the following fact (for notational convenience, we

state it for two operators, the extension to n operator being left to the reader).

Lemma 3.1. Let L1,L2 be self-adjoint differential operators on L2(Rd) with do-mains Dj = {f : Ljf ∈ L2(Rd)} and spectra F1, F2 ⊆ R. On (Rd)2 = R2d, withcoordinates (x1, x2) with xj ∈ Rd, consider the differential operator L acting as

Lf = (L1)x1f + (L2)x2

f ,

in the sense that each Lj acts on the corresponding variable xj ∈ Rd. Then L is

self-adjoint with domain D = {f : Lf ∈ L2(R2d)}, and with spectrum F = F1 + F2.

Proof. It is easy to verify that the operators (Lj)xjcommute as self-adjoint oper-

ators on L2(R2d), with domains D1 = {f(x1, x2) : f(·, x2) ∈ D1 for a.e. x2} and

similarly for D2. It is also easy to verify that their joint spectrum is the cartesianproduct F1 × F2 in R2. The conclusion follows by applying the spectral multiplierm(λ, µ) = λ+ µ. �

We prefer to use ad-hoc notations at this stage, and set

X = ∂x − i

2y , Y = ∂y +

i

2x ,

calling L the operator (3.1).We shall see that L has a discrete spectrum, and we can explicitely construct a

complete system of eigenfunctions. For this construction it is crucial to introducethe complex operators

(3.2)Zf =

1

2(X − iY)f = ∂zf +

z

4f

Zf =1

2(X + iY)f = ∂zf − z

4f .

For reasons that will be immediately clear, we call Z the annihilation operatorand Z the creation operator.

Since

(3.3) Zf = e−|z|2

4 ∂z(e|z|2

4 f) , Zf = e|z|2

4 ∂z(e−

|z|2

4 f) ,

it follows that

Zf = 0 ⇐⇒ f = e−|z|2

4 × an antiholomorphic function ,

Zf = 0 ⇐⇒ f = e|z|2

4 × a holomorphic function .

16 CHAPTER I

This implies that Z is injective on L2(C), whereas Z has a big null-space inL2(C). For j, k ∈ N, define

(3.4)hj,0(z) = zje−

|z|2

4 ,

hj,k(z) = Zkhj,0(z) .

Proposition 3.2. The functions hj,k, with j, k ∈ N, form a complete orthogonalsystem in L2(C), and

L1hj,k = (2k + 1)hj,k .

Proof. Since

ZZf =1

4(X + iY)(X − iY) =

1

4

(X 2 + Y2 − i[X ,Y]

),

and [X ,Y] = iI by (2.3), we have that

(3.5) L = −4ZZ + I .

But Zhj,0 = 0, hence Lhj,0 = hj,0. A similar computation shows that

(3.6) L = −4ZZ − I ,

so that

Lhj,k = (−4ZZ + I)Zhj,k−1

= Z(−4ZZ + I)hj,k−1

= Z(L + 2I)hj,k−1 .

By induction, Lhj,k = (2k + 1)hj,k.

It also follows by induction that hj,k equals a polynomial in z, z times e−|z|2

4 . Inparticular, hj,k ∈ S(C), hence in the domain of L1 in L2(C). We then have

(2k + 1)〈hj,k|hj′,k′〉 = 〈L1hj,k|hj′,k′〉= 〈hj,k|L1hj′,k′〉= (2k′ + 1)〈hj,k|hj′,k′〉 ,

so that the two functions are orthogonal if k 6= k′.

If k = k′ = 0, then

(3.7)

〈hj,0|hj′,0〉 =

∫

C

zjzj′

e−|z|2

2 dz

=

∫ +∞

0

rj+j′+1e−r2

2 dr

∫ 2π

0

ei(j′−j)θ dθ ,

which is zero if j 6= j′. The same conclusion follows by induction for k = k′ > 0,since

(3.8)

〈hj,k|hj′,k〉 = 〈Zhj,k−1|Zhj′,k−1〉= −〈hj,k−1|ZZhj′,k−1〉

=1

4〈hj,k−1|(L + I)hj′,k−1〉

=k

2〈hj,k−1|hj′,k−1〉 .

From (3.3) we obtain that

(3.9) hj,k(z) = e|z|2

4 ∂kz

(zje−

|z|2

2

),

and, by Leibniz’s formula,

(3.10)

hj,k(z) = e−|z|2

4

min{j,k}∑

`=0

(−1)k−`

(k

`

)j(j − 1) · · · (j − `+ 1)

2k−`zj−`zk−`

=

{zj−kPj,k

(|z|2

)e−

|z|2

4 if j ≥ k ,

zk−jPj,k

(|z|2

)e−

|z|2

4 if j < k ,

where Pj,k is a polynomial of degree equal to min{j, k}.This implies that the linear span of the hj,k contains all functions of the form

Q(z, z)e−|z|2

4 , where Q is an arbitrary polynomial in two variables. Switching back

to real coordinates, we find that for every m,n the function xmyne−x2+y2

4 is in thelinear span of the hj,k.

Assume that f ∈ L2(R2) is orthogonal to all the hj,k. Then

(3.11)

∫

R2

f(x, y)xmyne−x2+y2

4 dx dy = 0 ,

for all m,n. Consider the functions g(x, y) = f(x, y)e−x2+y2

4 and

G(ζ1, ζ2) =

∫

R2

g(x, y)e−i(xζ1+yζ2) dx dy .

Because of the rapid decay of g at infinity due to the Gaussian factor, G isdefined on all of C2, and holomorphic. Hence g = G|

R2, i.e. the Fourier transform

of g, is real-analytic on all R2. Condition (3.11) says that all derivatives of g at theorigin are zero. Hence g = 0, i.e. g = 0 and finally f = 0. �

This shows that L has a discrete spectrum, the eigenvalues being the odd positiveintegers. Combining this with Lemma 3.1 and with Lemma 2.3, we obtain thefollowing description of the spectral measure Eλ of Lλ on R2n.

18 CHAPTER I

Corollary 3.3. Let n = 1. If λ 6= 0, the spectrum of Lλ, as an operator on L2(R2),is

{|λ|(2k + 1) : k ∈ N

}.

If λ > 0, Eλ

({λ(2k + 1)}

)is the orthogonal projection onto span

{hj,k

(√λ z

):

j ∈ N}.

If λ < 0, Eλ

({|λ|(2k+1)}

)is the orthogonal projection onto span

{hj,k

(√|λ| z

):

j ∈ N}.

For general n, the spectrum of Lλ is{|λ|(2k+ n) : k ∈ N

}. If λ > 0, a complete

orthogonal system of eigenfunctions for the eigenvalue λ(2k + n) is given by theproducts

n∏

i=1

hji,ki

(√λ zi

),

with ji, ki ∈ N and k1 + · · · + kn = k (and similarly for λ < 0, replacing λ by |λ|and zi by zi).

One may observe that the coefficient in (3.10)

(−1)k−`

(k

`

)j(j − 1) · · · (j − `+ 1) =

(−1)k−`k!j!

`!(k − `)!(j − `)!

exhibits a symmetry in j and k, which give the identity

(3.12) hj,k(z) = (−2)j−khk,j(z) = (−2)j−khk,j(z) .

Modulo normalizations, the polynomials Pj,k appearing in (3.10) are the so-called

Laguerre polynomials L(α)m , with α = |j − k| and m = min{j, k}. The Laguerre

polynomial belong to the class of confluent hypergeometric functions, and they aredescribed, e.g., in the book Higher transcendental functions, vol. II, by A. Erdelyi,W. Magnus, F. Oberhettinger, F. Tricomi.

4. The joint spectrum of L and i−1T

We have already observed in Section 2 that

T = ∂t = [Xj, Yj] ,

this last equality holding for every j.Since T has constant coefficients, TLf = LTf for every f ∈ D(R2n+1). It takes

a little thought to realize that, as self-adjoint operators, L and i−1T commute.

Proposition 4.1. L and i−1T commute with each other. The joint spectrum of Land i−1T consist of the pairs

(λ, |λ|(2k + n)

)with λ ∈ R and k ∈ N, and of the

pairs (0, µ) with µ ≥ 0.

Proof. It is convenient to replace the two operators by the unitarily equivalentones obtained by conjugating both of them with (2π)−

12Ft. Instead of L, we then

consider L, introduced in Section 2, and instead of i−1T the multiplication operator

Sf(x, y, λ) = λf(x, y, λ) .

We call E and E′ the spectral measures of L and S respectively. The examplegiven in Section 1 shows that

E′(ω′)f(x, y, λ) = χω′(λ)f(x, y, λ) ,

and, by Proposition 2.2, calling fλ(x, y) = f(x, y, λ),

E(ω)f(x, y, λ) =(Eλ(ω)fλ

)(x, y) .

Then

(4.1) E(ω)E′(ω′)f(x, y, λ) = E′(ω′)E(ω)f(x, y, λ) = χω′(λ)(Eλ(ω)fλ

)(x, y) ,

hence L and i−1T commute.In order to determine the joint spectrum of L and S, we discuss the possibility

that

(4.2) E(ω)E′(ω′) = 0 .

We can take ω, ω′ open intervals; for the moment we assume that 0 6∈ ω′, sayω′ ⊂ R+, to fix the notation.

By (4.1), (4.2) happens if Eλ(ω) = 0 for every λ ∈ ω′. We show that the converseis also true.

Let λ0 ∈ ω′ be such that Eλ0(ω) 6= 0. Then ω contains a point λ0(2k + n) for

some k ∈ N. Then there is δ > 0 such that λ(2k + n) ∈ ω for |λ − λ0| < δ. Letω′′ = (λ0 − δ, λ0 + δ) ⊂ ω′.

Take h(x, y) a non-trivial function in the (2k+n)-eigenspace of L1. By Corollary

3.3, hλ(x, y) = h(√λx,

√λy

)is in the λ(2k+ n)-eigenspace of Lλ for λ > 0, hence

Eλ(ω)hλ = hλ for λ ∈ ω′′.Let f(x, y, λ) = χω′′(λ)hλ(x, y). By (4.1),

E(ω)E′(ω′)f = f ,

contradicting (4.2).This proves that a point (λ, µ) with λ 6= 0 is in the joint spectrum if and only

if µ = |λ|(2k + n) for some k ∈ N. Then the joint spectrum must also contain thepoints in the closure of this set, i.e. the points (0, µ) with µ ≥ 0.

On the other hand E is zero on the negative half-line, and this concludes theproof. �

The joint spectrum of L and i−1T is called the Heisenberg fan.

20 CHAPTER I

MIHLIN-HORMANDER MULTIPLIERS 21

CHAPTER II

MIHLIN-HORMANDER MULTIPLIERS

FOR CONSTANT COEFFICIENT OPERATORS

1. Spectral and Fourier Lp-multipliers

Definition. Let {T1, . . . , Tn} be a commuting system self-adjoint operators, actingon L2(X,µ). If 1 ≤ p ≤ ∞, we say that a measurable, a.e. finite, function m onRn is a spectral Lp-multiplier if the operator m(T1, . . . , Tn) extends to a boundedoperator from Lp(X,µ) to itself.

In particular, By Proposition 1.4 in Chapter I, m is a spectral L2-multiplier ifand only if it is bounded a.e. with respect to the joint spectral measure of the Tj .

In this chapter we shall take X = Rn, with µ the Lebesgue measure, and weshall discuss spectral multipliers for

(1) the system i−1∂xj, 1 ≤ j ≤ n, i.e. the Fourier multipliers on Rn;

(2) the Laplacian, i.e. the radial Fourier multipliers;(3) other constant coefficient operators, satisfying homogeneity conditions that

will be defined below.

We shall mainly restrict ourselves to p 6= 1,∞. In this section we make somepreliminary considerations, starting with case (1).

Lemma 1.1. If m is a Fourier Lp-multiplier on Rn for some p ∈ (1,∞), thenm ∈ L∞(Rn). The set of points 1/p ∈ (0, 1) such that m is a Fourier Lp-multiplieris an interval, symmetric w.r. to 1/2.

Proof. Assume that m is real and that Sm = m(i−1∂) is bounded on Lp(Rn). ByProposition 1.4 in Chapter I, Sm is self-adjoint. For f, g ∈ D(Rn),

〈Smf |g〉 = 〈f |Smg〉 ,

which implies that Sm is also bounded on the dual space Lp′

(Rn). By the Riesz-Thorin interpolation theorem, Sm is bounded on Lq(Rn) for every q between pand p′. This proves the second part of the statement. In particular Sm is L2-bounded, hence m ∈ L∞(Rn).

If m = m1 + im2 is not real and Sm is Lp-bounded, take f, g ∈ D(Rn) real. Then

∣∣〈Smf |g〉∣∣ ≤

∣∣〈Sm1f |g〉

∣∣ +∣∣〈Sm2

f |g〉∣∣ ≤ C‖f‖p‖g‖p′ ,

and this easily implies that both Sm1and Sm2

are Lp-bounded. �

One next remarks concern homogeneity.

Typeset by AMS-TEX

22 CHAPTER II

For r = (r1, . . . , rn) ∈ (R+)n, define

r · x = (r1x1, . . . , rnxn)

on Rn, and the dilation operator

δrf(x) = fr(x) = f(r1x1, . . . , rnxn) .

It is easy to verify that, if f is in the domain of i−1∂xj, the same is true for fr

and that(i−1∂xj

) ◦ δr = rjδr ◦ (i−1∂xj) .

Hence

i−1∂xj=

∫

Rn

ξj dE(ξ)

= δ−1r ◦

(r−1j

∫

Rn

ξj dE(ξ)

)◦ δr

= δ−1r ◦

( ∫

Rn

ξj dE(r · ξ))◦ δr ,

showing that

(1.1) E(ω) = δ−1r ◦E(r · ω) ◦ δr ,

for every Borel set ω.Hence, for every multiplier m,

(1.2)

Smr=

∫

Rn

m(r · ξ) dE(ξ)

=

∫

Rn

m(ξ) dE(r−1 · ξ)

= δ−1r ◦

( ∫

Rn

m(ξ) dE(ξ)

)◦ δr

= δ−1r ◦ Sm ◦ δr .

For every p, ‖fr‖p = (r1r2 · · · rn)−1/p‖f‖p, so that

‖Smrf‖p = ‖Smf‖p .

This proves the following statement.

Proposition 1.2. If m is a Fourier Lp-multiplier, and r ∈ (R+)n, then mr is alsoa Fourier Lp-multiplier, and the operators Sm and Smr

have the same norm.

Spectral multipliers of the Laplacians correspond to a special subclass of theFourier multipliers. In fact, if m(λ) is Borel measurable on the positive half-line,then

(1.3) m(∆) = m(i−1∂) ,

with m(ξ) = m(|ξ|2

). Observe that if E denotes now the spectral measure of ∆ on

the positive half-line, then (1.1) and (1.2) hold for r = (r, . . . , r), with r ·ω replacedby rω. As a consequence, we have the following analogue of Proposition 1.2.


Proposition 1.3. If m is a spectral Lp-multiplier of ∆, and r > 0, then mr(λ) =m(r−1λ) is also a spectral Lp-multiplier, and the operators Sm and Smr

have thesame norm.

This kind of homogeneity argument is quite general. Staying in the context ofRn, let L = P (i−1∂) be any constant coefficient, self-adjoint differential operator,and assume that there are positive exponents λ1, . . . , λn and k such that

P (rλ1ξ1, . . . , rλnxn) = rkP (x1, . . . , xn) .

The expressions non-isotropic dilations and non-isotropic homogeneity refer tothis more general situation5.

One example is

L = −∂2x1

+

( n∑

j=2

∂2xj

)2

,

corresponding to P (ξ) = ξ21 +

( ∑nj=2 ξ

2j

)2

, with λ1 = 2, λ2 = · · · = λn = 1, and

k = 4.

Again, spectral multipliers m of L correspond to special Fourier multipliers,because (1.3) holds with m(ξ) = m

(P (ξ)

). In this more general case, (1.1) and

(1.2) hold for r = (rd1 , . . . , rdn), with r > 0. Proposition 1.3 holds unchanged.

The rest of the chapter is devoted to the presentation of conditions on the multi-pliers which are invariant under dilations and which assure that they define boundedoperators on Lp for all p ∈ (1,∞). This requires the introduction of the Calderon-Zygmund theory of singular integrals.

2. The Hardy-Littlewood maximal function

Il punto di partenza della teoria di Calderon-Zygmund e l’analisi dell’operatoremassimale di Hardy-Littlewood. In questo paragrafo ne presentiamo gli aspettiche ci consentiranno piu avanti di inglobare in un unico contesto lo studio deimoltiplicatori spettrali di tutti gli operatori differenziali presentati nel Capitolo I.

Sia X un insieme. Una quasi-distanza su X e una funzione d da X×X in R taleche

(1) d(x, y) ≥ 0 per ogni x, y ∈ X;(2) d(x, y) = 0 se e solo se x = y;(3) d(x, y) = d(y, x) per ogni x, y ∈ X;(4) esiste una costante c ≥ 1 tale che

(2.1) d(x, z) ≤ c(d(x, y) + d(y, z)

)

per ogni x, y, z ∈ X.

5Despite the name, the isotropic case, λ1 = · · · = λn, is also included.

24 CHAPTER II

Una quasi-distanza induce in modo naturale una topologia su X, una cui basee costituita dalle palle B(x, r) = {y : d(x, y) < r}. Sia ora m una misura di Borelpositiva su X. Si dice che m e doubling se esiste una costante c′ tale che

(2.2) m(B(x, 2r)

)≤ c′m

(B(x, r)

)

per ogni x ∈ X e r > 0.

Definizione. Una terna (X, d,m), dove d e una una quasi-distanza su X e m euna misura doubling, si dice uno spazio di natura omogenea.

Esempi.

(1) Ovviamente Rn, con la distanza Euclidea e la misura di Lebesgue, e dinatura omogenea.

(2) Anche Z, con la distanza d(n,m) = |n − m| e la misura del conteggiom(E) = cardE, e di natura omogenea.

(3) Sia α > −n. Allora R, con la distanza Euclidea e la misura dm(x) = |x|αdx,e di natura omogenea.

(4) Si prenda X = Rn, e siano d1, . . . , dn > 0. Si ponga

d(x, y) = max{|x1 − y1|1/d1 , . . . , |xn − yn|1/dn

}.

Allora d e una quasi-distanza e, con la misura di Lebesgue, Rn e unospazio di natura omogenea.

(5) La sfera unitaria Sn−1, dotata della distanza indotta da Rn e della misuradi Hausdorff σ, e uno spazio di natura omogenea.

Sia X uno spazio di natura omogenea.

Definizione. Sia f localmente integrabile rispetto alla misura m. La funzione

(2.3) Mf(x) = supx∈B

1

m(B)

∫

B

|f(y)| dy ,

si chiama funzione massimale di Hardy-Littlewood di f e l’operatore M : f 7−→Mfoperatore massimale di Hardy-Littlewood.

Chiaramente M non e lineare (si osservi che Mf(x) ≥ 0 per ogni f), ma solosub-lineare, nel senso che

(2.4) M(f + g) ≤Mf +Mg , M(λf) = |λ|Mf .

Lemma 2.1. La funzione Mf e semicontinua inferiormente, e dunque misurabile.

Proof. Sia M(x0) > α. Esiste allora una palla B contenente x tale che

1

m(B)

∫

B

|f(y)| dy > α .

Ma allora per ogni x ∈ B Mf(x) > α. �

Osservazione. La definizione classica di funzione massimale di Hardy-Littlewood(nel contesto X = Rn, con distanza euclidea e misura di Lebesgue) e la seguente:

(2.5) M ′f(x) = supr>0

1

|B(x, r)|

∫

B(x,r)

|f(y)| dy ,

limitandosi quindi a considerare le medie di |f | sulle palle centrare in x. Chiara-mente M ′f(x) ≤Mf(x); tuttavia M ′f non e necessariamente semicontinua inferi-ormente. La misurabilita di M ′f segue dal fatto che la funzione

F (x, r) =1

|B(x, r)|

∫

B(x,r)

|f(y)| dy

e continua in r, per cui l’estremo superiore in (2.5) puo essere ristretto a r ∈ Q.In un generico spazio di natura omogenea, nulla assicura che F (x, r) sia continua

in r, per cui e preferibile ricorrere alla (2.3).

Ci interessa ora discutere la limitatezza di M sugli spazi Lp(X), ossia la validitadi disuguaglianze del tipo

‖Mf‖p ≤ C‖f‖p .

Si noti che dalle (2.4) segue che |Mf −Mg| ≤ M(f −g), e dunque la limitatezzasu Lp equivale alla continuita, come per gli operatori lineari.

Ovviamente M e limitato su L∞(X). All’altro estremo, per p = 1, non si halimitatezza in generale. Per esempio, nella situazione classica (X = Rn ecc.), se fe la funzione caratteristica della palla unitaria, si vede facilmente che

Mf(x) ≥ C

1 + |x|n ,

per cui Mf 6∈ L1.Tuttavia, M risulta limitato in un senso piu debole, fatto che costituisce il punto

fondamentale della teoria di Calderon-Zygmund.

Definizione. Sia T un operatore sub-lineare definito da Lp(X) a Lq(X) (con 1 ≤p, q < ∞) a valori nelle funzioni misurabili su X. Si dice che T e di tipo debole(p, q), se per ogni α > 0

(15.3) |{x : Mf(x) > α}| ≤ C

(‖f‖p

α

)q

.

Si noti che, se T e limitato da Lp(X) a Lq(X), allora T e di tipo debole (p, q)per la disuguaglianza di Chebishev.

Dimostreremo per prima cosa che M e di tipo debole (1, 1), basandoci sulseguente lemma di ricoprimento di Vitali.

Lemma 2.2. In uno spazio di natura omogenea X, sia {Bj}j∈J un ricoprimentofinito di un insieme misurabile E mediante palle. Esiste allora una sottofamiglia{Bj}j∈J ′ tale che Bj ∩Bk = ∅ per j, k ∈ J ′, j 6= k, e inoltre

∣∣∣∣⋃

j∈J ′

Bj

∣∣∣∣ ≥ κ|E| ,

26 CHAPTER II

dove κ dipende solo dalle costanti c, c′ in (2.1), (2.2).

Proof. Sia Bj1 una palla che abbia misura massima. Induttivamente si prendaBjk+1

in modo che abbia misura massima tra le palle disgiunte da Bj1 ∪ · · · ∪ Bjk.

Ovviamente il procedimento si arresta dopo un numero finito di passi, precisamentequando non ci sono piu palle disgiunte da Bj1∪· · ·∪Bjk

. Poniamo J ′ = {j1, . . . , jk}.Data una palla B di raggio r, sia B∗ la palla con lo stesso centro e raggio 2cr,

dove c e la costante che appare nella (2.1). Si noti che, se due palle B,B ′ hannointersezione non vuota e il raggio di B′ e minore o uguale al raggio di B, alloraB′ ⊆ B∗.

Sia B′ una delle palle avanzate. Necessariamente essa interseca almeno una diquelle selezionate. Sia ¯ il primo intero ` tale che B′ ∩ Bj`

6= ∅. Allora il raggio diBj¯ e maggiore o uguale al raggio di B′, e dunque

B′ ⊆ B∗j¯.

Di conseguenza

E ⊆⋃

j∈J

Bj ⊆⋃

j∈J ′

B∗j ,

Se 2ν ≥ 2c, si ha allora

m(B∗j ) ≤ m

(B(x, 2kr)

)≤ c′

νm(B) .

per cui, con κ = c′−ν

,

|E| ≤∑

j∈J ′

|B∗j | = κ−1

∑

j∈J ′

|Bj| = κ−1

∣∣∣∣∣∣⋃

j∈J ′

Bj

∣∣∣∣∣∣. �

Teorema 2.3. L’operatore M e di tipo debole (1, 1).

Proof. Data f ∈ L1(X) e dato α > 0, sia Eα = {x : Mf(x) > α}. Per il Lemma 2.1,Eα e aperto e la sua misura e l’estremo superiore delle misure dei suoi sottoinsiemicompatti.

Sia E un sottoinsieme compatto di Eα. Preso x ∈ E, essendo Mf(x) > α, esisteuna palla Bx contenente x tale che

1

|Bx|

∫

Bx

|f(y)| dy > α ,

ossia

|Bx| ≤1

α

∫

Bx

|f(y)| dy .

Per la compattezza di E, si puo estrarre da {Bx} un sottoricoprimento finito,e quindi, applicando il Lemma 2.2, un sottoinsieme finito {Bxj

} di palle disgiuntetali che

∑j |Bxj

| ≥ κ|E|. Si ha allora

|E| ≤ κ−1∑

j

|Bxj| ≤ κ−1

α

∑

j

∫

Bxj

|f(y)| dy ≤ κ−1 ‖f‖1

α.

Di conseguenza |Eα| ≤ κ−1‖f‖1/α. �

Avendo a disposizione la limitatezza su L∞ e il tipo debole (1,1), possiamo ap-plicare il seguente Teorema di interpolazione di Marcinkiewicz, la cui dimostrazionesi trova su E.M. Stein, G. Weiss, An introduction to Fourier analysis on Euclideanspaces.

Teorema 2.4. Sia T un operatore sub-lineare che sia di tipo debole (p0, q0) e ditipo debole (p1, q1), con 1 ≤ p0 ≤ q0, 1 ≤ p1 ≤ q1, p0 6= p1, q0 6= q1

Allora, dato t ∈ [0, 1] e posto

1

pt=

1 − t

p0+

t

p1

1

qt=

1 − t

q0+

t

q1,

T e di tipo forte (pt, qt).La stessa tesi vale se uno o entrambi dei qj e infinito, e T e limitato da Lpj in

Lqj .

Applicando questo teorema con p0 = q0 = 1 e p1 = q1 = ∞, si ottiene immedi-atamente il seguente risultato.

Corollario 2.5. Se 1 < p ≤ ∞, M e limitato su Lp(X).

3. Calderon-Zygmund operators on spaces of homogeneous type

Gli operatori di Calderon-Zygmund (o a integrali singolari) costituiscono unaclasse piu ampia di quella degli operatori integrali. In questo paragrafo essi ven-gono presentati su generici spazi di natura omogenea; in questa generalita la lorodefinizione risulta alquanto implicita, mentre nei contesti piu comuni, come Rn o levarieta differenziabili, la teoria delle distribuzioni ne consente una descrizione piuprecisa.

Cominciamo con il considerare, dato uno spazio di natura omogenea X, un op-eratore lineare T con le seguenti proprieta:

(1) e definito sullo spazio Cc(X) delle funzioni continue a supporto compatto,o su un suo sottospazio denso V ;

(2) e a valori in L1loc(X);

(3) esiste una funzione K(x, y), localmente integrabile su (X×X)\{diagonale},tale che, se f, g ∈ V e supp f ∩ supp g = ∅, allora

(3.1)

∫

X

(Tf)(x)g(x) dx =

∫∫

X×X

K(x, y)f(y)g(x) dydx ,

o, equivalentemente,

(3.2) Tf(x) =

∫

X

K(x, y)f(y) dy

per quasi ogni x 6∈ supp f .

Si noti che K non determina univocamente T : se, per esempio, Tf = ϕf , letre proprieta sono soddifatte con K = 0, indipendentemente da ϕ. Infatti, ilnucleo K non descrive completamente l’operatore, perche non dice quanto valga∫

X(Tf)(x)g(x) dx se i supporti di f e g hanno intersezione non vuota.

Definizione. Si chiama operatore di Calderon-Zygmund un operatore lineare Tche

(i) sia limitato su Lq(X) per qualche q ∈ (1,∞);

28 CHAPTER II

(ii) sia soddisfatta la condizione (3), ed esista una costante C > 0 tale che perogni coppia di punti distinti y, y′ di X

(3.3)

∫

{x:d(x,y)>4cd(y,y′)}

∣∣K(x, y)−K(x, y′)∣∣ dx < C ,

dove c indica la costante nella disuguaglianza triangolare (2.1).

La condizione (3.3) si chiama condizione di Calderon-Zygmund6.

La (3.3) va interpretata come una forma integrale di regolarita. Una condizionedi carattere puntuale che implica la (3.3) e la seguente: indichiamo con v(x, y) il vol-ume della palla di centro x e raggio r = d(x, y), e supponiamo che per qualche α > 0e per ogni terna di punti x, y, y′ con d(x, y) > 4cd(y, y′) valga la disuguaglianza

(3.4)∣∣K(x, y)−K(x, y′)

∣∣ ≤ Cd(y, y′)α

v(x, y)d(x, y)α,

(che appare come una condizione di Lipschitz7 in y, con una costante che dipendedalla distanza da x).

Allora, posto Ej = {x : 4c2jd(y, y′) ≤ d(x, y) < 4c2j+1d(y, y′)},∫


∣∣K(x, y)−K(x, y′)∣∣ dx ≤

≤ Cd(y, y′)α

∫


1

v(x, y)d(x, y)αdx

≤∞∑

j=0

Cd(y, y′)α

∫

Ej

1

v(x, y)d(x, y)αdx

≤∞∑

j=0

C ′2−αj

∫

Ej

1

v(x, y)dx .

Se x ∈ Ej, posto d = d(y, y′) e r = d(x, y), allora B(x, r) ⊇ B(x, 4c2jd), per cuiv(x, y) ≥ m

(B(x, 4c2jd)

). D’altra parte,

m(Ej) = m(B(x, 4c2j+1d)

)−m

(B(x, 4c2jd)

)≤ (c′ − 1)m

(B(x, 4c2jd)

),

per la (2.2). In conclusione,

∫


∣∣K(x, y)−K(x, y′)∣∣ dx ≤

∞∑

j=0

C ′′2−αj ,

il che fornisce la condizione di Calderon-Zygmund.

Dimostreremo ora che gli operatori di Calderon-Zygmund sono di tipo debole(1,1). La dimostrazione richiede due strumenti. Il primo e il lemma di ricoprimentodi Whitney per spazi di natura omogenea.

6Il coefficiente 4 nella (3.3) puo essere sostituito da un qualunque numero maggiore di 1.7Qui e nel seguito, diremo “lipschitziana di ordine α” in luogo della dizione piu comune

“holderiana di ordine α”.

Lemma 3.1. Sia F un chiuso non vuoto di X, e sia A il suo complementare.Esistono costanti 1 < k < k′, indipendenti da F , e una famiglia numerabile di palleBj = B(xj , rj) ⊂ A tali che

(i) le palle Bj sono a due a due disgiunte;(ii) l’unione delle palle B∗

j = B(xj, krj) e uguale a A;(iii) ogni palla B∗∗

j = B(xj, k′rj) ha intersezione non vuota con F .

Proof. Sia c la costante nella (2.1). Per ogni x ∈ A, sia dx = d(x, F ) e si prendaBx = B(x, δdx), con δ < 1 da determinarsi. Si scelga quindi una famiglia {Bj =Bxj

}j∈J di tali palle che sia massimale rispetto alla proprieta di essere a due a duedisgiunte.

La famiglia {Bj} e numerabile. Cio segue dal fatto che, fissati x0 ∈ X e unintero n, le palle Bj di misura maggiore di 1/n e contenute in B(x0, n) possonoessere solo in numero finito. Si noti anche che se una palla avesse misura nulla,tutto X avrebbe misura nulla per la (2.2).

La (i) e dunque verificata. Per la (iii), basta prendere B∗∗j = B(xj, 2dxj

). Pas-siamo quindi alla (ii).

Si considerino le palle B∗j = B(xj, dxj

/2). Esse sono chiaramente contenute in

A. Sia ora x ∈ A. Per la massimalita della famiglia {Bj}, la palla Bx interseca unadelle Bj . Vogliamo mostrare che x ∈ B∗

j , se δ e stato scelto opportunamente. Siay un punto in Bx ∩Bj e sia z ∈ F tale che d(xj , z) < 2dxj

. Allora

dx ≤ d(x, z)

≤ c(d(x, y) + d(y, z)

)≤ c2

(d(x, y) + d(y, xj) + d(xj , z)

)

< c2(δdx + δdxj

+ 2dxj

).

Quindi, se δc2 < 1,

dx <(δ + 2)c2

1 − δc2dxj

= σdxj.

Ora

d(x, xj) ≤ c(d(x, y) + d(y, xj)

)

< cδ(dx + dxj)

< cδ(1 + σ)dxj.

Si tratta ora di prendere δ tale che

{δ < 1

c2

δ(1 + (δ+2)c2

1−δc2

)< 1

2 ,

cioe δ < 1/(2 + 5c2). �

Il secondo risultato e la decomposizione di Calderon-Zygmund. Per semplicitasupponiamo che m(X) = ∞.

30 CHAPTER II

Lemma 3.2. Sia f ∈ L1(X,m) e sia α > 0. E possibile decomporre f come

f(x) = g(x) +∞∑

j=0

bj(x)

in modo che

(i) |g(x)| ≤ α;(ii) le funzioni bj hanno supporto in palle B′

j e sono tali che

1

m(B′j)

∫|bj(x)| dm(x) ≤ α ,

∫bj(x) dx = 0 ;

(iii)∑

j m(B′j) ≤ C

α ‖f‖1.

Proof. Sia A = {x : Mf(x) > κα}, con κ da determinarsi, e si costruiscano le palleBj come nel Lemma 2.1. Si ponga quindi

Q0 = B∗0 \

⋃

`≥1

B`

Q1 = B∗1 \

Q0 ∪

⋃

`≥2

B`

. . .

Qj = B∗j \

⋃

`<j

Q` ∪⋃

`>j

B`

.

I Qj danno una partizione di A e Bj ⊂ Qj ⊂ B∗j . Quindi

1

m(Qj)

∫

Qj

|f(x)| dm(x) ≤ 1

m(Bj)

∫

B∗j

|f(x)| dm(x)

≤ c′

m(B∗j )

∫

B∗j

|f(x)| dm(x)

≤ c′M1f(xj)

≤ c′κα ,

se c′ e la costante nella (2.2) e xj ∈ B∗j ⊂ A.

In particolare, se

βj =1

m(Qj)

∫

Qj

f(x) dm(x) ,

risulta|βj | ≤ c′κα .

Si ponga allora

g = fχX\A

+∑

j

βjχQj.


Poiche |f(x)| ≤ Mf(x) quasi ovunque, risulta

|g(x)| ≤ c′κα

quasi ovunque.Si ponga ora

bj = (f − βj)χQj.

Allora bj ha supporto nella palla B∗j , ha integrale nullo e

1

m(B∗j )

∫|bj(x)| dm(x) ≤ 1

m(B∗j )

∫

Qj

|f(x)| dm(x) +m(Qj)

m(B∗j )

|βj |

≤ (1 + c′)κα .

Quindi se κ = 1/(1+c′) e B′j = B∗

j , le condizioni (i) e (ii) sono verificate. Quantoalla (iii), abbiamo

∑

j

m(B∗j ) ≤ c′

∑

j

m(Bj) ≤ c′m(A) ≤ C‖f‖1

α

perche M e di tipo debole (1,1). �

Teorema 3.3. Un operatore T di Calderon-Zygmund e di tipo debole (1, 1).

Proof. Si supponga T limitato su Lq(X), e sia f ∈ L1. Dato α > 0, si consideri ladecomposizione di Calderon-Zygmund

f(x) = g(x) +∞∑

j=0

bj(x) ,

come dal Lemma 3.2, corrispondente al valore di α fissato. Se b(x) =∑

j bj(x), siha

m({x : |Tf(x)| > 2α}

)≤ m

({x : |Tg(x)| > α}

)+m

({x : |Tb(x)| > α}

).

Osserviamo ora che g ∈ Lq; piu precisamente, segue da (ii) e (iii) che

∑

j

‖bj‖1 ≤ C‖f‖1 ,

per cui anche ‖g‖1 ≤ C‖f‖1. Usando quindi anche (i), si ha

‖g‖qq ≤ Cαq−1‖f‖1

Poiche T e limitato su Lq(X) e per la disuguaglianza di Chebishev,

m({x : |Tg(x)| > α}

)≤

‖Tg‖qq

αq

≤ C‖g‖q

q

αq

≤ C‖f‖1

α.

32 CHAPTER II

Passiamo ora a Tb. Sia B′j = B(xj, rj). Poniamo B

′′

j = B(xj, 4crj). Se x 6∈ B′j ,

si ha

Tbj(x) =

∫

B′j

K(x, y)bj(y) dy =

∫

B′j

(K(x, y)−K(x, xj)

)bj(y) dy ,

in quanto bj ha integrale nullo. Allora

∫

X\B′′j

|Tbj(x)| dx ≤∫

X\B′′j

∫

B′j

∣∣K(x, y)−K(x, xj)∣∣|bj(y)| dy dx

=

∫

B′j

|bj(y)|∫

X\B′′j

∣∣K(x, y)−K(x, xj)∣∣ dx dy

≤∫

B′j

|bj(y)|∫

{x:d(x,xj)>4cd(y,xj)}

∣∣K(x, y)−K(x, xj)∣∣ dx dy

≤ C

∫

B′j

|bj(y)| dy

≤ Cα|B′j | .

Di conseguenza

∫

X\S

j B′′j

|Tb(x)| dx ≤ Cα∑

j

|B′j| ≤ C‖f‖1 .

Per la disuguaglianza di Chebishev,

∣∣{x 6∈⋃

j

B′′

j : |Tb(x)| > α}∣∣ ≤ C

‖f‖1

α.

Rimane da considerare la misura dell’insieme {x ∈ ⋃j B

′′

j : |Tb(x)| > α}. Maquesta e sicuramente minore o uguale a

m( ⋃

j

B′′

j

)≤

∑

j

m(B′′

j ) ≤ C∑

j

m(B′j) ≤ C

‖f‖1

α. �

Corollario 3.4. Un operatore di Calderon-Zygmund limitato su Lq(X) e anche

limitato su Lp(X) se 1 < p ≤ q. Se anche k∗(x, y) = k(y, x) soddisfa la condizionedi Calderon-Zygmund (3.3), allora T e limitato su Lp(X) per ogni p ∈ (1,∞).

Proof. La prima parte dell’enunciato segue direttamente dal Teorema di interpo-lazione di Marcinkiewicz. Se q = ∞ non c’e altro da dimostrare.

Se q < ∞ e k∗ soddisfa la (3.3), allora T ∗, che e limitato su Lq′

(X), e pure unoperatore di Calderon-Zygmund, e dunque e limitato su Lr(X) per 1 < r ≤ q′. Maallora T e limitato su Lp(X) per q ≤ p <∞. �

4. Integral Lipschitz conditions

Lasciamo per questo paragrafo gli spazi di natura omogenea, e introduciamoalcune nozioni preliminari al seguito del capitolo. Indicheremo con ‖x‖ la normaeuclidea di x ∈ Rn.

Si dice che una funzione f ∈ Lp(Rn) soddisfa una condizione Lp-Lipschitz diordine α ∈ (0, 1), o che f ∈ Λα

p (Rn), se esiste una costante c tale che

(4.3)

( ∫|f(x− h) − f(x)|p dx

)1/p

≤ c‖h‖α

per ogni h ∈ Rn. Si pone in tal caso

(4.4) ‖f‖Λαp

= ‖f‖p + suph6=0

‖h‖−α

( ∫|f(x− h) − f(x)|p dx

)1/p

.

La norma (4.4) puo essere sostituita da ciascuna delle norme equivalenti

(4.5) ‖f‖′Λαp

= ‖f‖p + sup0<‖h‖<a

‖h‖−α

( ∫|f(x− h) − f(x)|p dx

)1/p

.

Infatti basta maggiorare, per ‖h‖ ≥ a,

‖h‖−α

( ∫|f(x− h) − f(x)|p dx

)1/p

≤ 2a−α‖f‖p .

Le condizioni di Lipschitz integrali sono localmente piu deboli delle ordinariecondizioni di Lipschitz. Per esempio, sia f(x) = ‖x‖−n

p +αϕ(x), dove 0 < α < 1 eϕ e una funzione C∞ a supporto compatto uguale a 1 in un intorno di 0. Allora

( ∫ ∣∣‖x− h‖− np +αϕ(x− h) − ‖x‖−n

p +αϕ(x)∣∣p dx

)1/p

≤(∫

‖x‖>2‖h‖

∣∣‖x− h‖− np +αϕ(x− h) − ‖x‖−n

p +αϕ(x)∣∣p dx

)1/p

+ 2

(∫

‖x‖<3‖h‖

‖x‖−n+αp dx

)1/p

≤ C‖h‖( ∫

‖x‖>2‖h‖

‖x‖−n+αp−p dx

)1/p

+ C

( ∫ 3‖h‖

0

rαp−1 dr

)1/p

= C|h|( ∫ ∞

2‖h‖

rαp−p−1 dr

)1/p

+ C

( ∫ 3‖h‖

0

rαp−1 dr

)1/p

= C|h|α .

Dunque f ∈ Λαp (Rn).

34 CHAPTER II

Piu in generale, lo spazio di Besov Bαp,q(R

n) e definito, per α > 0 e 1 ≤ p, q ≤ ∞,come lo spazio delle funzioni f ∈ Lp tali che

‖f‖Bαp,q

=

{‖f‖p +

(∫Rn (‖h‖−α‖τhf − f‖p)

q dh‖h‖n

)1/q

se q <∞‖f‖p + suph6=0 ‖h‖−α‖τhf − f‖p se q = ∞ ,

dove τhf(x) = f(x− h). Ovviamente Λαp (Rn) = Bα

p,∞(Rn).

Diamo ora due risultati riguardanti Λα1 (Rn) e che useremo nel prossimo para-

grafo.

Lemma 4.1. Se f ∈ Λα1 (Rn), allora

|f(ξ)| ≤ C‖f‖Λα1(1 + ‖ξ‖)−α .

Proof. Si osservi che∫f

(x− π

ξ

‖ξ‖2

)e−ix·ξ dx = −

∫f(x)e−ix·ξ dx = −f(ξ) .

Quindi

|f(ξ)| =1

2

∣∣∣∣∫f

(x− π

ξ

‖ξ‖2

)e−ix·ξ dx−

∫f(x)e−ix·ξ dx

∣∣∣∣

≤ 1

2

∫ ∣∣∣∣f(x− π

ξ

‖ξ‖2

)− f(x)

∣∣∣∣ dx

≤ C‖ξ‖−α‖f‖Λα1.

D’altra parte,|f(ξ)| ≤ ‖f‖1 ≤ ‖f‖Λα

1.

Quindi

(1 + ‖ξ‖)α|f(ξ)| ≤ C(1 + ‖ξ‖α)|f(ξ)| ≤ C‖f‖Λα1. �

Lemma 4.2. Se f ∈ Λα1 , allora f ∈ Lp per ogni p < n

n−α e ‖f‖p ≤ Cp‖f‖Λα1.

Proof. Sia ϕ ∈ D(Rn) tale che∫

Rn ϕ = 1 e suppϕ sia contenuto nella palla unitaria.Poniamo

ψ0(x) = ϕ(x) , ψj(x) = 2njϕ(2jx) − 2n(j−1)ϕ(2j−1x) ,

cosı che

f = limj→+∞

f ∗(2njϕ(2j ·)

)=

∞∑

j=0

f ∗ ψj .

Si ha ‖ψj‖1 ≤ 2 e ‖ψj‖∞ ≤ C2nj . Inoltre suppψj ⊂ B(0, 2−(j−1)) e∫

Rn ψj = 0per j ≥ 1. Quindi, se j ≥ 1,

‖f ∗ ψj‖1 =

∫

Rn

∣∣∣∣∫

Rn

(f(x− y) − f(x)

)ψj(y) dy

∣∣∣∣dx

≤∫

Rn

∫

Rn

∣∣f(x− y) − f(x)∣∣|ψj(y)| dx dy

≤ ‖f‖Λα1

∫

Rn

|y|α|ψj(y)| dy

≤ C2−αj‖f‖Λα1.

Inoltre ‖f ∗ ψj‖∞ ≤ C2nj‖f‖1 ≤ C2nj‖f‖Λα1. Quindi

‖f ∗ ψj‖pp ≤ C‖f ∗ ψj‖p−1

∞

∫

Rn

|f ∗ ψj(x)| dx ≤ C2nj(p−1)−αj‖f‖Λα1.

Sommando su j, la serie delle norme di ordine p converge se p < n/(n− α). �

5. Non-isotropic dilations in Rn and Calderon-Zygmund kernels

Per la parte restante di questo capitolo l’insieme ambiente X sara Rn, dotatodella misura di Lebesgue. A completare la struttura di spazio di natura omogenea,prenderemo in esame diverse quasi-distanze (tra cui quella euclidea), associate afamiglie di dilatazioni.

Dati numeri positivi (non necessariamente interi) λ1, . . . , λn, si chiamano di-latazioni non isotropiche di Rn relative agli esponenti λj le trasformazioni lineari

r · x = (rλ1x1, . . . , rλnxn) .

Se Q = λ1 + · · ·+ λn, si ha chiaramente d(r · x) = rQ dx. Il numero Q si chiamala dimensione omogenea di Rn rispetto alle date dilatazioni.

Si chiama norma omogenea associata alle date dilatazioni una funzione continuax 7−→ |x| da Rn a [0,+∞) tale che

(1) |x| = 0 se e solo se x = 0;(2) | − x| = |x| per ogni x;(3) |r · x| = r|x|Un esempio e dato da

|x| = |x1|1/λ1 + · · ·+ |xn|1/λn .

La norma euclidea sara indicata con ‖ ‖.Lemma 5.1. Sia | | una norma omogenea.

(i) Gli insiemi Br = {x : |x| ≤ r} sono compatti.(ii) Esiste una costante c ≥ 1 tale che

|x+ y| ≤ c(|x| + |y|

)

per ogni x, y ∈ Rn.(iii) Se λ′ = mini λi e λ′′ = maxi λi, allora esistono due costanti A,B > 0 tali

che, per ogni x con |x| > 1,

A|x|λ′ ≤ ‖x‖ ≤ B|x|λ′′

.

(iv) Esistono due costanti A′, B′ > 0 tali che, per ogni x con |x| < 1,

A′|x|λ′′ ≤ ‖x‖ ≤ B′|x|λ′

.

(v) Se | |′ e un’altra norma omogenea, allora | | e | |′ sono equivalenti, nel sensoche esistono costanti a, b > 0 tali che

a|x| ≤ |x|′ ≤ b|x|per ogni x ∈ Rn.

36 CHAPTER II

Proof. Sia S la sfera unitaria chiusa nella norma euclidea, e sia m > 0 il minimodella funzione | | su S. Dimostriamo che Bm e contenuto nella palla unitaria chiusa

euclidea B1. Sia x tale che ‖x‖ > 1. Poiche limr→0 r · x = 0 e per la continuita

della norma omogenea, esiste δ < 1 tale che δ · x ∈ B1 e |δ · x| = δ|x| < m. Poichel’applicazione r 7−→ r · x e continua, esiste r ∈ [δ, 1) tale che r · x ∈ S. Allora|r · x| ≥ m, da cui |x| = m/r > m.

Quindi Bm e limitato e dunque compatto. Ma allora Br = (r/m) · Bm e purecompatto per ogni r > 0.

Sia c = max{|x+ y| : x, y ∈ B1}. Dati x, y 6= 0, sia t−1 = |x| + |y| > 0. Allorat · x e t · x sono in B1, per cui

t|x+ y| = |t · (x+ y)| = |t · x+ t · y| ≤ c ,

e questo dimostra la (ii).Per dimostrare la (iii), osserviamo che esistono s, σ > 0 tali che

Bs ⊆ B1 ⊆ Bσ .

Quindi

r · Bs ⊆ Br ⊆ r · Bσ ,

per ogni r > 0. Se r > 1,

r · Bs ⊇ Brλ′s , r ·Bσ ⊆ Brλ′′

σ ,

e da questo segue facilmente la tesi. La (iv) si dimostra in modo analogo.Per la (v), siano a, b rispettivamente il minimo e il massimo di |x|′ sulla sfera

S′ = {x : |x| = 1}. Se x = r · y con y ∈ S ′, allora

a|x| = ar ≤ r|y|′ = |x|′ ≤ rb = b|x| . �

In particolare ogni norma omogenea e equivalente a

|x| =n∑

j=1

|xj|1/λj .

Proposizione 5.2. Sia | | una norma omogenea associata a una famiglia di di-latazioni non isotropiche. Allora Rn, dotato della misura di Lebesgue e della quasi-distanza d(x, y) = |x− y|, e uno spazio di natura omogenea.

Proof. Poiche la quasi-distanza e invariante per traslazioni, basta confrontare lemisure di Br e B2r. Ma m(Br) = crQ, dove c = m(B1), per cui la (2.2) vale conc′ = 2Q. �

Supporremo ora fissata una famiglia di dilatazioni non isotropiche e una cor-rispondente norma omogenea. Gli operatori che discuteremo saranno operatori diconvoluzione Tf = f ∗ k, con k ∈ S ′(Rn).


La (3.1) equivale allora alla condizione che k coincida, su Rn \ {0}, con unafunzione k(x) ∈ L1

loc(Rn \ {0}). In tal caso8

K(x, y) = k(x− y) .

La condizione di Calderon-Zygmund (3.3) equivale a richiedere che per ognih ∈ Rn, h 6= 0,

(5.1)

∫

|x|>4c|h|

∣∣k(x+ h) − k(x)∣∣ dx ≤ C .

Allo stesso modo, la (3.4) equivale a richiedere che k(x) sia lipschitziana9 diordine α > 0 fuori dall’origine e che esista C > 0 tale che per |x| > 4c|h| sia

(5.2)∣∣k(x+ h) − k(x)

∣∣ ≤ C|h|α

|x|Q+α.

Si noti che una distribuzione che soddisfi la condizione (ii) nella definizione dioperatore di Calderon-Zygmund non soddisfa necessariamente la condizione (i): siprenda, per es., come k la funzione identicamente uguale a 1. La (ii) e banalmenteverificata, ma Tf(x) = f ∗ 1(x) =

∫f non e limitato su nessun Lq. Per operatori

di convoluzione, l’ipotesi piu naturale da imporre e la limitatezza su L2(Rn), che

equivale a richiedere che k ∈ L∞(Rn).Chiameremo quindi nucleo di Calderon-Zygmund una distribuzione k ∈ S ′(Rn)

per cui valga la (5.1) e con k ∈ L∞(Rn).

Vedremo ora un procedimento abbastanza generale di costruzione di nuclei diCalderon-Zygmund. La distribuzione k si ottiene come “somma diadica”, a partireda una successione di funzioni integrabili ϕj che, al variare di j ∈ Z, hanno normeuniformemente limitate in qualche Λα

1 (Rn), soddisfano una condizione di decadi-mento all’infinito, e hanno tutte media nulla. Ogni ϕj viene poi dilatata di unfattore 2j .

Dati una funzione f e j ∈ Z, poniamo f (j)(x) = 2−Qjf(2−j · x).Teorema 5.3. Sia {ϕj}j∈Z ⊂ L1(Rn) una famiglia di funzioni tali che esistanocostanti ε > 0, α ∈ (0, 1), C > 0 per cui valgano le seguenti proprieta:

(a)∫|ϕj(x)|(1 + ‖x‖)ε dx ≤ C;

(b)∫ϕj(x) dx = 0;

(c) ‖ϕj‖Λα1≤ C.

Allora la serie∑

j∈Zϕ

(j)j converge in S ′ a un nucleo di Calderon-Zygmund.

Proof. Consideriamo la serie delle trasformate di Fourier

(5.3)∑

j∈Z

ϕ(j)j (ξ) =

∑

j∈Z

ϕj(2j · ξ) ,

8Questa notazione crea un’ambiguita tra la distribuzione k e la funzione k(x). Occorre non

confondere le due entita. Per esempio, quando si parla di trasformata di Fourier di k, questa va

intesa come distribuzione.9Nel senso della quasi-distanza d.

38 CHAPTER II

e dimostriamone la convergenza assoluta.Se |2j · ξ| ≥ 1, usiamo il Lemma 4.1 e il Lemma 5.1 (iii) per ricavare che

∣∣ϕj(2j · ξ)

∣∣ ≤ C(1 + ‖2j · ξ‖

)−α ≤ C ′(1 + |2j · ξ|

)−αλ′

.

Se invece |2j ·ξ| < 1, usando la (b), la disuguaglianza |eit−1| ≤ C|t|ε, e il Lemma5.1 (iv), si ha

∣∣ϕj(2j · ξ)

∣∣ =

∣∣∣∣∫

Rn

ϕj(x)(e−i(2j ·ξ)·x − 1) dx

∣∣∣∣

≤ C‖2j · ξ‖ε

∫

Rn

|ϕj(x)|‖x‖ε dx

≤ C ′|2j · ξ|ελ′

.

Allora

∑

j∈Z

∣∣∣ϕ(j)j (ξ)

∣∣∣ ≤ C|ξ|−αλ′ ∑

j:2j |ξ|≥1

2−αλ′j + C|ξ|ελ′ ∑

j:2j |ξ|<1

2ελ′j ≤ C ′ .

Poniamo dunque

u(ξ) =∑

j∈Z

ϕ(j)j (ξ) ∈ L∞(Rn) .

Per convergenza dominata, data f ∈ S(Rn),

∫

Rn

uf =∑

j∈Z

∫

Rn

ϕ(j)j f ,

cioe u =∑

j∈Zϕ

(j)j nel senso delle distribuzioni.

Se k = F−1u, si ha dunque

k =∑

j∈Z

ϕ(j)j

nel senso delle distribuzioni.Rimane da dimostrare che, fuori dall’origine, k coincide con una funzione che

soddisfa la (5.1).Mostriamo che la serie ∑

j∈Z

ϕ(j)j (x)

converge in L1(K) per ogni compatto K ⊂ Rn \ {0}.Possiamo supporre che K = {x : 2m ≤ |x| ≤ 2m+1}, con m ∈ Z. Cambiando

variabile,

∫

2m≤|x|≤2m+1

∣∣ϕ(j)j (x)

∣∣ dx =

∫

2m−j≤|x|≤2m+1−j

∣∣ϕj(x)∣∣ dx .

Per j ≤ m, usiamo la (a) e il Lemma 5.1 (iii) per ottenere che

∫

2m−j≤|x|≤2m+1−j

∣∣ϕj(x)∣∣ dx ≤ 2(j−m)ελ′

∫

2m−j≤|x|≤2m+1−j

∣∣ϕj(x)∣∣|x|ελ′

dx

≤ 2(j−m)ελ′

∫

2m−j≤|x|≤2m+1−j

∣∣ϕj(x)∣∣‖x‖ε dx

≤ C2(j−m)ελ′

.

Per j > m, usiamo invece la (c) e il Lemma 4.2 per ottenere che

∫

2m−j≤|x|≤2m+1−j

∣∣ϕj(x)∣∣ dx ≤

( ∫

|x|≤2m+1−j

∣∣ϕj(x)∣∣p dx

)1/p

m(B2m+1−j )1/p′

≤ C2(m−j)Q/p′

,

per un opportuno p > 1. Quindi

∑

j∈Z

‖ϕ(j)j ‖L1(K) ≤ C

∑

j≤m

2(j−m)ελ′

+ C∑

j>m

2(m−j)Q/p′

,

dove entrambe le somme convergono.Verifichiamo ora la condizione di Calderon-Zygmund. Sia h ∈ Rn \ {0}, e si

supponga 2m ≤ 4c|h| < 2m+1. Allora

∫

|x|>4c|h|

∣∣k(x+ h) − k(x)∣∣ dx ≤

∑

j∈Z

∫

|x|>2m

∣∣ϕ(j)j (x+ h) − ϕ

(j)j (x)

∣∣ dx

=∑

j∈Z

∫

|y|>2m−j

∣∣ϕj(y + 2−j · h) − ϕj(y)∣∣ dy

≤ 2∑

j<m

∫

|y|> 2m−j

2c

|ϕj(y)| dy+∑

j≥m

∫ ∣∣ϕj(y + 2−j · h) − ϕj(y)∣∣ dy

≤ C∑

j<m

2(j−m)ελ′

∫

|y|> 2m−j

2c

|ϕj(y)||y|ελ′

dy + C∑

j≥m

‖2−j · h‖α

≤ C∑

j<m

2(j−m)ελ′

+ C∑

j≥m

2(m−j)αλ′

.

Nel passaggio alla terza riga sie utilizzato il fatto che

2m−j < |y| ≤ c(|y + 2−j · h| + 2−j |h|

)< c

(|y + 2−j · h| + 2m−j

2c

),

da cui segue che

|y + 2−j · h| > 2m−j

2c. �

40 CHAPTER II

6. Mihlin-Hormander conditions on Fourier multipliers

Si consideri una famiglia di dilatazioni non isotropiche in Rn. La condizione diMihlin-Hormander su un moltiplicatore di Fourier m(ξ) ha le seguenti caratteris-tiche:

(1) e invariante rispetto alle dilatazioni fissate, nel senso che se m(ξ) la soddisfa,anche m(r · ξ) la soddisfa per ogni r > 0;

(2) implica che la distribuzione k = F−1m e un nucleo di Calderon-Zygmundadattato alle dilatazioni fissate.

Di conseguenza, l’operatore Smf = m(i−1∂)f = f ∗ k e limitato su Lp(Rn) perogni p ∈ (1,∞).

Premettiamo la definizione e alcune proprieta degli spazi di Sobolev con espo-nente frazionario.

Definizione. Lo spazio di Sobolev Hs(Rn), con s ∈ R, consiste delle distribuzioni

f ∈ S ′(Rn) tali che f e una funzione localmente integrabile e

‖f‖2Hs =

∫

Rn

|f(τ)|2(1 + ‖τ‖2

)sdτ <∞ .

Vale l’inclusione Hs(Rn) ⊂ Ht(Rn) per s > t. In particolare, Hs(Rn) ⊂H0(Rn) = L2(Rn) per s > 0.

Lemma 6.1. Se s ∈ N, Hs(Rn) coincide con lo spazio delle funzioni f ∈ L2(Rn)tali che ∂αf ∈ L2(Rn) per ogni multiindice α con |α| ≤ s.

Tralasciamo la dimostrazione, che segue facilmente dalla formula di Plancherel.

Lemma 6.2. Sia ϕ tale che ϕ ∈ Hs(Rn) con s > n2 , allora ϕ ∈ L1(Rn). Inoltre

per ogni ε < s− n2

∫

Rn

|ϕ(x)|(1 + ‖x‖

)εdx ≤ Cε‖ϕ‖Hs .

Proof. Per la disuguaglianza di Holder,

∫

Rn

|ϕ(x)|(1 + ‖x‖)ε dx ≤∫

Rn

|ϕ(x)|(1 + ‖x‖2)ε/2 dx

≤ C

(∫

Rn

|ϕ(x)|2(1 + ‖x‖2)s dx

)1/2 (∫

Rn

1

(1 + ‖x‖2)s−εdx

)1/2

≤ Cε‖ϕ‖Hs .

La conclusione segue dal fatto che∫(1 + ‖x‖2)−s+ε dx converge. �

Si osservi che il Lemma 6.2 e il Teorema di Riemann-Lebesgue implicano laimmersione di Sobolev Hs(Rn) ⊆ C0(R

n) per s > n2

e la relativa disuguaglianza

(6.1) ‖f‖∞ ≤ C‖f‖Hs .

Lemma 6.3. Siano f ∈ Hs(Rn) e g ∈ S(Rn). Allora fg ∈ Hs(Rn) e ‖fg‖Hs ≤Cg‖f‖Hs

.

Proof. Poiche f g = (2π)−nf ∗ g, si trova che

∫

Rn

|f g(τ)|2(1 + ‖τ‖2

)sdτ = (2π)−2n

∫

Rn

∣∣∣∣∫

Rn

f(τ − τ ′)g(τ ′) dτ ′∣∣∣∣2(

1 + ‖τ‖2)sdτ

≤ C‖g‖1

∫

Rn

∫

Rn

|f(τ − τ ′)|2|g(τ ′)| dτ ′(1 + ‖τ‖2

)sdτ

≤ C‖g‖1

∫

Rn

|g(τ ′)|∫

Rn

|f(τ)|2(1 + ‖τ + τ ′‖2

)sdτ dτ ′ .

Ma

1 + ‖τ + τ ′‖2 ≤ 1 + 2(‖τ‖2 + ‖τ ′‖2

)≤ 2

(1 + ‖τ‖2

)(1 + ‖τ ′‖2

),

per cui

‖fg‖2Hs ≤ C‖g‖1

∫

Rn

|g(τ ′)|(1 + ‖τ ′‖2

)dτ ′

∫

Rn

|f(τ)|2(1 + ‖τ‖2

)sdτ .

I due termini contenenti g sono finiti, in quanto g ∈ S(Rn), e questo dimostra latesi. �

Siano ora a0 < a1 < b1 < b0 numeri positivi. Indichiamo con η una funzione inD(Rn) tale che10

(i) supp η ⊆ {ξ : a0 ≤ |ξ| ≤ b0},(ii) η(ξ) ≥ 0 per ogni ξ e η(ξ) > 0 per a1 ≤ |ξ| ≤ b1.

Poniamo inoltre mr(ξ) = m(r · ξ).Definizione. Si chiama moltiplicatore di Mihlin-Hormander, adattato alle dilata-zioni fissate, una funzione m(ξ) tale che

supr>0

‖mrη‖Hs = ‖m‖MHs<∞

per qualche s > n2 .

Indicheremo con MHs(Rn) la classe dei moltiplicatori su Rn con ‖m‖MHs<∞.

Si noti che la definizione stessa implica che m e mr hanno la stessa norma MHs

per ogni r > 0. Inoltre, per la (6.1), le norme ‖mrη‖∞ sono uniformemente limitate,da cui segue che m ∈ L∞(Rn).

Per s intero, la condizione puntuale

(6.2)∣∣∂αm(ξ)

∣∣ ≤ C|ξ|−P

λiαi

per ξ 6= 0 e |α| ≤ s implica che m ∈MHs(Rn). Questa e la condizione inizialmenteenunciata da Mihlin (per dilatazioni isotropiche).

10Si noti che nelle condizioni (i) e (ii) si puo sostituire la norma euclidea alla norma omogenea.

42 CHAPTER II

Si puo dimostrare, usando il Lemma 6.3, che scelte diverse di η inducono normeMHs equivalenti, per cui la condizione di Mihlin-Hormander non dipende dallascelta di η. Per i nostri scopi e utile scegliere η in modo che

(6.2)∑

j∈Z

η(2j · ξ) = 1

per ogni ξ 6= 0. Per ottenere una tale η, si parta da una η0 ∈ D(Rn) soddisfacente(i) e (ii) con a1 ≤ 1 e b1 ≥ 2. Posto

η(ξ) =∑

j∈Z

η0(2j · ξ) ,

la funzione η = η0/η soddisfa (i), (ii) e la (6.2).

Teorema 6.4. Sia m un moltiplicatore di Mihlin-Hormander, adattato a una fis-sata famiglia di dilatazioni. Allora l’operatore Sm = m(i−1∂) e limitato su Lp(Rn)per 1 < p <∞.

Proof. Sia mj(ξ) = m(2−j · ξ)η(ξ). Allora

∑

j∈Z

mj(2j · ξ) = m(ξ)

quasi ovunque e nel senso delle distribuzioni. Ne segue che, se poniamo

k = F−1m ,ϕj = F−1mj ,

allorak =

∑

j∈Z

ϕ(j)j

nel senso delle distribuzioni. Mostriamo ora che le ϕj soddisfano le ipotesi (a), (b),(c) del Teorema 5.3.

Poiche ‖mj‖Hs ≤ ‖m‖MHs, la (a) segue dal Lemma 6.2. La (b) segue dal fatto

che mj(0) = 0.Dimostriamo ora la (c) con α = 1, osservando preliminarmente che per ogni

i = 1, . . . , n e per ogni j ∈ Z, ξimj(ξ) ∈ Hs(Rn) e che ‖ξimj‖Hs ≤ C‖m‖MHs.

Infatti, sia ω(ξ) ∈ D(Rn), con ω = 1 sul supporto di η. Allora ξimj = (ξiω)mj , epossiamo dunque applicare il Lemma 6.3.

Usando la disuguaglianza di Holder come nella dimostrazione del Lemma 6.2, siottiene che

∫

Rn

∣∣ϕj(x− h) − ϕj(x)∣∣ dx ≤ C

( ∫

Rn

∣∣ϕj(x− h) − ϕj(x)∣∣2(1 + ‖x‖2

)sdx

) 12

.

Essendo mj a supporto compatto, ϕj e C∞, per cui

∣∣ϕj(x− h) − ϕj(x)∣∣2 =

∣∣∣∣∫ 1

0

h · ∇ϕj(x− th) dt

∣∣∣∣2

≤ ‖h‖2

∫ 1

0

‖∇ϕj(x− th)‖2 dt .

Quindi, poiche ∂xiϕj = iξimj ,

sup0<‖h‖<1

‖h‖−1

∫

Rn

∣∣ϕj(x− h) − ϕj(x)∣∣ dx ≤

≤ sup0<‖h‖<1

‖h‖−1

( ∫

Rn

∣∣ϕj(x− h) − ϕj(x)∣∣2(1 + ‖x‖2

)sdx

) 12

≤ sup0<‖h‖<1

( ∫

Rn

∫ 1

0

‖∇ϕj(x− th)‖2 dt(1 + ‖x‖2

)sdx

) 12

≤ sup0<‖h‖<1

( ∫ 1

0

∫

Rn

‖∇ϕj(x)‖2(1 + ‖x+ th‖2

)sdx dt

) 12

≤ 2

( ∫

Rn

‖∇ϕj(x)‖2(1 + ‖x‖2

)sdx

) 12

= 2

( n∑

i=1

‖ξimj‖2Hs

) 12

≤ C‖m‖MHs.

Usando la norma (4.5) e il fatto che ‖ϕj‖1 ≤ C‖m‖MHsper il Lemma 6.2, si ha

che ‖ϕj‖Λ11≤ C‖m‖MHs

.Applicando allora il Teorema 5.3 si ha la conclusione. �

7. Applications

Results about Lp-boundedness for Fourier multipliers have important conse-quences for differential operator with constant coefficients.

Our first application concerns spectral multipliers. We shall make the followingtwo assumptions on the symbol P (ξ) of the operator L = P (i−1∂):

(1) there are dilations r · ξ = (rλ1ξ1, . . . , rλnξn) such that P is homogeneous of

degree k > 0, i.e. P (r · ξ) = rkP (ξ);(2) P (ξ) > 0 for ξ 6= 0.

Condition (1) is equivalent to saying that each α such that the monomial ξα hasa non-zero coefficient in P has a non-isotropic degree

d(α) =n∑

j=1

λjαj

equal to k.Condition (2) implies that L is self-adjoint and positive with domain D = {f ∈

L2 : Lf ∈ L2}.Theorem 7.1. Let L = P (i−1∂) with P satisfying (1) and (2), and let m(λ) ∈MHs(R) with s > n

2 . Then m(L) is bounded on Lp(Rn) for 1 < p <∞.

Observe that the Mihlin-Hormander condition on the real line makes sense alsofor multipliers defined only on R+ or on R−. Moreover, a multiplier m on the whole

44 CHAPTER II

line satisfies the Mihlin-Hormander condition if and only if both m± = mχR± do.Since the spectrum of L is the positive half line, it is sufficient to assume that m isdefined for λ > 0.

The proof of Theorem 7.1 requires some further remarks on Sobolev spaces andnon-isotropic norms.

Lemma 7.2. Let T be a linear operator, bounded from Hs0(Rn) to Hs0(Rm) andfrom Hs1(Rn) to Hs1(Rm), with 0 ≤ s0 < s1, and let C0, C1 be the correspondingoperator norms. Then, for s0 < s < s1, T is bounded from Hs(Rn) to Hs(Rm). Ifs = (1 − θ)s0 + θs1, then the operator norm of T acting between the Hs spaces is

not larger than C1−θ0 Cθ

1 .

Proof. Consider T ′ = FTF−1. By assumption, for j = 0, 1, T ′ is bounded fromthe weighted L2 spaces L2

sj= L2

(Rn, (1 + ‖τ‖2)sj dτ

)to the same space on Rm,

and we want to prove that it is bounded between the L2s spaces, with the stated

bound on the norm.Like in the proof of the Riesz-Thorin theorem, we use the three-lines theorem.

It is sufficient to prove to prove that, if g, h are continuous functions with compactsupport in Rn and Rm respectively, and ‖g‖L2

s= ‖h‖L2

s= 1, then

(7.1)

∣∣∣∣∫

Rm

(T ′g)(x)h(x)(1 + ‖x‖2

)sdx

∣∣∣∣ ≤ C1−θ0 Cθ

1 .

For z ∈ C, define

gz(x) = g(x)(1 + ‖x‖2

)z, hz(x) = h(x)

(1 + ‖x‖2

)z,

and let

F (z) =

∫

Rm

(T ′gz)(x)hz(x)(1 + ‖x‖2

)s+2zdx .

Since gz, hz have compact support, F is defined and holomorphic in the wholeplane. We restrict F to the vertical strip S where

s0 − s

2≤ <ez ≤ s1 − s

2.

For z ∈ S,

|F (z)| ≤∫

Rm

∣∣(T ′gz)(x)∣∣∣∣hz(x)

∣∣(1 + ‖x‖2)s1

dx

≤ C1‖gz‖L2s1‖hz‖L2

s1.

If g and h are supported on the ball of radius r, using the normalization of gand h in L2

s,

‖gz‖2L2

s1=

∫

‖x‖<r

|g(x)|2(1 + ‖x‖2

)2s1−sdx

≤ (1 + r2)2(s1−s) ,

and similarly for hz. Hence F is bounded on S.

For <ez = σj =sj−s

2 ,

|F (z)| ≤ Cj‖gz‖L2σj‖hz‖L2

σj

= Cj‖g‖L2s‖h‖L2

s

= Cj .

By the three lines theorem, if <ez = (1 − θ)σ0 + θσ1 = 0,

|F (z)| ≤ C1−θ0 Cθ

1 .

For z = 0 this gives (7.1). �

Lemma 7.3. Given a family of dilations on Rn, with homogeneous dimension Q.Let 〈x〉 be a continuous function from Rn to R, homogeneous of degree 1 with respectto the given dilations, and strictly positive for x 6= 0 (e.g. a homogeneous norm11).Let S be the set where 〈x〉 = 1. There is a positive Borel measure σ on S such that

∫

Rn

f(x) dx =

∫ ∞

0

∫

S

f(r · x) dσ(x) rQ−1 dr ,

for every integrable function f .

Proof. If E is a Borel subset of S, let

E] = {r · x : x ∈ E , r ≤ 1} ,

and defineσ(E) = Qm(E]) .

For 0 < a < b, let

Ea,b = {r · x : x ∈ E , a < r ≤ b} = (b ·E]) \ (a ·E]) .

Then

m(Ea,b) =bQ − aQ

Qσ(E) =

∫

E×[a,b]

rQ−1dr dσ .

Standard measure-theoretic arguments give the conclusion. �

Proposition 7.4. Let f ∈ Hs(R), s ≥ 0, be supported on interval [b, 2b], withb > 0. If P satisfies (1), (2), then f ◦ P ∈ Hs(Rn), and

‖f ◦ P‖Hs(Rn) ≤ C(b, P )‖f‖Hs(R) .

Proof. If s = m ∈ N, we use the characterization of Hm as the space of L2 functionswith derivatives in L2 up to order m.

Set 〈ξ〉 = P (ξ)1k . By Lemma 7.3,

‖f ◦ P‖22 = σ(S)

∫ (2b)1/k

b1/k

∣∣f(rk)∣∣2rQ−1 dr = C

∫ 2b

b

∣∣f(r)∣∣2rQ

k −1 dr ≤ CbQk −1‖f‖2

2 .

11The missing hypothesis is that 〈−x〉 = 〈x〉.

46 CHAPTER II

Similar estimates hold for the L2 norms of ∂α(f ◦P ), with |α| ≤ m, by the chainrule and Leibniz’s rule.

Assume now that m < s < m + 1. Let ω be a smooth function on the positivehalf-line, equal to 1 on [b, 2b] and with compact support. Define the operator

Tg(ξ) = (gω)(P (ξ)

),

mapping functions on R to functions on Rn. By the first part of the proof andLemma 6.3,

‖Tg‖Hm(Rn) ≤ C(b, P )‖gω‖Hm(R) ≤ C(b, P, ω)‖g‖Hm(R) ,

and similarly for Hm+1. By Lemma 7.2,

‖Tg‖Hs(Rn) ≤ C(b, P, ω)‖g‖Hs(R) .

If supp f ⊆ [b, 2b], then Tf = f ◦P and does not depend on the choice of ω. �

We can prove now Theorem 7.1.

Proof of Theorem 7.1. Let m(ξ) = m(P (ξ)

). If η satisfies (i) and (ii) of Section 6

on R+, then η ◦ P satisfies the same conditions on Rn. Since

mr(ξ) = m(P (r−1 · ξ)

)= m

(r−1P (ξ)

)= mr

(P (ξ)

),

we have that

‖mr(η ◦ P )‖Hs(Rn) = ‖(mrη) ◦ P‖Hs(Rn) ≤ C‖mrη‖Hs(R) .

Therefore m ∈MHs(Rn). �

Here is a corollary which shows the importance of looking for minimal assump-tions on the multipliers (in terms of the Sobolev spaces they must belong to).

Corollary 7.5. Let γ ∈ R and 1 0,

‖Liγ‖Lp→Lp ≤ Cp,ε

(1 + |γ|

)(n+ε)∣∣ 12−

1p

∣∣.

Proof. We apply Theorem 7.1 to m(λ) = λiγ . If η ∈ D(R+) and r > 0,

‖mrη‖Hs = ‖mη‖Hs .

If s = k ∈ N, ‖mη‖Hk ≤ Ck(1 + |γ|)k, by estimating L2-norms of derivatives.Assume now that k < s < k+1, s = θk+(1− θ)(k+1). Setting u = mη and usingHolder’s inequality,

‖mη‖2Hs =

∫

R

|u(τ)|2(1 + |τ |2)s dτ

≤( ∫

R

|u(τ)|2(1 + |τ |2)k dτ

)θ( ∫

R

|u(τ)|2(1 + |τ |2)k+1 dτ

)1−θ

= ‖mη‖2θHk‖mη‖2(1−θ)

Hk+1

≤ Cs

(1 + |γ|

)2s.

Then ‖m‖MHs≤ Cs(1 + |γ|)s. By Theorem 7.1, ‖Liγ‖Lp→Lp ≤ C(1 + |γ|)s for

every p ∈ (1,+∞). This is not yet the required estimate, but we shall use thispartial result to complete the proof.

It is sufficient to take p > 2, by duality. The Plancherel formula gives that‖Liγ‖L2→L2 = 1. Take p0 > p, and let θ ∈ (0, 1) be such that

1

p=

θ

p0+

1 − θ

2=

1

2− θ

(1

2− 1

p0

).

By the Riesz interpolation theorem,

‖Liγ‖Lp→Lp ≤ Cp(1 + |γ|)sθ = Cp(1 + |γ|)s

12− 1

p12− 1

p0 .

If we let p0 tend to ∞, the exponent decreases to 2s(

12 − 1

p

), and if s tends to

n/2, this quantity dcereases to n(

12 − 1

p

). Given ε > 0, it is then possible to find s

and p0 such that the required estimates holds. �

Our second application concerns a-priori estimates. We keep condition (1) atthe beginning of this Section, and replace (2) by

(2’) P (ξ) 6= 0 for ξ 6= 0.

Theorem 7.6. Assume that P satisfies (1) and (2′), and let α = (α1, . . . , αn) bea multi-index with d(α) ≤ k. Then for every f ∈ S(Rn) and 1 < p <∞,

‖∂αf‖p ≤ Cp

(‖f‖p + ‖Lf‖p

).

Proof. Take ϕ ∈ D(Rn) such that ϕ(ξ) = 1 on some neighborhood of 0. Then

∂αf(ξ) = (iξ)αϕ(ξ)f(ξ) +(iξ)α

(1 − ϕ(ξ)

)

P (ξ)Lf(ξ)

= m1(ξ)f(ξ) +m2(ξ)Lf(ξ) .

The multiplier m1 is in D(Rn), so that u = F−1m1 ∈ S(Rn) and

∥∥F−1(m1f)∥∥

p≤ ‖f ∗ u‖p ≤ ‖u‖1‖f‖p .

We verify now that m2 satisfies (6.2) for arbitrary multi-indices β. Because m2

is smooth, these estimates are trivial when ξ is in the support of ϕ. We can thenrestrict ourselves to ξ large enough so that ϕ(ξ) = 0.

In this region, m2 is homogeneous of degree −k+ d(α) ≤ 0, hence it is bounded.Any derivative ∂βm2 is homogeneous of degree −k + d(α) − d(β) ≤ −d(β). Hence

∣∣∂βm2(ξ)∣∣ ≤ Cβ |ξ|−d(β) .

It follows that ∥∥F−1(m2f)∥∥

p≤ Cp‖f‖p . �

48 CHAPTER II

MARCINKIEWICZ MULTIPLIERS 49

CHAPTER III

LITTLEWOOD-PALEY THEORY

AND MARCINKIEWICZ MULTIPLIERS

1. Square functions

Sia I = [0, 1]. La funzione di Rademacher rn ∈ L2(I) e definita, per n ≥ 0, da

rn(t) = (−1)[2nt] .

In altri termini, decomponendo I nell’unione degli intervalli

[j2−n, (j + 1)2−n] , j = 0, . . . , 2n − 1 ,

rn assume il valore costante (−1)j su ciascun intervallo.

Lemma 1.1. Se k ≥ 1 e 0 < n1 < n2 < · · · < nk, allora

∫ 1

0

rn1(t)rn2

(t) · · · rnk(t) dt = 0 ,

e le funzioni di Rademacher formano un sistema ortonormale, ma non completo,in L2(I).

Proof. Il primo asserto e ovvio per k = 1. Supponiamo dunque k ≥ 2. Su og-nuno degli intervalli [j2−nk−1 , (j + 1)2−nk−1 ] il prodotto rn1

(t)rn2(t) · · · rnk−1

(t) ecostante, mentre rnk

(t) assume i valori ±1 su sottoinsiemi di uguale misura. Quindil’integrale dell’intero prodotto e nullo su ciascuno di tali intervalli.

L’ortonormalita e dunque ovvia. Si osservi infine che la funzione r1r2 e ortogonalea tutte le rn per concludere che il sistema non e completo. �

La rilevanza delle funzioni di Rademacher e dovuta al seguente risultato, notocome teorema di Khintchin.

Teorema 1.2. Sia f(t) =∑∞

n=0 anrn(t) ∈ L2(I). Allora per ogni p <∞, la normadi f in Lp e equivalente alla norma di f in L2, ossia

cp

( ∞∑

n=0

|an|2)1/2

≤ ‖f‖p ≤ Cp

( ∞∑

n=0

|an|2)1/2

.

Typeset by AMS-TEX

50 CHAPTER III

Proof. Supponiamo inizialmente p > 2. Per la disuguaglianza di Holder, ‖f‖2 ≤‖f‖p. Per dimostrare la disuguaglianza opposta, e sufficiente prendere p = 2k ed freale. Si ha

(1.1)

‖f‖2k2k =

∫ 1

0

( ∞∑

n=0

anrn(t)

)2k

dt

=∑

(n1,...,n2k)∈N2k

∫ 1

0

an1an2

· · ·an2krn1

(t)rn2(t) · · · rn2k

(t) dt .

Per il Lemma 1.1, gli addendi non nulli nella (1.1) possono essere solo quelli incui uno stesso indice compare un numero pari di volte. In tal caso l’integrando euna costante. Pertanto

‖f‖2k2k ≤ Ck

∑

n1≤···≤nk

a2n1a2

n2· · ·a2

nk,

dove Ck e un maggiorante del numero di elementi di N2k in cui compaiono ripetutidue volte gli indici n1 ≤ · · · ≤ nk.

Ma allora

‖f‖2k2k ≤ Ck

∑

(n1,...,nk)∈Nk

a2n1a2

n2· · ·a2

nk

= Ck

( ∞∑

n=0

a2n

)k

,

che fornisce la tesi per p > 2.Se 1 < p < 2, dalla disuguaglianza di Holder segue che ‖f‖p ≤ ‖f‖2. Sempre

per la disuguaglianza di Holder, e per la parte precedente della dimostrazione,

‖f‖22 ≤ ‖f‖p‖f‖p′ ≤ Cp′‖f‖p‖f‖2 .

Quindi ‖f‖2 ≤ Cp′‖f‖p.Rimane da considerare il caso p = 1. Procedendo come sopra,

‖f‖24/3 ≤ ‖f‖1‖f‖2 ≤ C‖f‖1‖f‖4/3 ,

da cui ‖f‖4/3 ≤ C‖f‖1. Essendo anche ‖f‖1 ≤ ‖f‖4/3 per la disuguaglianza diHolder, la dimostrazione e completata. �

Corollario 1.3. Sia Tn una successioni di operatori limitati su Lp(X), dove X euno spazio di misura e p < ∞. Se esiste una costante A tale che per ogni sceltapossibile dei segni εn = ±1, risulta

(1.2) ‖∞∑

n=0

εnTn‖pp ≤ A ,

allora vale la maggiorazione

∥∥∥∥( ∞∑

n=0

|Tnf |2)1/2∥∥∥∥

p

≤ CpA‖f‖p .

Proof. Preso t ∈ [0, 1], si consideri l’operatore

Tt =

∞∑

n=0

rn(t)Tn ,

dove rn e l’n-esima funzione di Rademacher. Per ipotesi,

‖Ttf‖pp ≤ Ap‖f‖p

p .

Allora anche

∫ 1

0

∫

X

|Ttf(x)|p dx dt =

∫ 1

0

‖Ttf‖pp dt ≤ Ap‖f‖p

p .

Cambiando ordine di integrazione, si ha internamente

∫ 1

0

|Ttf(x)|p dt =

∫ 1

0

∣∣∣∣∞∑

n=0

rn(t)Tnf(x)

∣∣∣∣p

dt

≥ cp

( ∞∑

n=0

|Tnf(x)|2)p/2

per il Teorema 1.2. Quindi

∫

X

( ∞∑

n=0

|Tnf(x)|2)p/2

dx ≤ c−1p

∫

X

∫ 1

0

|Ttf(x)|p dt dx

≤ c−1p Ap‖f‖p

p ,

come da dimostrarsi. �

Il Corollario 1.3 puo essere visto nel modo seguente. Si consideri lo spazio Lp(`2)costituito dalle successioni F = {fn} di funzioni misurabili su X tali che F (x) ={fn(x)} ∈ `2 per quasi ogni x ∈ X e inoltre

‖F‖Lp(`2) =

( ∫

X

‖F (x)‖p`2 dx

)1/p

< ∞ .

Il Corollario 1.3 afferma che, sotto l’ipotesi (1.2), l’operatore

Tf = {Tnf}

e limitato da Lp a Lp(`2). Per dualita si ha allora il seguente corollario.

Corollario 1.4. Sia Tn una successione di operatori che soddisfino la (1.2), e sia1 < p < ∞. Allora, data F = {fn} ∈ Lp(`2), risulta

∥∥∥∥∞∑

n=0

Tnfn

∥∥∥∥p

≤ CpA‖F‖Lp(`2) .

52 CHAPTER III

Proof. La (1.2) implica la stessa maggiorazione per gli operatori T ∗n e con p′ al

posto di p. Essendo p > 1, p′ e finito. Dunque l’operatore Uf = {T ∗nf} e limitato

da Lp′

a Lp′

(`2). Di conseguenza, U∗ e limitato dal duale di Lp′

(`2) a Lp.Data G ∈ Lp(`2), si ponga

〈F |G〉 =

∫

X

∞∑

n=0

fn(x)gn(x) dx .

Si verifica facilmente che le applicazioni lineari F 7→ 〈F,G〉 sono tutti e soli

i funzionali continui su Lp′

(`2), ossia lo spazio duale di Lp′

(`2) si identifica conLp(`2). Inoltre

〈U∗F |g〉 = 〈F |Ug〉

=∞∑

n=0

∫

X

fn(x)T ∗ng(x)dx

=∞∑

n=0

∫

X

Tnfn(x)g(x)dx

=

∫

X

( ∞∑

n=0

Tnfn

)g(x) dx .

Quindi U∗F =∑∞

n=0 Tnfn, da cui la segue la tesi. �

2. Littlewood-Paley functions

Combining the results in the previous Section with the Calderon-Zygmund the-ory, we shall obtain the basic properties of Littlewood-Paley functions.

on Rn we consider a family of dilations x 7−→ r · rx, and call Q the resultinghomogeneous dimension of Rn. If f is defined on Rn, we set f (j)(x) = 2−Qjf(2−j ·x).

We shall take at various stages functions ψ ∈ S(Rn) satisfying

(2.1) 0 ≤ ψ ∈ D(Rn) and 0 6∈ supp ψ ;

sometimes we shall also impose one of the two following conditions:

∑

j∈Z

ψ(2j · ξ) > 0 , for ξ 6= 0 ,(2.2)

∑

j∈Z

ψ(2j · ξ) = 1 , for ξ 6= 0 ,(2.2’)

Observe that condition (2.2) can be obtained by imposing that ψ(ξ) > 0 for1 ≤ |ξ| ≤ 2, if | | is a homogeneous norm for the given dilations.

There are different ways to obtain a ψ satisfying (2.1) and (2.2’). Starting with ψ0

satisfying (2.1) and (2.2), and denoting by s(ξ) the sum in (2.2), set ψ = F−1(ψ0/s).Another way is to take ϕ ∈ D(Rn) so that ϕ(ξ) = 1 on a neighborhood of 0, and set

ψ(ξ) = ϕ(ξ) − ϕ(2 · ξ). We remark that, conversely, if ψ satisfies (2.1) and (2.2’),the function

(2.3) ϕ(ξ) =

∑

j≥0

ψ(2j · ξ) if ξ 6= 0 ,

1 if ξ = 0 ,

is in D(Rn), ϕ(ξ) = 1 on a neighborhood of 0, and ψ(ξ) = ϕ(ξ) − ϕ(2 · ξ).

Proposition 2.1. Suppose ψ satisfies (2.1) and (2.2′). If f ∈ Lp(Rn), and 1 <p <∞, the series

∑j∈Z

f ∗ ψ(j) converges to f in Lp.

Proof. Let ϕ be the function in (2.3) and let u = F−1ϕ. The u(j) form an approx-imate identity for j → −∞, so that

limj→−∞

f ∗ u(j) = f

in Lp. We prove now that

limj→∞

‖f ∗ u(j)‖p = 0 .

For f continuous with compact support, this follows from

‖f ∗ u(j)‖p ≤ ‖f‖1‖u(j)‖p ≤ C2−Qj/p′

.

For a general f ∈ Lp, given δ > 0, take g continuous with compact support suchthat ‖f − g‖p < δ. If j is large enough, ‖g ∗ u(j)‖p < δ, so that

‖f ∗ u(j)‖p ≤ ‖(f − g) ∗ u(j)‖p + ‖g ∗ u(j)‖p

≤ ‖f − g‖p‖u(j)‖1 + ‖g ∗ u(j)‖p

< 2δ .

Therefore

N∑

j=−M

f ∗ ψ(j) =N∑

j=−M

f ∗ (u(j) − u(j+1))

= f ∗ u(−M) − f ∗ u(N+1) ,

and the conclusion follows. �

If {fj}j∈Z is a sequence of Lp-functions on Rn, we set

∥∥{fj}∥∥

Lp(`2)=

( ∫

Rn

( ∑

j∈Z

|fj(x)|2) p

2

dx

) 1p

,

i.e. the norm in Lp(Rn, `2(Z)

).

54 CHAPTER III

Teorema 2.2. Assume that ψ ∈ S(Rn) satisfies (2.1). Then, if 1 < p < ∞ andf ∈ Lp(Rn),

(2.4) ‖{f ∗ ψ(j)}‖Lp(`2) ≤ C(ψ, p)‖f‖p .

If, in addition, ψ also satisfies (2.2), then

(2.5) ‖f‖p ∼ ‖{f ∗ ψ(j)}‖Lp(`2) ,

i.e. the two norms are equivalent for f ∈ Lp(Rn). In other words, defining theLittlewood-Paley function

Sf(x) =

( ∑

j∈Z

∣∣f ∗ ψ(j)(x)∣∣2

) 12

,

we have ‖Sf‖p ∼ ‖f‖p.

Proof. We apply Corollary 1.3 to the operators Tjf = f ∗ ψ(j). By Theorem 5.3 of

Chapter II, for every choice of the signs εj , the series∑

j∈Zεjψ

(j) converges to a

Calderon-Zygmund kernel, provided ψ satisfies (2.1). Since the constants appearingin the estimates do not depend on the choice of signs,, the operator norms of the∑

j∈ZεjTj are uniformly bounded. This gives (2.4).

Assume now that ψ also satisfies (2.2) and consider the function

(2.6) b(ξ) =∑

j∈Z

ψ(j)(ξ)2 =∑

j∈Z

ψ(2j · ξ)2 ,

By (2.1), the support of ψ is contained in a rim 0 < a ≤ |ξ| ≤ b, so that for everyξ 6= 0 at most N ∼ log2(b/a) terms in the series (2.7) are different from 0. Thisimplies that b(ξ) is smooth for ξ 6= 0. Since b(2 · ξ) = b(ξ),

infξ 6=0

b(ξ) = min1≤|ξ|≤2

b(ξ) > 0 .

Let η ∈ S(Rn) be the function such that

η(ξ) =ψ(ξ)

b(ξ).

Then η too satisfies (2.1), so that, arguing as before, the operators T ′jf = f ∗η(j)

satisfy (1.2). By Corollary 1.4,

∥∥∥∥∑

j∈Z

f ∗ ψ(j) ∗ η(j)

∥∥∥∥p

≤ C‖{f ∗ ψ(j)}‖Lp(`2) .

Consider finally ψ ∗ η. Since

∑

j∈Z

ψ ∗ η(2j · ξ) =∑

j∈Z

ψ(2j · ξ)2b(2j · ξ) = 1 ,

it satisfies (2.2’) together with (2.1), so that

∑

j∈Z

f ∗ (ψ ∗ η)(j) =∑

j∈Z

f ∗ ψ(j) ∗ η(j) = f .

This proves (2.5). �

We shall prove below a multi-parameter version of Theorem 2.2. But beforethat, we present the general aspects of the “multi-parameter theory”.

The Calderon-Zygmund theory is often referred to as the one-parameter singularintegral theory, because the assumptions made on the kernels are adapted to a givenfamily of dilations depending on one parameter r > 0. From the point of view ofFourier multipliers, the same can be said for the Mihlin-Hormander condition.

In the multi-parameter theory (also called product theory) one has a finite familyof spaces Rni , each with its own dilations xi 7−→ r · xi, and on the product RN =Rn1 × · · · × Rnk one defines

(2.7) r(x1, . . . , xk) = (r1 · x1, . . . , rk · xk) ,

for r = (r1, . . . , rk) ∈ (R+)k.The simplest example of an operator arising in the multi-parameter theory is the

convolution operator Tf = f ∗K, where K is the tensor product

(2.8) K(x) = K1(x1) · · ·Kk(xk)

of Calderon-Zygmund kernels on the various Rni . Since each Ki is singular (i.e.non-locally integrable) only at the origin, the product kernel K is singular on theunion of the “coordinate subspaces” xi = 0.

However, we shall not discuss product kernels, but we shall instead restrict our-selves to the Fourier multipliers connected with the product theory, the Marcin-kiewicz multipliers. The simplest example is the product

(2.9) m(ξ) = m1(ξ1) · · ·mk(ξk)

of Mihlin-Hormander multipliers on the Rni .The general product theory is however less trivial than what these examples

maight suggest12. The proofs are based on the one-parameter theory and on Little-wood-Paley decompositions.

On each Rni fix a ψi satisfying (2.1), and, for J = (j1, . . . , jk) ∈ Zk, let

ψ(J)(x) = ψ(j1)1 (x1) · · ·ψ(jk)

k (xk) .

We then construct

Sf(x) =

( ∑

J∈Zk

∣∣f ∗ ψ(J)(x)∣∣2

) 12

,

12In fact, there is no difficulty in proving that operators f 7−→ f ∗ K, with K as in (2.8), or

f 7−→ F−1(mf), with m as in (2.9), are bounded on Lp(RN ) for 1 < p < ∞. It is a general

fact that if Ti is a bounded operator on Lp(Xi, µi), then T = T1 ⊗ · · · ⊗ Tk is bounded onLp(X1×,×Xk, µ1 × · · · × µk).

56 CHAPTER III

Theorem 2.3. For 1 < p < ∞, ‖Sf‖p ≤ C(ψ, p)‖f‖p. If, in addition, the ψi

satisfy (2.2), f and Sf have equivalent Lp-norms.

Proof. The proof is based on an iteration argument. We only discuss the case when(2.2) is satisfied.

For each i and j, let ψ(j)i be the measure on RN obtained by tensoring the

function ψ(j)i (xi) on Rni with the Dirac measure δ0 on the other Rni′ .

Consider the operator T1 mapping f into the sequence {f ∗ ψ(j1)1 }j1∈Z. For a.e.

(x2, . . . , xk), denoting by ∗Rm convolution in Rm, we have

(f ∗RN ψ

(j1)1

)(x) =

(f(·, x2, . . . , xk) ∗Rn1 ψ

(j1)1

)(x1) ,

so that, by Theorem 2.2,

∫

Rn1

( ∑

j1∈Z

∣∣f ∗ ψ(j1)1 (x)

∣∣2) p

2

dx1 ∼∫

Rn1

∣∣f(x)∣∣p dx1 .

Integrating in the other variables, we obtain that T1 is bounded from Lp(RN ) toLp

(RN , `2(Z)

), with equivalence between the norm of f and T1f .

Consider next T2, mapping a sequence {gj1}j1∈Z into the double sequence {gj1 ∗ψ

(j2)2 }(j1,j2)∈Z2 . The same argument as before shows that T2 is bounded from

Lp(RN , `2(Z)

)to Lp

(RN , `2(Z2)

), with equivalence between the norms of {gj1}

and T2

({gj1}

).

Iterating this argument k times and considering the composition TkTk−1 · · ·T1,we obtain that

∫

RN

( ∑

j∈Zk

∣∣f ∗ ψ(j1)1 ∗ · · · ∗ ψ(jk)

k (x)∣∣2

) p2

dx ∼∫

RN

∣∣f(x)∣∣p dx .

But this is the required estimate, because ψ(j1)1 ∗ · · · ∗ ψ(jk)

k = ψ(J). �

3. Marcinkiewicz multipliers

Given s = (s1, . . . , sk) ∈ Rk, we define the product Sobolev space Hs(RN ) as the

space of tempered distributions f such that f is locally integrable and

‖f‖2Hs =

∫

RN

(1 + ‖τ1‖2

)s1 · · ·(1 + ‖τk‖2

)sk |f(τ)|2 dτ < +∞ ,

where, as before, τi ∈ Rni and∑k

i=1 ni = N . Lemmas 6.1 and 6.2 of Chapter IIhave the following analogues.

Lemma 3.1. If si ∈ N for every i, Hs(RN ) consists of the L2-functions f suchthat ∂α1

x1· · ·∂αk

xkf ∈ L2(RN ) for every choice of the multi-indices αi with |αi| ≤ si

for every i.

Lemma 3.2. Let ϕ be such that ϕ ∈ Hs(RN) with si >ni

2 for every i. Then

ϕ ∈ L1(RN ) and, if 0 ≤ ε < si − ni

2 for every i,

∫

RN

|ϕ(x)|(1 + ‖x1‖

)ε · · ·(1 + ‖xk‖

)εdx ≤ Cε‖ϕ‖Hs .

In particular, under these hypotheses on s, Hs(Rn) ⊂ C0(Rn).

The proofs are direct adaptations of those given in Chapter II.On each Rni we fix a homogeneous norm (denoted by | | for each i), and a

function ηi ∈ D(Rni) such that

(i) supp ηi ⊆ {ξi : a0 ≤ |ξi| ≤ b0},(ii) ηi(ξi) ≥ 0 for every ξi e ηi(ξi) > 0 for a1 ≤ |ξi| ≤ b1,

with 0 < a0 < a1 < b1 < b0 are given constants.We set η(ξ) = η1(ξ1) · · ·ηk(ξk). For r ∈ (R+)k, we also set mr(ξ) = m(r · ξ),

with the notation of (2.7).

Definition. A Marcinkiewicz multiplier on RN , adapted to the k-parameter dila-tions (2.7), is a function m such that

(3.1) supr∈(R+)k

‖mrη‖Hs = ‖m‖Ms<∞ ,

for some s with si >ni

2 for every i.

We shall denote by Ms(RN ) the class of such multipliers. A simpler pointwisecondition implying that m is a Marcinkiewicz multiplier is that for some s ∈ Nk

with si >ni

2for every i, and for every ξ ∈ RN with ξi 6= 0 for every i,

(3.2)∣∣∂α1

ξ1· · ·∂αk

ξkm(ξ)

∣∣ ≤ Cα|ξ1|−d1(α1)|ξ2|−d2(α2) · · · |ξk|−dk(αk) ,

where the di are the degrees of the multi-indices αi w.r. to the dilations in Rni .Condition (3.1) does not depend on the choice of the ηi, and we shall choose the

ηi in such a way that for every i

(3.3)∑

j∈Z

ηi(2j · ξi) > 0

for ξi 6= 0.

Theorem 3.3. Let m be a Marcinkiewicz multiplier on RN . Then the operatorSm = m(i−1∂) is bounded on Lp(RN ) for 1 < p <∞.

Proof. Let ψi = F−1ηi, ψ(x) =∏k

i=1 ψi(xi). We write 2J ·ξ for (2j1 ·ξ1, . . . , 2jk ·ξk),

with J ∈ Zk. DefinemJ (ξ) = m(2−J · ξ)η(ξ) ,

and KJ = F−1mJ . The Marcinkiewicz condition implies that

(3.4)

∫

RN

|KJ(x)|2k∏

i=1

(1 + ‖xi‖2

)sidx ≤ C

58 CHAPTER III

uniformly in J .Let

TJf = (Smf) ∗ ψ(J) = Sm(f ∗ ψ(J)) = F−1(fmψ(J)

)= f ∗K(J)

J ,

where, consistently with our previous notation and calling Qi the homogeneousdimension of Rni w.r. to the given dilations,

K(J)J (x) = 2−Q1j1 · · · 2−QkjkKJ(2−j1 · x1, . . . , 2

−jkxk) .

Since each ψi ∗ ψi satisfies (2.1) and (2.2), we have, assuming 2 ≤ p <∞,

‖Smf‖2p ≤ C

∥∥∥∥( ∑

J∈Zk

|(Smf) ∗ ψ(J) ∗ ψ(J)|2))

12

∥∥∥∥2

p

= C

∥∥∥∥( ∑

J∈Zk

∣∣TJ(f ∗ ψ(J))∣∣2

) 12∥∥∥∥

2

p

= C

∥∥∥∥∑

J∈Zk

∣∣TJ (f ∗ ψ(J))∣∣2

∥∥∥∥p/2

.

Call fJ = f ∗ ψ(J) and w(x) =∏k

i=1

(1 + ‖xi‖2

)−si. By Holder’s inequality

and (3.4),

∣∣TJfJ (x)∣∣2 =

∣∣∣∣∫

RN

K(J)J (x− y)fJ(y) dy

∣∣∣∣2

≤( ∫

RN

∣∣K(J)J (x− y)

∣∣2

w(J)(x− y)dy

)( ∫w(J)(x− y)

∣∣fJ(y)∣∣2 dy

)

≤ C(|fJ |2 ∗ w(J)

)(x) .

Take g ∈ L(p/2)′(RN ). Then

∫

RN

∑

J∈Zk

∣∣TJfJ(x)∣∣2g(x) dx ≤ C

∑

J∈Zk

∫

RN

(|fJ |2 ∗ w(J)

)(x)|g(x)| dx

= C∑

J∈Zk

∫

RN

|fJ(x)|2(w(J) ∗ |g|

)(x) dx

≤ C

∫

RN

( ∑

J∈Zk

|fJ(x)|2)(

supJ∈Zk

(w(J) ∗ |g|

)(x)

)dx

≤ C

∥∥∥∥∑

J∈Zk

|fJ |2∥∥∥∥

p/2

∥∥∥ supJ∈Zk

w(J) ∗ |g|∥∥∥

(p/2)′.

If we prove that the maximal operator

(3.5) Mwf(x) = supJ∈Zk

(w(J) ∗ |f |

)(x)

is bounded on L(p/2)′(RN ), it follows that∥∥∥∥

∑

J∈Zk

∣∣TJfJ(x)∣∣2

∥∥∥∥p/2

≤ C

∥∥∥∥∑

J∈Zk

∣∣fJ(x)∣∣2

∥∥∥∥p/2

and we can conclude that

‖Smf‖2p ≤ C

∥∥∥∥∑

J∈Zk

∣∣TJfJ(x)∣∣2

∥∥∥∥p/2

≤ C ′

∥∥∥∥∑

J∈Zk

∣∣fJ(x)∣∣2

∥∥∥∥p/2

= C ′

∥∥∥∥( ∑

J∈Zk

∣∣fJ (x)∣∣2

) 12∥∥∥∥

2

p

≤ C ′′‖f‖2p .

We have used the fact that fj = f ∗ ψ(J) and that the ψi satisfies (2.1).The proof of Lp-boundedness of (3.5) is given as a separate lemma. �

Lemma 3.4. The maximal operator Mw is bounded on Lp(RN ) for 1 < p ≤ ∞.

Proof. We need at this stage to introduce scalar coordinates x = (t1, . . . , tN ) in RN ,and we do this in such a way that the first n1 coordinates determine the componentof x in Rn1 , etc. Following the notation used in the rest of this chapter, this meansthat

(t1, . . . , tn1) = x1 , · · · · · · (tN−nk+1, . . . , tN ) = xk .

Inside each Rni we choose the coordinates so that the dilations are diagonal.This implies that for each index ` ∈ {1, . . . , N} there are i = i(`) ∈ {1, . . . , k} andλ` > 0 such that

2J · x = (2ji(`)λ`x`)`=1,...,N .

Write si = ni

2(1 + εi) with εi > 0. Then, taking i = 1 to simplify the notation,

(1 + ‖x1‖2

)s1 ≥n1∏

`=1

(1 + |t`|2

) 12 (1+ε1) ≥ C

n1∏

`=1

(1 + |t`|

)1+ε1.

Therefore, if ε = mini εi,

w(x) =k∏

i=1

(1 + ‖xi‖2

)−si ≤ CN∏

`=1

(1 + |t`|

)−1−ε= Cw(x) .

Therefore Mwf(x) ≤ CMwf(x) for every f and every x, where

Mwf(x) = supJ∈Zk

(w(J) ∗ |f |

)(x)

= supJ∈Zk

∫

RN

N∏

`=1

2−ji(`)λ`(1 + |2−ji(`)λ`t′`|

)−1−ε∣∣f(t1 − t′1, . . . , tN − t′N )∣∣ dt′

≤ supJ∈ZN

∫

RN

N∏

`=1

2−j`λ`(1 + |2−j`λ`t′`|

)−1−ε∣∣f(t1 − t′1, . . . , tN − t′N )∣∣ dt′ .

60 CHAPTER III

Consider first the one-dimensional integral

Ij(t) =

∫

R

2−jλ(1 + |2−jλt′|

)−1−ε∣∣f(t− t′)∣∣ dt′ .

We decompose the real line as the union of the interval [−2jλ, 2jλ], where 1 +

|2−jλt′| can be bounded by 1 from below, and the outer dyadic regions {t : 2j′λ <

|t′| < 2(j′+1)λ}, where 1 + |2−jλt′| can be bounded from below by 2(j′−j)λ. Eachpartial integral can be estimated by the Hardy-Littlewood maximal function Mf ,using the trivial inequality

∫

|t′|<r

|f(t− t′)| dt′ ≤ 2rMf(t) .

Therefore

Ij(t) ≤ 2−jλ

∫

|t′|<2jλ

∣∣f(t− t′)∣∣ dt′

+∑

j′≥j

2−jλ

∫

2j′λ<|t′|<2(j′+1)λ

2(1+ε)(j−j′)λ∣∣f(t− t′)

∣∣ dt′

≤ 2 · 2−jλ2jλMf(t) + 2∑

j′≥j

2−jλ2(j′+1)λ2(1+ε)(j−j′)λMf(t)

≤ Cλ,εMf(t) .

For each `, define M`f as the one-dimensional Hardy-Littlewood maximal func-tion of f regarded as a function of t` only:

M`f(t1, . . . , tN ) = supa<t`<b

1

b− a

∫ b

a

|f(t1, . . . , t`−1, t, t`+1, . . . , tN )| dt .

The estimate obtained for Ij(t) implies that

∫

R

N∏

`=1

2−j`λ`(1 + |2−j`λ`t′`|

)−1−ε∣∣f(t1 − t′1, . . . , tN − t′N )∣∣ dt′1 ≤

≤ CN∏

`=2

2−j`λ`(1 + |2−j`λ`t′`|

)−1−εM1f(t1, t2 − t′2 . . . , . . . , tN − t′N ) .

Integrating one variable at the time, we find that

∫

RN

N∏

`=1

2−j`λ`(1 + |2−j`λ`t′`|

)−1−ε∣∣f(t1 − t′1, . . . , tN − t′N )∣∣ dt′ ≤

≤ CMNMN−1 · · ·M1f(x) .

Taking the supremum over J ∈ Zk, we obtain that

Mwf(x) ≤ CMwf(x) ≤ C ′MNMN−1 · · ·M1f(x) .

By Corollary 2.5 in Chapter II, if 1 < p ≤ ∞,

∫ +∞

−∞

M`f(x)p dx` ≤ C

∫ +∞

−∞

|f(x)|p dx` ,

for a.e. choice of the x`′ with `′ 6= `. Integrating in the remaining variables, weobtain that

‖M`f‖p ≤ C‖f‖p .

Therefore

‖Mwf‖p ≤ C‖MNMN−1 · · ·M1f‖p ≤ C ′‖f‖p . �

4. Applications

We discuss two different applications of the product theory to differential opera-tors. One is to estimates of the kind given in Theorem 7.6 of Chapter II, involvingfractional powers of differential operators, the other is to multipliers of a system ofconstant coefficient differential operators.

If L = P (i−1∂) is a constant coefficient differential operator on Rn, define, forγ ∈ C,

|L|γf = F−1(|P |γ f) .

In particular, the fractional derivative of order γ in xj (xj being a scalar com-ponent of x) is

Dγj f = F−1(|ξj|γ f) .

The statement of Theorem 7.6 in Chapter II can then be completed as follows.

Theorem 4.1. In the notation of Section 7 in Chapter II, assume that P satisfies(1) and (2′), and let α1, . . . , αn ∈ C be such that

∑j λj<eαj ≤ k. Then for every

f ∈ S(Rn) and 1 < p <∞,

‖Dα11 · · ·Dαn

n f‖p ≤ Cp

(‖f‖p + ‖Lf‖p

).

Proof. We give the proof for n = 2, the general case being essentially the same,with the extra disadvantage of more complicated notations.

Let ϕ ∈ D(R) be equal to 1 on a neighborhood of 0, and write the Fouriertransform of Dαf = Dα1

1 · · ·Dαnn f as

Dαf(ξ) =(|ξ1|α1ϕ(ξ1)

)(|ξ2|α2ϕ(ξ2)

)f(ξ)

+|ξ1|α1

(1 − ϕ(ξ1)

)

P (ξ1, 0)

(|ξ2|α2ϕ(ξ2)

)P (ξ1, 0)f(ξ)

+(|ξ1|α1ϕ(ξ1)

) |ξ2|α2(1 − ϕ(ξ2)

)

P (0, ξ2)P (0, ξ2)f(ξ)

+|ξ1|α1

(1 − ϕ(ξ1)

)|ξ2|α2

(1 − ϕ(ξ2)

)

P (ξ)P (ξ)f(ξ)

= m1(ξ)f(ξ) +m2(ξ)P (ξ1, 0)f(ξ) +m3(ξ)P (0, ξ2)f(ξ) +m4(ξ)P (ξ)f(ξ) .

62 CHAPTER III

We begin with m2 and verify condition (3.2), taking s1 = s2 = 1. Because m2 isthe product of two factors in splitted variables, we need to observe that

(1) m2 is bounded (the hypotheses on P imply that |P (ξ1, 0)| ≥ C|ξ1|k/λ1);(2) the derivative of |ξ2|α2ϕ(ξ2) is bounded by a constant times |ξ2|−1 for ξ2 6= 0;

(3) the derivative of|ξ1|

α1

(1−ϕ(ξ1)

)P (ξ1,0) is smooth and, away from the support of ϕ,

it is homogeneous (in the ordinary sense) of degree one less than the degreeof this fraction itself (this is α1 − k

λ1, which has a non-positive real part);

hence the derivative is bounded by a constant times |ξ1|−1 for ξ1 6= 0.

The same remarks imply that also m1 and m3 satisfy (3.2) with s1 = s2 = 1.The verification that m4 also verifies (3.2) with s1 = s2 = 1 cannot be done by

separation of variables.Observe that m4 equals

(1−ϕ(ξ1)

)(1−ϕ(ξ2)

)times a function which is continu-

ous and homogeneous, w.r. to the non-isotropic dilations, of degree λ1α1+λ2α2−k,whose real part is non-positive. Since ξ1 and ξ2 are bounded away from 0 on thesupport of m4, we conclude that m4 is bounded.

We skip now to estimating that ∂ξ1∂ξ2

m4 is bounded by |ξ1|−1|ξ2|−1 times aconstant. We apply Leibniz’s rule.

If both derivatives fall on(1 − ϕ(ξ1)

)(1 − ϕ(ξ2)

), this term is supported on

a compact set and the estimate is trivial. Consider then the case where the ξ1-derivative falls on 1 − ϕ(ξ1) and the ξ2-derivative on the homogeneous function.The derivation in ξ2 reduces the non-isotropic degree of homogeneity by λ2. Hencewe obtain −ϕ′(ξ1) times a function whose non-isotropic degree of homogeneity isλ1α1 + λ2(α2 − 1) − k, whose real part is ≤ −λ2.

Considering that this term is supported on a set where a < |ξ1| cfor appropriate a, b, c > 0, this term is bounded by a constant times

|ξ|−λ2 ∼(|ξ1|1/λ1 + |ξ2|1/λ2

)−λ2 ∼ |ξ2|−1 ∼ |ξ1|−1|ξ2|−1 .

The same applies if the role of ξ1 and ξ2 is interchanged. Suppose finally thatboth derivatives fall on the homogeneous factor. We then obtain a function ho-mogeneous of degree λ1(α1 − 1) + λ2(α2 − 1) − k, whose real part is ≤ −λ1 − λ2.Considering that the other factor restrict the support to |ξ1|, |ξ2| > c > 0, we getthe bound

|ξ|−λ1−λ2 ∼(|ξ1|1/λ1 + |ξ2|1/λ2

)−λ1−λ2 ≤ C|ξ1|−1|ξ2|−1 .

The remaining verifications follow the same lines. Therefore all the mi areMarcinkiewicz multipliers. If we call L1 = P (i−1∂x1

, 0), and L2P (0, i−1∂x2), we

have proved that

‖Dαf‖p ≤ Cp

(‖f‖p + ‖L1f‖p + ‖L2f‖p + ‖Lf‖p

),

for 1 < p <∞. But we can now apply Theorem 7.6 in Chapter II directly to showthat ‖Lif‖p ≤ C

(‖f‖p + ‖Lf‖p

)for i = 1, 2. �

We pass now to the second application. On Rn are given k differential operatorsLi = Pi(i

−1∂), where

(1) for each i there is a subspace Vi of Rn of dimension ni ≤ n such thatPi(ξ) = Pi(ξi), where ξi is the orthogonal projection of ξ on Vi;

(2) Pi(ξi) > 0 for ξi ∈ Vi \ {0};(3) each Pi is homogeneous of degree ki w.r. to some non-isotropic 1-parameter

dilations on Vi.

Theorem 4.2. Let m(λ1, . . . , λk) be a Marcinkiewicz multiplier on Rk = R×· · ·×R, with the standard dilations on each coordinate line, of order s = (s1, . . . , sk) withsi >

ni

2 for every i. Then m(L1, . . . , Lk) is bounded on Lp(Rn) for 1 < p <∞.

We point out that this is not the optimal result, but is what can be obtainedby our method of proof. The process of lifting to a higher–dimensional space, asdescribed below, forces us to impose the Marcinkiewicz conditions adapted to thehigher dimension.

Proof. On each Vi we fix coordinates that reduce the dilations to diagonal form.For each i, this choice of coordinates determines a linear bijection τi : Rni → Vi.Let Pi = Pi ◦ τi.

On Rn1 × · · · × Rnk = RN we consider the operators

Li = Pi(i−1∂xi

) ,

where ∂xiis the gradient in Rni ∼ Vi. For convenience, we use the symbol ηi ∈ Rni

to denote the variable for Fourier transforms on Rni .We claim that the Fourier multiplier on RN

m(η1, . . . , ηk) = m(P1(η1), . . . , Pk(ηk)

)

is a Marcinkiewicz multiplier of order s w.r. to the k-parameter dilations on RN

induced by the 1-parameter dilations on each Rni (we skip the details of the proof,which is a refinement of the proofs of Theorem 7.1 and Proposition 7.4 in ChapterII). Therefore m(L1, . . . , Lk) is bounded on Lp(RN ) for 1 < p <∞.

Call σi = τ−1i ◦ πi : Rn → Rni , where πi is the orthogonal projection onto Vi in

Rn, and σ : Rn → RN the map

σ(ξ) =(σi(ξ), . . . , σk(ξ)

).

Then the Fourier multiplier of m(L1, . . . , Lk) is

m(P1(ξ), . . . , Pk(ξ)

)= m ◦ σ(ξ) .

We give the conclusion after the next lemma. �

What follows is part of a general principle, called transference, whose generalidea is that an Lp-estimate for a convolution operator (or, equivalently, for a Fouriermultiplier) induces other Lp-estimates for a certain class of induced (or transferred)operators. The statement we give concerns transference from Euclidean space toanother, by means of a linear map.

Lemma 4.3. Let σ : Rn → Rd be a linear map, and let m be a Fourier Lp-multiplier on Rd, continuous on the image of A. Then m ◦ σ is a Fourier Lp-multiplier on Rn, and, if Sm and Sm◦σ are the corresponding operators,

‖Sm◦σ‖L(Lp(Rn)) ≤ ‖Sm‖L(Lp(Rd)) .

Proof. Changing coordinates on Rd and Rn if necessary, we can assume that

(1) Rd = Rν × Rd′

, and imσ = Rν × {0};(2) Rn = Rν × Rn′

, and kerσ = {0} × Rn′

.

64 CHAPTER III

Introducing coordinates (η, η′) on Rd and (ξ, ξ′) on Rn compatible with thesesplittings, we then have σ(ξ, ξ′) = (Aξ, 0) for some invertible ν × ν matrix A.

The proof follows from the combination of three facts.First fact: if m(η, η′) is a Fourier Lp-multiplier on Rd, continuous on Rν × {0},

then m0(η, η′) = m(η, 0) is also a Fourier Lp-multiplier on Rd, with no increase in

the norm. In order to see this, for ε > 0 consider mε(η, η′) = m(η, εη′). Then, for

f, g ∈ S(Rd),

〈Smεf |g〉 = (2π)

d2

∫

Rd

m(η, εη′)f(η, η′)g(η, η′) dη dη′

= (2π)d2

∫

Rd

m(η, η′)ε− d′

p′ f(η, ε−1η′)ε−d′

p g(η, ε−1η′) dη dη′

= 〈Smfε|gε〉 ,

where fε(y, y′) = ε

d′

p f(x, εx′) and gε(y, y′) = ε

d′

p′ g(x, εx′). Therefore

∣∣〈Smεf |g〉

∣∣ ≤ ‖Sm‖L(Lp(Rd))‖fε‖p‖gε‖p′ = ‖Sm‖L(Lp(Rd))‖f‖p‖g‖p′ .

If ε tends to 0, 〈Smεf |g〉 tends to 〈Sm0

f |g〉, and this proves the first fact.Second fact: a bounded function m(η, η′) = µ(η) is a Fourier Lp-multiplier on Rd

if and only if µ is a Fourier Lp-multiplier on Rν and the two norms coincide. To seethis, let k = F−1µ ∈ S ′(Rν). Since m = µ⊗ 1, we have F−1m = k ⊗ δ0 ∈ S ′(Rd).

Calling fy′

(y) = f(y, y′), we then have

Smf(y, y′) = f ∗Rd (k ⊗ δ0)(y, y′) = fy′ ∗Rν k(y) = Sµf

y′

(y) .

Therefore,

‖Smf‖pLp(Rd)

=

∫

Rd′‖Sµf

y′‖pLp(Rν) dy

′

≤ ‖Sµ‖pL(Lp(Rν))

∫

Rd′‖fy′‖p

Lp(Rν) dy′

≤ ‖Sµ‖pL(Lp(Rν))‖f‖p

p .

This proves that ‖Smf‖Lp(Rd) ≤ ‖Sµ‖L(Lp(Rν )). For the opposite inequality, takef(y, y′) = f1(y)f2(y

′). Then Smf(y, y′) = (Sµf1)(y)f2(y′). With f2 6= 0 fixed,

‖Sµf1‖p =‖Smf‖p

‖f2‖p≤ ‖Smf‖Lp(Rd)‖f1‖p .

Third fact: the statement is true if d = n = ν. In This case m(ξ) = m(Aξ) withA invertible. Taking again f, g ∈ S(Rν), we have

〈Smf |g〉 = (2π)ν2

∫

Rν

m(Aξ)f(ξ)g(ξ)dξ

= (2π)ν2 | detA|−1

∫

Rν

m(η)f(A−1η)g(A−1η) dη

= | detA|−1〈SmfA|gA〉 ,

where fA(x) = | detA|f(Ax), gA(x) = | detA|g(Ax). Then

∣∣〈Smf |g〉∣∣ ≤ | detA|−1‖Sm‖L(Lp(Rν ))‖fA‖p‖gA‖p′ = ‖Sm‖L(Lp(Rν ))‖f‖p‖g‖p′ .

Going back to the general statement, from the Lp-boundedness of Sm on Rd

we deduce the Lp-boundedness of µ(η) = m(η, 0) on Rν (facts 1 and 2). Fromthis we deduce the Lp-boundedness of Sµ◦A on Rν (fact 3), and from this theLp-boundedness of Sm◦σ on Rn (fact 2 again). �

End of the proof of Theorem 4.2. Lemma 4.3 implies thatm(L1, . . . , Lk) is boundedon Lp for 1 0 and ϕ ∈ D(R) with suppϕ ⊂ [−2, 2] and ϕ = 1 on [−1, 1], let

mε(λ) = m(λ)k∏

i=1

(1 − ϕ

(λi

ε

)).

Then mε is continuous. If we prove that the Marcinkiewicz norm (3.1) ofthe mε is uniformly bounded in ε, we can conclude that the operator normsof mε(L1, . . . , Lk) are uniformly bounded. It is then easy to observe, using thePlancherel formula, that, for f, g ∈ S(Rn),

〈m(L1, . . . , Lk)f |g〉 = limε→0

〈mε(L1, . . . , Lk)f |g〉 ,

which would allow to conclude. Fix η ∈ D(R) satisfying (i) and (ii) in Section 3.We must estimate the Hs-norm of

m(r1λ1, . . . , rkλk)k∏

i=1

(1 − ϕ

(riλi

ε

)) k∏

i=1

η(λi) .

Observe that(1−ϕ(riλi/ε)

)η(λi) is identically zero if ri/ε ≤ 1/4, and it equals

η(λi) if ri/ε ≥ 4. Matters reduce therefore to verifying that multiplication byϕ(tλi) is a continuous operation on Hs, uniformly for 1/4 ≤ t ≤ 4. This fact canbe proved by adapting the proof of Lemma 6.3 in Chapter II. �

66 CHAPTER III

HEISENBERG GROUP 67

CHAPTER IV

FOURIER ANALYSIS ON THE HEISENBERG GROUP

1. The Heisenberg group

On R2n+1 = Rn × Rn × R consider the composition law

(x, y, t)(x′, y′, t′) =(x+ x′, y + y′, t+ t′ +

1

2(x · y′ − y · x′)

).

In complex coordinates (z, t) ∈ Cn × R, this becomes

(z, t)(z′, t′) =(z + z′, t+ t′ − 1

2=m〈z|z′〉

),

where 〈z|z′〉 =∑n

j=1 zjz′j is the Hermitean product on Cn.One can verify that this is a non-commutative group law, with neutral element

(0, 0), and with inverse (z, t)−1 = (−z,−t). This is the Heisenberg group Hn.The elements (0, t) form the center Z(Hn) of Hn, and Hn/Z(Hn) is isomorphic

to the additive group R2n. The following transformations are automorphisms ofHn:

(1) the dilations (z, t) 7−→ (rz, r2t) = r · (z, t), with r > 0;(2) the rotations (z, t) 7−→ (Uz, t), with U a unitary transformation of Cn (i.e.

UU∗ = I);(3) the conjugation (z, t) 7−→ (z,−t).One finds the vector fields Xj , Yj in (2.1), Chapter I, by means of the following

operations:

(1.1)

Xjf(x, y, t) =d

ds |s=0

f((x, y, t)(sej, 0, 0)

)

=d

ds |s=0

f(x+ sej , y, t−

1

2syj

)

= ∂xjf(x, y, t)− yj

2∂tf(x, y, t) ,

Yjf(x, y, t) =d

ds |s=0

f((x, y, t)(0, sej, 0)

)

=d

ds |s=0

f(x, y + sej , t+

1

2sxj

)

= ∂yjf(x, y, t) +

xj

2∂tf(x, y, t) .

Typeset by AMS-TEX

68 CHAPTER IV

Therefore these vector fields are left-invariant: denoting by

L(z′,t′)f(z, t) = f((z′, t′)−1(z, t)

)

the left-translate of f by (z′, t′), then

Xj

(L(z′,t′)f

)= L(z′,t′)(Xjf) , Yj

(L(z′,t′)f

)= L(z′,t′)(Yjf) .

It follows that also

[Xj , Yj] = T = ∂t

is left-invariant and that any left-invariant vector field onHn is a linear combinationof the Xj , Yj, T . The other Lie brackets are

[Xj, Xk] = [Yj, Yk] = [Xj, T ] = [Yj, T ] = 0 , [Xj, Yk] = δj,kT .

More generally, one says that a linear operator T acting on functions on Hn isleft-invariant if

T (L(z,t)f) = L(z,t)(Tf)

for all (z, t) ∈ Hn and every f . We impose here and in the sequel that T maps13

S(Hn) into S ′(Hn) boundedly, in the sense that the map14

(f, g) 7−→ 〈Tf, g〉

is continuous on S(Hn) × S(Hn). This is a very mild initial assumption. It issatisfied, e.g., by the convolution operators

(1.2) Tf = f ∗ k(z, t) =

∫

Hn

f((z, t)(w, u)−1

)k(w, u) dw du ,

with k ∈ L1(Hn) (which must be understood w.r. to the Lebesgue measure).

Convolution on Hn is a non-commutative operation, and in order to have a left-invariant operator, the kernel must be on the right of f . The convolution (1.1) canbe extended to distributional kernels k ∈ S ′(Hn), provided f ∈ S(Hn); the integralon the right-hand side of (1.2) must then be interpreted as the pairing between kand the function (w, u) 7−→ f

((z, t)(w, u)−1

).

It follows from the Schwartz kernel theorem that any left-invariant operator Tmapping S(Hn) boundedly into S ′(Hn) has the form (1.2) for some k ∈ S ′(Hn).

The sub-Laplacian L = −∑nj=1(X

2j + Y 2

j ) studied in Chapter I is also left-invariant, and such are its spectral projections, the joint spectral projections ofL and i−1T , and the operators defined by spectral multipliers of L and i−1T .Therefore they can all be realized as convolution operators on Hn.

13S(Hn) is the same as S(R2n+1), and similarly for S ′.14Recall that the notation 〈 , 〉 stands for the bilinear pairing.

HEISENBERG GROUP 69

Obviously, besides the left-invariant vector fields, one has the right-invariantones. We shall put a superscript (r) to denote the right-invariant versions ofthe Xj, Yj:

(1.3)

X(r)j f(x, y, t) =

d

ds |s=0

f((sej , 0, 0)(x, y, t)

)

=d

ds |s=0

f(x+ sej , y, t+

1

2syj

)

= ∂xjf(x, y, t) +

yj

2∂tf(x, y, t) ,

Y(r)j f(x, y, t) =

d

ds |s=0

f((0, sej, 0)(x, y, t)

)

=d

ds |s=0

f(x, y + sej , t−

1

2sxj

)

= ∂yjf(x, y, t)− xj

2∂tf(x, y, t) .

Observe that T is both right- and left-invariant, and that

[X(r)j , Y

(r)j ] = −T .

We shall also write

L(r) = −n∑

j=1

((X

(r)j )2 + (Y

(r)j )2

).

One easily verifies that, if f(z, t) = f((z, t)−1

)= f(−z,−t), then

X(r)j f = −(Xj f ) ,

(similarly for the Yj) and that

L(r)f = (Lf ) .

2. The group Fourier transform

For a few sections we shall restrict ourselves to H1 = C × R. We recall that inChapter I we have introduced the functions

(2.1) hj,k(z) = Zk(zje−|z|2

4 ) = e|z|2

4 ∂kz (zje−

|z|2

2 )

on C, with j, k ∈ N, and discussed their role in the spectral analysis of the sub-Laplacian L. As we shall see, these functions are the key ingredients to constructthe group Fourier transform on H1 (later on we shall extend all this to Hn).

The general notion of Fourier transform on locally compact groups involves no-tions of abstract representation theory that we do not want to develop here15.

15See the notes of the course “Analisi di Fourier non commutativa”.

70 CHAPTER IV

We shall use instead the spectral analysis of L to study the specific case of theHeisenberg group.

To begin with, it is appropriate to normalize the hj,k in L2(C). From (3.8) inChapter I, we deduce that

‖hj,k‖22 =

k!

2k‖hj,0‖2

2

=k!

2k2π

∫ ∞

0

r2j+1e−r2

2 dr

= (2π)j!k!2j−k .

We then define

(2.2) ϕj,k(z) =1√

j!k!2j−khj,k(z) .

By Proposition 3.2 in Chapter 1, {(2π)−1/2ϕj,k} is an orthonormal basis ofL2(C). From (3.10) in Chapter I, it follows that the ϕj,k are in S(C).

By (3.10) and (3.12) in Chapter I,

(2.3)ϕj,k(0) = δj,k ,

ϕj,k(z) = (−1)j−kϕj,k(−z) = (−1)j−kϕk,j(z) = (−1)j−kϕk,j(z) .

In analogy with Corollary 3.3 in Chapter I, we define

(2.4) ϕλj,k(z) =

ϕj,k(λ12 z) if λ > 0 ,

ϕj,k

(|λ| 12 z

)if λ < 0 .

We also set

(2.5) Φλj,k(z, t) = eiλtϕλ

j,k(z) .

Given f ∈ L1(H1), we define16, for λ 6= 0,

(2.6) f(λ, j, k) =

∫

H1

f(z, t)Φλj,k(z, t) dz dt = 〈Ftf(·,−λ)|ϕλ

j,k〉 .

Proposition 2.1. If f ∈ L1(H1) ∩ L2(H1), then the following Plancherel formulaholds:

(2.7) ‖f‖22 =

1

(2π)2

∫ +∞

−∞

∑

j,k∈N

∣∣f(λ, j, k)∣∣2 |λ| dλ .

The map f 7−→ (2π)−1f extends to an isometric bijection from L2(H1) ontoL2

(R, |λ| dλ, `2(N2)

).

16It will be clear soon (Theorem 2.3) that it is preferable not to take complex conjugates ofthe basis elements.

HEISENBERG GROUP 71

Proof. By the Plancherel formula on R,

∫

H1

|f(z, t)|2 dt =1

2π

∫ +∞

−∞

∫

C

∣∣Ftf(z,−λ)∣∣2 dz dλ .

Then, for a.e. λ, Ftf(z,−λ) ∈ L2(C). But{(2π)−1/2|λ| 12ϕλ

j,k

}j,k

is an orthonor-

mal basis of L2(C); hence

∫

C

∣∣Ftf(z,−λ)∣∣2 dz =

|λ|2π

∑

j,k∈N

∣∣〈Ftf(·,−λ) |ϕλj,k〉

∣∣2

=|λ|2π

∑

j,k∈N

∣∣f(λ, j, k)∣∣2 . �

We shall give a more elegant presentation of this result, introducing at the sametime the representations of H1.

Using (2.1) above and the identity (3.6) in Chapter I, we obtain the followingidentities:

(2.8)Zϕj,k =

√k + 1

2ϕj,k+1 ,

Zϕj,k =

√2

kZZϕj,k−1 = −1

4

√2

k(L + I)ϕj,k−1 = −

√k

2ϕj,k−1 .

Lemma 2.2. The following identity holds:

∑

`∈N

ϕj,`(z)ϕ`,k(w) = e−i2=m(zw)ϕj,k(z + w) .

Proof. By (3.9) in Chapter I and by (2.3) above,

∑

`∈N

ϕj,`(z)ϕ`,k(w) =∑

`∈N

(−1)j−` ϕ`,j(z)ϕ`,k(w)

= (−1)j

√2j+k

j!k!

∑

`∈N

(−1)`

`!2`h`,j(z)h`,k(w)

= (−1)j

√2j+k

j!k!e

|z|2+|w|2

4

∑

`∈N

(−1)`

`!2`∂j

z

(zè−

|z|2

2

)∂k

w

(wè−

|w|2

2

).

The series∑

`∈N

(−1)`

`!2` zè−

|z|2

2 wè−|w|2

2 converges with all its derivatives. Thesum equals

∑

`∈N

(−1)`

`!2`zè−

|z|2

2 wè−|w|2

2 = e−|z|2+|w|2+zw

2 = e−|z+w|2−zw

2 .

72 CHAPTER IV

Therefore,

∑

`∈N

ϕj,`(z)ϕ`,k(w) = (−1)j

√2j+k

j!k!e

|z|2+|w|2

4 ∂jz∂

kwe

− |z+w|2−zw2

=1√

j!k!2j−ke

|z|2+|w|2+2zw4 ∂k

w

((z + w)je−

|z+w|2

2

)

=1√

j!k!2j−ke

|z+w|2+2i=mzw4 ∂k

w

((z + w)je−

|z+w|2

2

)

=1√

j!k!2j−ke−

12=mzwhj,k(z + w)

= e−12=mzwϕj,k(z + w) . �

It follows immediately that

(2.9) Φλj,k

((z, t)(w, u)

)=

∑

`∈N

Φλj,`(z, t)Φ

λ`,k(w, u) ,

for every λ 6= 0.It is then natural to arrange the Φλ

j,k in the infinite matrix

(2.10) Φλ(z, t) =(Φλ

j,k(z, t))j,k

.

Theorem 2.3. The following identities hold:

Φλ(z, t)∗Φλ(z, t) = Φλ(z, t)Φλ(z, t)∗ = I

Φλ((z, t)−1

)= Φλ(z, t)∗ = Φλ(z, t)−1 ,

Φλ((z, t)(w, u)

)= Φλ(z, t)Φλ(w, u) .

In particular, Φλ(z, t) defines, for every (z, t) ∈ H1 and every λ 6= 0, a unitaryoperator πλ(z, t) on `2 = `2(N). The map

πλ : H1 −→ L(`2)

is a continuous homomorphism, w.r. to the strong topology on L(`2), i.e. a unitaryrepresentation of H1.

For f ∈ L1(H1), the integral

(2.11) πλ(f) =

∫

H1

f(z, t)πλ(z, t) dz dt

converges in the strong topology to a bounded operator on `2. The identity

(2.12) πλ(f ∗ g) = πλ(f)πλ(g)

holds for every f, g ∈ L1(H1).

HEISENBERG GROUP 73

The proof requires a few verifications that we leave to the reader. Observe thatthe operator πλ(f) in (2.11) is represented, w.r. to the canonical basis of `2, by thematrix ∫

H1

f(z, t)Φλ(z, t) dz dt =(f(λ, j, k)

)j,k

def= f(λ) .

Then (2.12) takes the form

(2.13) f ∗ g(λ) = f(λ)g(λ) ,

in analogy with the ordinary Fourier transform. These identities justify the choice ofnot conjugating Φλ

j,k in (2.6). If we had done so, the order of the two factors in (2.13)should have been changed. Recall that both convolution in H1 and composition oflinear operators on `2 are non-commutative.

We obtain the following reformulation of Proposition 2.1.

Theorem 2.4. For f ∈ L1(H1)∩L2(H1), the operator πλ(f) is a Hilbert-Schmidtoperator for a.e. λ, and the Plancherel formula can be written as:

(2.14) ‖f‖22 =

1

(2π)2

∫ +∞

−∞

‖πλ(f)‖2HS |λ| dλ .

The map f 7−→ (2π)−1πλ(f) extends to an isometric bijection from L2(H1) ontoL2

(R, |λ| dλ,HS(`2)

).

This is the standard form of a Plancherel formula on a non-commutative group,invoking Hilbert-Schmidt norms of operator-valued Fourier transforms.

We write explicitely the polarized form of Plancherel’s formula:

(2.16)

∫

H1

f(z, t)g(z, t)dz dt =1

(2π)2

∫ +∞

−∞

tr(πλ(f)πλ(g)∗

)|λ| dλ

=1

(2π)2

∫ +∞

−∞

tr(f(λ)g(λ)∗

)|λ| dλ

=1

(2π)2

∫ +∞

−∞

∑

j,k∈N

f(λ, j, k)g(λ, j, k) |λ| dλ .

Like in commutative Fourier analysis, the Plancherel formula has a companioninversion formula. We give the inversion formula for a narrow class of functions(the Schwartz class). It can however be extended to a larger class17. The proofrequires a few lemmas.

Lemma 2.5. For every j, k, ‖Φλj,k‖∞ ≤ 1. If L = −X2 − Y 2 is the sub-Laplacian

on H1, and T = ∂t,

(2.17) LΦλj,k = |λ|(2k + 1)Φλ

j,k , TΦλj,k = iλΦλ

j,k .

Proof. The first identity in Theorem 2.3 implies that, for each j and each (z, t),

(2.18)∑

k∈N

|Φλj,k(z, t)|2 = 1 .

This obviously proves the first statement. The first identity in (2.17) is a conse-quence of Corollary 3.3 in Chapter I, and the second is obvious. �

17The formula below shows that it is sufficient that πλ(f) be of trace class for a.e. λ and that

the function λ 7→ tr`

|πλ(f)|´

be integrable on R. The proof below shows that Schwartz functionssatisfy this property.

74 CHAPTER IV

Lemma 2.6. The following identities hold:

Lf(λ, j, k) = |λ|(2k + 1)f(λ, j, k) , T f(λ, j, k) = −iλf(λ, j, k) .

If f∗(z, t) = f(−z,−t) then

f∗(λ, j, k) = f(λ, k, j) ,

i.e.πλ(f∗) =

(πλ(f)

)∗.

Proof. The first two identities follow directly from Lemma 2.5, by integration by

parts. The last one follows from the fact that Φλj,k(−z,−t) = Φλ

k,j(z, t). �

Proposition 2.7. For each N ∈ N, there is a Schwartz norm ‖ ‖N such that, iff ∈ S(H1),

∣∣f(λ, j, k)∣∣ ≤ ‖f‖N(

1 + |λ|(1 + j + k))N

.

Moreover f(λ, j, k) is smooth in λ for λ 6= 0.

Proof. If f ∈ S(H1), we have in the first place

∣∣f(λ, j, k)∣∣ =

∣∣∣∣∫

H1

f(z, t)Φλj,k(z, t) dz dt

∣∣∣∣ ≤ ‖f‖1‖Φλj,k‖∞ ≤ ‖f‖1 .

By Lemma 2.6,

|λ|N∣∣f(λ, j, k)

∣∣ =∣∣TNf(λ, j, k)

∣∣ ≤ ‖TNf‖1 ,

|λ|N (2k + 1)N∣∣f(λ, j, k)

∣∣ =∣∣LNf(λ, j, k)

∣∣ ≤ ‖LNf‖1 ,

|λ|N (2j + 1)N∣∣f(λ, j, k)

∣∣ = |λ|N (2j + 1)N∣∣f∗(λ, k, j)

∣∣ ≤ ‖LNf∗‖1 ,

Putting all these estimates together,

(1 + |λ|(1 + j + k)

)N |f(λ, j, k)| ≤ C(‖f‖1 + ‖Tf‖1 + ‖Lf‖1 + ‖L∗f‖1

),

and this last expression is controlled by a Schwartz norm.

The smoothness of f(λ, j, k) for λ 6= 0 is obvious. �

Theorem 2.8. For f ∈ S(H1) the following inversion formula holds:

(2.19)

f(z, t) =1

(2π)2

∫ +∞

−∞

tr(πλ(f)πλ(z, t)∗

)|λ| dλ

=1

(2π)2

∫ +∞

−∞

∑

j,k∈N

f(λ, j, k)Φλj,k(z, t) |λ| dλ .

Proof. Since

f(λ, j, k) = 〈Ftf(·,−λ) |ϕλj,k〉 ,

HEISENBERG GROUP 75

for every λ 6= 0, and Ftf(·,−λ) ∈ L2(C),

Ftf(z,−λ) =|λ|2π

∑

j,k∈N

f(λ, j, k)ϕλj,k(z) ,

with convergence in L2(C). The convergence is also pointwise. In fact, by (2.18),|ϕj,k(z)| ≤ 1, so that, by Proposition 2.7,

∑

j,k∈N

|f(λ, j, k)||ϕλj,k(z)| ≤

∑

j,k∈N

‖f‖N(1 + |λ|(1 + j + k)

)N<∞ ,

taking N > 2.Therefore,

Ftf(z,−λ)e−iλt =|λ|2π

∑

j,k∈N

f(λ, j, k)Φλj,k(z, t) ,

and the Fourier inversion formula on R does the rest. �

We give a first application of this formula, which will be used in the sequel.

Corollary 2.9. Let S0(H1) ⊂ S(H1) be the space of the functions f such that

f(λ, j, k) is non-zero only for j, k varying in a finite set, and, for these values of

j, k, f(λ, j, k) is C∞ in λ and supported on a compact subset of R \ {0}. ThenS0(H1) is dense in L2(H1).

Given any finite family of functions vj,k ∈ D(R \ {0}), with (j, k) ∈ B, thefunction

f(z, t) =1

(2π)2

∫

R

∑

(j,k)∈B

vj,k(λ)Φλj,k(z, t) |λ| dλ

is in S0(H1) and f(λ, j, k) equals vj,k(λ) if (j, k) ∈ B and zero otherwise.

Proof. It is sufficient to prove that any g ∈ S(H1) can be approximated in theL2-norm by functions in S0(H1). Given ε > 0, take K ⊂ R \ {0} and N ∈ N suchthat ∫

R\K

∑

j+k>N

|g(λ, j, k)|2 |λ| dλ < ε2 ,

and fix η ∈ D(R \ {0}) equal to 1 on K, with 0 ≤ η(λ) ≤ 1 for every λ. LetK ′ = supp η and define

fε(z, t) =1

(2π)2

∫

K′

∑

j+k≤N

g(λ, j, k)Φλj,k(z, t) η(λ)|λ| dλ

=1

(2π)2

∑

j+k≤N

∫

K′

g(λ, j, k)ϕλj,k(z)η(λ)|λ|e−iλt dλ

=1

2π

∑

j+k≤N

F−1t

(g(λ, j, k)ϕλ

j,k(z)η(λ)|λ|)(−t) .

It follows from Proposition 2.7 that g(λ, j, k)ϕλj,k(z)η(λ)|λ| ∈ S(C×R). Therefore

fε ∈ S(H1).

76 CHAPTER IV

Moreover, fε(λ, j, k) = η(λ)g(λ, j, k) if j + k ≤ N , and 0 otherwise. Hencefε ∈ S0(H1). Finally, by the Plancherel formula, ‖g − fε‖2 < Cε.

The proof of the last statement is now obvious. �

We conclude this section by discussing the possibility of defining the Fouriertransform for a general distribution u ∈ S ′(H1). In analogy with (2.6), we aretempted to define u(λ, j, k) = 〈u,Φλ

j,k〉. But Φλj,k is not in S(H1) because it does

not have any decay in t, even though it satisfies the required decay conditions in z.We can however apply u to a “packet” of Φλ

j,k.

Given ψ ∈ D(R \ {0}), define

ψj,k(z, t) =

∫

R

Φλj,k(z, t)ψ(λ) dλ = 2πF−1

t

(ϕλ

j,k(z)ψ(λ))(t) .

Since ϕλj,k(z)ψ(λ) ∈ S(C × R) as a function of z and λ, it follows that ψj,k ∈

S(H1). We can then defined distributions uj,k ∈ D′(R \ {0}) by setting

(2.20) 〈uj,k, ψ〉 = 〈u, ψj,k〉 .It is a simple verification that, if u ∈ L1(H1), then uj,k coincides with f(λ, j, k)

as a function of λ.The convolution formula in (2.13) has the following extension.

Lemma 2.10. If u ∈ S ′(H1), and f ∈ S0(H1),

(f ∗ u)j,k =∑

`∈N

f(·, j, `)u`,k .

Proof. By (2.20), if ψ ∈ D(R \ {0}) and f(z, t) = f(−z,−t),〈(f ∗ u)j,k, ψ〉 = 〈f ∗ u, ψj,k〉

= 〈u, f ∗ ψj,k〉 .By (2.9),

f ∗ ψj,k(z, t) =

∫

H1

f(w, u)ψj,k

((w, u)(z, t)

)dw du

=

∫

H1

∫

R

f(w, u)Φλj,k

((w, u)(z, t)

)ψ(λ) dλ dw du

=∑

`∈N

∫

R

f(λ, j, `)Φλ`,k(z, t)ψ(λ) dλ

=∑

`∈N

(ψf(·, j, `)

)˜,k .

But, according to (2.20),⟨u,

(ψf(·, j, `)

)˜ ,k

⟩= 〈u`,k, ψf(·, j, `)〉 = 〈f(·, j, `)u`,k, ψ〉 ,

and this concludes the proof. �

A complete description of the image of S ′(H1) under Fourier transform is possi-ble18, but we do not go into it because it will not be needed.

18See D. Geller, ....

HEISENBERG GROUP 77

3. Fourier multipliers

It follows from Lemma 2.6 that the matrices f(λ) ={f(λ, j, k)

}j,k

and Lf(λ)

are related by the identity

(3.1) Lf(λ) = f(λ)

|λ| 0 · · · 0 · · ·0 3|λ| · · · 0 · · ·...

.... . .

0 0 (2k + 1)|λ|...

.... . .

.

Similarly,

(3.2) T f(λ) = −iλf(λ) = f(λ)(−iλI) .Similar formulas can be obtained for the other left-invariant vector fields. In

analogy with (3.2) in Chapter I, define the complex vector fields

Z =1

2(X − iY ) = ∂z −

i

4z∂t , Z =

1

2(X + iY ) = ∂z +

i

4z∂t ,

on H1. Observe that, by (2.8), for λ > 0,

ZΦλj,k(z, t) =

(∂z −

i

4z∂t

)eiλtϕj,k(λ

12 z)

= eiλt(λ

12 (∂zϕj,k)(λ

12 z) +

1

4λzϕj,k(λ

12 z)

)

= eiλt(λ

12Zϕj,k(λ

12 z) − λ

z

4ϕj,k(λ

12 z) +

1

4λzϕj,k(λ

12 z)

)

= −√kλ

2Φλ

j,k−1(z, t) .

Integrating by parts, we find that

Zf(λ, j, k) =

√kλ

2f(λ, j, k − 1) .

Similarly,

Zf(λ, j, k) = −√

(k + 1)λ

2f(λ, j, k + 1) .

For λ < 0, the two expressions are interchanged:

Zf(λ, j, k) = −√

(k + 1)|λ|2

f(λ, j, k + 1) , Zf(λ, j, k) =

√k|λ|2f(λ, j, k − 1) .

Define the matrices

Uλ =

0√

|λ|/2 0 · · · 0 · · ·0 0

√|λ| · · · 0 · · ·

......

. . .. . .

0 0 0√k|λ|/2

......

. . .. . .

,

78 CHAPTER IV

Lλ =

0 0 · · · 0 · · ·−

√|λ|/2 0 · · · 0 · · ·0 −

√|λ| . . .

......

. . . 0

0 0 −√kλ/2

.. ....

.... . .

.

Then, for λ > 0,

(3.3) Zf(λ) = f(λ)Uλ ,Zf(λ) = f(λ)Lλ ,

and, for λ < 0,

(3.4) Zf(λ) = f(λ)Lλ ,Zf(λ) = f(λ)Uλ ,

In all these cases, we see that the action of a left-invariant differential operatoron f is reflected on the Fourier transform side by right multiplication by a matrixdepending on λ. The same holds for the integral operators Tf = f ∗ k in (1.1). By(2.13) we know in fact that

T f(λ) = f(λ)k(λ) ,

at least for k ∈ L1(H1). We prove now that this is a general fact for boundedoperators on L2(H1).

Definition. We say that a matrix-valued funtion M(λ) =(m(λ, j, k)

)j,k∈N

is a

bounded Fourier multiplier for H1 if m(·, j, k) ∈ L∞(R) for every j, k ∈ N, and if

‖M‖∞ = ess sup ‖M(λ)‖L(`2) <∞ .

Theorem 3.1. If M is a bounded Fourier multiplier for H1, the requirement that

(3.5) TMf(λ) = f(λ)M(λ)

defines a bounded left-invariant operator TM on L2(H1), with ‖TM‖L(L2(H1)

) =

‖M‖∞.Conversely, for any bounded left-invariant operator T on L2(H1), there is a

bounded Fourier multiplier M such that T = TM .

Proof. Assume that M is a bounded Fourier multiplier. Since, for any pair ofoperators A,B on a Hilbert space H,

‖AB‖HS ≤ ‖A‖HS‖B||L(H) ,

taking f ∈ L2(H1),

∫

R

‖πλ(f)M(λ)‖2HS |λ| dλ ≤ ‖M‖2

∞

∫

R

‖πλ(f)‖2HS |λ| dλ .

HEISENBERG GROUP 79

By Plancherel’s formula, TM is bounded on L2(H1), and ‖TM‖L(L2(H1)

) ≤‖M‖∞. The opposite inequality will follow from the second part of the proof.

By (2.11),

L(w,u)f(λ) = Φλ(w, u)f(λ) .

Therefore(L(w,u)Tf)(λ) = Φλ(w, u)T f(λ) = (L(w,u)f)(λ)k(λ) ,

which implies that T commutes with L(w,u) for every (w, u) ∈ H1.

Suppose now that T commutes with left translations and is bounded on L2(H1).As a consequence of the Schwartz kernel theorem, as it has already been mentioned,there is a distribution u ∈ S ′(H1) such that Tf = f ∗ u for f ∈ S(H1). We wantto show that the distributions uj,k defined in (2.20) are in fact bounded functions,and that M(λ) = {uj,k(λ)}j,k is a bounded Fourier multiplier.

Take f ∈ S0(H1) such that f(λ, j, k) = ψ(λ)δj,0δk,p, where ψ ∈ D(R \ {0}) andp ∈ N. Then, by Lemma 2.9,

(f ∗ u)j,k = δj,0ψup,k .

If f ∗ u ∈ L2(H1), as we are assuming, a necessary condition is that for everyψ as above and every p, k ∈ N, ψup,k be square integrable in λ. This implies thateach uj,k is locally integrable on R \ {0}. We can then define M(λ) for a.e. λ.

Take now an infinite matrix A with only finite many entries different from 0, and

ψ ∈ D(R \ {0}). Let f ∈ S0(H1) be such that f(λ) = ψ(λ)A. We have

∫

R

‖f(λ)M(λ)‖2HS |λ| dλ =

∫

R

|ψ(λ)|2‖AM(λ)‖2HS |λ| dλ

≤ ‖T‖2L(L2(H1))

∫

R

‖f(λ)‖2HS |λ| dλ

= ‖T‖2L(L2(H1))

∫

R

|ψ(λ)|2‖A‖2HS |λ| dλ .

Since this must hold for every ψ, it follows that for a.e. λ,

‖AM(λ)‖HS ≤ ‖T‖L(L2(H1))‖A‖HS .

Given v ∈ `2 with only finitely many components different from 0, take Av asthe matrix having the components of v on the top row, and 0 on the others. Then

‖AvM(λ)‖HS = ‖M(λ)∗v‖`2 ≤ ‖T‖L(L2(H1))‖Av‖HS = ‖T‖L(L2(H1))‖v‖`2 .

Therefore ‖M(λ)‖L(`2) = ‖M(λ)∗‖L(`2) ≤ ‖T‖. That each uj,k ∈ L∞(R) followseasily from the fact that it is measurable, and

uj,k = 〈M(λ)ek|ej〉`2 .

Hence M is a bounded Fourier multiplier and ‖M‖∞ ≤ ‖T‖L(L2(H1)). �

80 CHAPTER IV

How to transfer all this discussion to right-invariant operators is rather clear.The right-invariant analogues of the Zj , Zj are

Z(r) =1

2(X(r) − iY (r)) = ∂z +

i

4z∂t , Z(r) =

1

2(X(r) + iY (r)) = ∂z −

i

4z∂t .

Then Z(r)f(λ) can be expressed as in (3.3) and (3.4), only with the order of thetwo factors on the right-hand side reversed. This fact goes together with identitieslike

(3.6) Z(r)Φλj,k(z, t) = −

√(j + 1)λ

2Φλ

j+1,k(z, t) ,

valid for λ > 0. Precisely, we have

(3.7) Z(r)f(λ) = Uλf(λ) , Z(r)f(λ) = Lλf(λ) ,

and, for λ < 0,

(3.8) Z(r)f(λ) = Lλf(λ) , Z(r)f(λ) = Uλf(λ) ,

Theorem 3.1 has the same formulation, except for the order of the two factorsin (3.5), for right-invariant convolution operators, Tf = k ∗ f .

4. Radial functions and diagonal multipliers

In Section 1 we presented the rotations (z, t) 7−→ (Uz, t) of Hn, with U a unitarytransformation of Hn. For a function f defined on Hn, we set

fU (z, t) = f(Uz, t) .

The fact that rotations are automorphisms of Hn implies that

(4.1) (f ∗ g)U = fU ∗ gU .

If n = 1, rotations are just scalar multiplications by complex numbers of modulus1. We then write

fθ(z, t) = f(eiθz, t) .

We say that a function on H1 is radial if it depends only on |z| and t, or,equivalently, if fθ = f for every θ. More generally, we say that f is of type m ∈ Z if

fθ = eimθf

for every θ, or, equivalently, if e−im arg zf(z, t) is radial.These notions can be adapted to distributions as follows: a distribution u is of

type m if〈u, fθ〉 = e−imθ〈u, f〉 ,

for every test function f .Clearly, by expansion in Fourier serie in the angular variable, every function

(or distribution) decomposes as a sum of functions of the different types (withconvergence in a sense that depends on the function itself).

The function Φλj,k is of type k− j if λ > 0 and of type j − k if λ < 0. This gives

the following result.

HEISENBERG GROUP 81

Lemma 4.1. A function u is of type m if and only if f(λ, j, k) = 0, unless λ > 0and j − k = m, or λ < 0 and k − j = m. In particular, f is radial if and only if

f(λ) is diagonal for every λ 6= 0.If u is a radial tempered distributions, then uj,k = 0 for j 6= k.

Proof. Suppose that f is of type m, and take λ > 0. Then

eimθf(λ, j, k) =

∫

H1

f(eiθz, t)Φλj,k(z, t) dz dt

=

∫

H1

f(z, t)Φλj,k(e

−iθz, t) dz dt

= e−i(k−j)θ f(λ, j, k) ,

for every θ. So, if m 6= j−k, necessarily f(λ, j, k) = 0. The rest of the proof followsin the same way. �

Consider now a convolution operator Tf = f ∗ u, with u. If u is of type m,

(Tf)θ = fθ ∗ uθ = eimθfθ ∗ u = eimθT (fθ) .

Conversely, if (Tf)θ = eimθT (fθ) for every θ and every test function f , then

u ∗ f = eimθ(u ∗ fθ)−θ = eimθu−θ ∗ f .

Hence u = eimθu−θ, i.e. u is of type m.The special case m = 0 concerns the left-invariant operators that also commute

with rotations.

Theorem 4.2. Let Tf = f ∗ u be a bounded operator on L2(H1). The followingconditions are equivalent:

(i) T commutes with rotations;(ii) u is radial;(iii) the Fourier multiplier M(λ) is diagonal for a.e. λ;(iv) T = µ(iT, L), for some bounded spectral multiplier µ(λ, ξ) on the Heisenberg

fan F1.

If these conditions are satisfied, then µ(λ, ξ) and the diagonal entries m(λ, k, k)of M(λ) are related by the identity

m(λ, k, k) = µ(λ, |λ|(2k + 1)

).

Proof. The equivalence between (i) and (ii) follows from the previous remarks. Theimplication (ii)⇒(iii) follows from Lemma 4.1, since uj,k = m(·, j, k).

Given a Borel subset ω in R2, define the bounded Fourier multiplier Mω =(Mω(λ, j, k)

)j,k

as

Mω(λ, j, k) =

{1 if

(λ, |λ|(2k+ 1)

)∈ ω and j = k ,

0 otherwise ,

and let E(ω) be the corresponding orthogonal projection on L2(H1). This define aresolution of the identity and its support is the Heisenberg fan F1.

82 CHAPTER IV

For f, g ∈ S(H1), it follows from the Plancherel formula that

νf,g(ω) = 〈E(ω)f |g〉 =1

(2π)2

∫

R

∑{

(j,k):(λ,|λ|(2k+1))∈ω} f(λ, j, k)g(λ, j, k) |λ| dλ .

This implies that

(4.2)

∫

F1

ϕ(λ, ξ)dνf,g(λ, ξ) =

=1

(2π)2

∫

R

∑

j,k∈N

ϕ(λ, |λ|(2k + 1)

)f(λ, j, k)g(λ, j, k) |λ| dλ .

Let

A =

∫

F1

λ dE(λ, ξ) , B =

∫

F1

ξ dE(λ, ξ) .

The domain D(A) of A consists of the functions f such that∫

F1

λ2 dνf,f (λ, ξ) =1

(2π)2

∫

R

∑

j,k∈N

λ2|f(λ, j, k)|2 |λ| dλ <∞ .

Let V be the space of finite families v = {vj,k}(j,k)∈Bvof functions vj,k ∈ D(R \

{0}). By Corollary 2.9, V consists of the finite families {g(·, j, k)} with g ∈ S0(H1).Then

( ∫

R

∑

j,k∈N

λ2|f(λ, j, k)|2 |λ| dλ) 1

2

= supv∈V

∫R

∑(j,k)∈Bv

λf(λ, j, k)vj,k(λ) |λ| dλ( ∫

R

∑(j,k)∈Bv

|vj,k(λ)|2 |λ| dλ) 1

2

= 2π supg∈S0(H1)

〈f | − iTg〉‖g‖2

= 2π supg∈S0(H1)

〈iTf |g〉‖g‖2

Since S0(H1) is dense in L2(H1), f ∈ D(A) if and only if iTf (defined as adistribution) is in L2, i.e. D(A) = D(iT ). Moreover, for f, g ∈ S(H1),

〈Af |g〉 =

∫

F1

λ dνf,g(λ, ξ)

=1

(2π)2

∫

R

∑

j,k∈N

λf(λ, j, k)g(λ, j, k) |λ| dλ

= 〈iTf, g〉 .Hence A = iT . A similar argument shows that B = L.Applying (4.2) with ϕ = µ, we then have, by (2.16),

〈µ(iT, L)f |g〉 =1

(2π)2

∫

R

∑

j,k∈N

µ(λ, |λ|(2k+ 1)

)f(λ, j, k)g(λ, j, k) |λ| dλ

=1

(2π)2

∫

R

tr(f(λ)M(λ)g(λ)∗

)|λ| dλ

= 〈TMf |g〉 .

HEISENBERG GROUP 83

This proves the equivalence between (iii) and (iv).

Finally, assume that T = TM , with M satisfying (iii). Given f ∈ S(Hn), let fm,m ∈ Z, be the components of f of type m. Then Tfm is also of type m by Lemma4.1. Therefore,

T (fθ) =∑

m∈Z

T((fm)θ

)

=∑

m∈Z

eimθT (fm)

=∑

m∈Z

T (fm)θ

= (Tf)θ .

This shows that (iii)⇒(i). �

5. Radiality in Hn

We extend the Fourier analysis on H1 presented in the last three sections to Hn.A large part of what we are going to say is a straightforward adaptation of whathas been presented in detail for H1 (only with more complicated notation), and weleave the verification to the reader. The new facts will arise when we will presentthe different notions of radiality.

Let j = (j1, . . . , jn),k = (k1, . . . , kn) be two n-tuples in Nn. For λ ∈ R \ {0}, weset

(5.1) Φλj,k(z, t) = eiλt

n∏

i=1

ϕλji,ki

(zi) .

For f ∈ L1(Hn), define

(5.2) f(λ, j,k) =

∫

Hn

f(z, t)Φλj,k(z, t) dz dt ,

and, with Φλ(z, t) =(Φλ

j,k(z, t))j,k∈Nn ,

(5.3) f(λ) =(f(λ, j,k)

)j,k∈Nn =

∫

Hn

f(z, t)Φλ(z, t) dz dt .

Theorem 5.1.

(1) Φλ(z, t) is unitary for every (z, t);(2) Φλ

((z, t), (w, u)

)= Φλ(z, t)Φλ(w, u), Φλ

((z, t)−1

)= Φλ(z, t)∗;

(3) LΦλj,k = |λ|

(2|k| + n

)Φλ

j,k and TΦλj,k = iλΦλ

j,k;

(4) f ∗ g(λ) = f(λ)g(λ) for every f, g ∈ L1(Hn) and every λ 6= 0;

(5) Lf(λ, j,k) = |λ|(2|k| + n

)f(λ, j,k) and T f(λ, j,k) = −iλf(λ, j,k);

84 CHAPTER IV

(6) if λ > 0 and ei = (0, 0, · · · , 1, . . . , 0), with the 1 in the i-th position,

Zif(λ, j,k) =

√kiλ

2f(λ, j,k− ei) ,

Zif(λ, j,k) = −√

(ki + 1)λ

2f(λ, j,k + ei) ,

Z

(r)i f(λ, j,k) =

√(ji + 1)λ

2f(λ, j + ei,k) ,

Z

(r)i f(λ, j,k) = −

√jiλ

2f(λ, j − ei,k) ,

(7) if λ < 0,

Zif(λ, j,k) = −√

(ki + 1)|λ|2

f(λ, j,k + ei) ,

Zif(λ, j,k) =

√ki|λ|

2f(λ, j,k− ei) ,

Z

(r)i f(λ, j,k) = −

√ji|λ|

2f(λ, j− ei,k) ,

Z

(r)i f(λ, j,k) =

√(ji + 1)|λ|

2f(λ, j + ei,k) ,

(8) the following Plancherel formula holds, for f ∈ L2(Hn):

∫

Hn

|f(z, t)|2 dz dt =1

(2π)n+1

∫

R

‖f(λ)‖2HS |λ|n dλ .

(9) the following inversion formula holds, for f ∈ S(Hn):

f(z, t) =1

(2π)n+1

∫

R

tr(f(λ)Φλ(z, t)∗

)|λ|n dλ .

Moreover, Theorem 3.1 has the same formulation on Hn.The formulas at points (6) and (7) can be expressed in analogy with (3.3) and

(3.4). Define the matrices Uλ,i, Lλ,i with indices (j,k) ∈ (Nn)2 by

(5.4) (Uλ,i)j,k =

√ki|λ|

2δj,k−ei

, (Lλ,i)j,k = −√

(ki + 1)|λ|2

δj,k+ei.

Then, for λ > 0,

(5.5) Zif(λ) = f(λ)Uλ,i ,Zif(λ) = f(λ)Lλ,i ,

and, for λ < 0,

(5.6) Zif(λ) = f(λ)Lλ,i ,Zif(λ) = f(λ)Uλ,i ,

HEISENBERG GROUP 85

The representation-theoretic formulation of (1) and (2) can be given in terms ofthe unitary representations πλ (for λ 6= 0) of Hn on `2(Nn), such that πλ(z, t) isthe operator defined by the matrix Φλ(z, t) in the canonical basis.

If n > 1, we must distinguish between two notions of radiality.We say that a function f(z, t) is radial if it only depends on |z| and t. This is

equivalent to saying that f(Uz, t) = f(z, t) for every unitary n × n matrix U andevery (z, t).

We say that a function f(z, t) is poliradial if it depends on |z1|, . . . , |zn| and t.This is equivalent to saying that f(Uθz, t) = f(z, t) for every unitary diagonal n×nmatrix,

Uθ =

eiθ1 0 · · · 00 eiθ2 · · · 0...

.... . .

...0 0 · · · eiθn

,

and every (z, t). We shall write fθ(z, t) for f(Uθz, t).The natural extensions of Lemma 4.1 and Theorem 4.2 concerns poliradial func-

tions and joint spectral multipliers of T and the partial sub-Laplacians Li = −X2i −

Y 2i . We group them in one statement, disregarding the first part of Lemma 4.119.

Theorem 5.2. A function f ∈ L1(Hn) is poliradial if and only if f(λ) is diagonalfor every λ 6= 0. For a bounded operator Tf = f ∗ u on L2(H1), the followingconditions are equivalent:

(i) T commutes with the rotations Uθ, i.e. T (fθ) = (Tf)θ for every θ;(ii) u is poliradial;(iii) the Fourier multiplier M(λ) is diagonal for a.e. λ;(iv) T = µ(iT, L1, . . . , Ln), for some bounded spectral multiplier µ(λ, ξ1, . . . , ξn).

If these conditions are satisfied, then µ(λ, ξ1, . . . , ξn) and the diagonal entriesm(λ,k,k) of M(λ) are related by the identity

m(λ,k,k) = µ(λ, |λ|(2k1 + 1), . . . , |λ|(2kn + 1)

).

The condition on f(λ) characterizing radial functions must be more restrictive.

We shall show that it consists in the fact that f(λ,k,k) = f(λ,k′,k′) if |k| = |k′|.Lemma 5.3. A polyradial function f ∈ S(Hn) is radial if and only if for everyi, i′,

ZiZ(r)i′ f = Zi′Z

(r)i f .

Proof. Write f(z, t) = g(r1, . . . , rn, t), with ri = |zi|2. Then f is radial if and onlyif g only depends on r1 + · · · + rn and t, i.e. if and only if ∂ri

g = ∂ri′g for every

i, i′. Since

zi∂zi′f − zi′∂zi

f = zizi′∂ri′g − zi′ zi∂ri

g = zizi′(∂ri′− ∂ri

)g ,

it follows that f is radial if and only if

(zi∂zi′− zi′∂zi

)f = 0

19One can introduce the notion of type m, for m ∈ Zn, and extend that part too.

86 CHAPTER IV

for every i, i′. Now,

(ZiZ(r)i′ − Zi′Z

(r)i )f =

(∂zi

− i

4zi∂t

)(∂zi′

+i

4zi′∂t

)f

−(∂zi′

− i

4zi′∂t

)(∂zi

+i

4zi∂t

)f

=i

2(zi′∂zi

− zi∂zi′)∂tf .

Therefore, if f is radial, (ZiZ(r)i′ − Zi′Z

(r)i )f = 0. Conversely, if ZiZ

(r)i′ f =

Zi′Z(r)i f , we obtain that ∂tf is radial. Since ∂tf ∈ S(Hn),

f(z, t) =

∫ t

−∞

∂tf(z, u) du

is also radial. �

Theorem 5.4. A function f ∈ S(Hn) is radial if and only if f(λ) is diagonal and

f(λ,k,k) only depends on |k|.

Proof. Assume that f is radial. In particular, it is poliradial, hence f(λ) is diagonal.

Moreover, by Lemma 5.3, ZiZ(r)i′ f = Zi′Z

(r)i f for every i, i′. By Theorem 5.1, (6)

and (7), we obtain that, for λ > 0,

(ZiZ

(r)i′ f)(λ,k− ei′ ,k + ei) =

√(ki + 1)ki′λ

2f(λ,k,k) ,

and

(Zi′Z(r)i f)(λ,k′ − ei,k

′ + ei′) =

√k′i(k

′i′ + 1)λ

2f(λ,k′,k′) .

Fix two different indices i, i′and take k,k′ such that k + ei = k′ + ei′ . Then thetwo left-hand sides coincide, and so do the expressions under square root. We mustthen have

f(λ,k,k) = f(λ,k′,k′) .

This means that, moving one unit from one entry of k to another entry, the valuefo the Fourier coefficient does not change. Repeating this operation, we can passfrom any k to any other k′ with the length.

For λ < 0 the argument is the same, and this proves the first part of the state-ment.

Suppose conversely that f ∈ S(Hn) and f(λ, j,k) = δj,kν(λ, |k|). By the inver-sion formula,

f(z, t) =1

(2π)n+1

∫

R

∑

k∈N

ν(λ, k)Ψλk(z, t) |λ|n dλ ,

withΨλ

k(z, t) =∑

|k|=k

Φλk,k(z, t) .

HEISENBERG GROUP 87

Since each Φλk,k is poliradial, Ψλ

k is poliradial too. As in Section 3, for λ > 0,

ZiΦλk,k = −

√kiλ

2Φλ

k,k−ei

and

Z(r)i Φλ

k,k = −√

(ki + 1)λ

2Φλ

k+ei,k.

Therefore

ZiZ(r)i′ Φλ

k,k =

√ki(ki′ + 1)λ

2Φλ

k+ei′ ,k−ei.

Summing over k and setting k′ = k+ei′−ei, we find that ZiZ(r)i′ Ψλ

k = Zi′Z(r)i Φλ

k .As in the proof of Lemma 5.3, this implies that (zi′∂zi

− zi∂zi′)∂tΨ

λk = 0. But, since

Ψλk(z, t) = ψλ

k (z)eiλt, we conclude that ψλk only depends on |z|.

Finally, repeating the same argument for λ < 0, we conclude that Ψλk is radial

for every λ 6= 0, and hence f is radial. �

We need at this point to describe the orthogonal projection P from L2(Hn) ontothe subspace of radial functions. This requires some notions concerning the groupU(n) of unitary transformations of Cn. With the natural topology, induced from

Cn2

, U(n) is compact and the proup operations are continuous. The basic fact isthe existence of a unique Borel probability measure m (called the Haar measure)which is invariant under left and right translations20, i.e. such that

m(E) = m(gE) = m(Eg) = m(E−1) ,

for every Borel set E and every g ∈ U(n). As a consequence,∫

U(n)

f(x) dm(x) =

∫

U(n)

f(gx) dm(x)

=

∫

U(n)

f(xg) dm(x) =

∫

U(n)

f(x−1) dm(x) ,

for every integrable function f and every g ∈ U(n).

Lemma 5.5. The orthogonal projection P from L2(Hn) onto the subspace of radialfunctions L2

rad(Hn) is given by

(5.7) Pf(z, t) =

∫

U(n)

f(Uz, t) dm(U) ,

and

(5.8) P f(λ, j,k) = δj,k

(|k| + n− 1

n− 1

)−1 ∑

k′:|k′|=|k|

f(λ,k′,k′) .

P is well-defined and continuous on the following spaces:

(i) Lp(Hn), for 1 ≤ p ≤ ∞;(ii) S(Hn) into itself continuously, and it can therefore be extended by duality

to S ′(Hn);(iii) S0(Hn), defined as in Corollary 2.9.

20See, e.g., the notes of the course Analisi di Fourier non commutativa.

88 CHAPTER IV

If X denotes each of these spaces, the image of P in X is the subspace Xrad ofall radial functions (or distributions) in the space itself. In particular Srad(Hn) andS0,rad(Hn) are dense in Lp

rad(Hn) for 1 ≤ p <∞.

Proof. Since

Pf(Uz, t) =

∫

U(n)

f(U ′Uz, t) dm(U ′) =

∫

U(n)

f(U ′z, t) dm(U ′) = Pf(z, t) ,

the image of P consists of radial functions. If f is already radial, it is clear thatPf = f . Hence P 2 = P and the image consists of all radial functions. Givenf, g ∈ L2(Hn),

〈Pf |g〉 =

∫

Hn

∫

U(n)

f(Uz, t)g(z, t)dm(U) dz dt

=

∫

U(n)

∫

Hn

f(Uz, t)g(z, t)dz dt dm(U)

=

∫

U(n)

∫

Hn

f(z, t)g(U−1z, t) dz dt dm(U)

= 〈f |Pg〉 .

Hence P is an orthogonal projection.Formula (5.8) is a direct consequence of Theorem 5.4 and the Plancherel formula,

once we have observed that the binomial coefficient in front of the sum gives thenumber of k′ with the same length of k..

If f ∈ Lp(Hn), the Minkowski integral inequality gives

‖Pf‖p =

∥∥∥∥∫

U(n)

fU dm(U)

∥∥∥∥p

≤∫

U(n)

‖fU‖p dm(U) = ‖f‖p .

Boundedness of P on S(Hn) follows from (5.7) by differentiation under the in-tegral sign, and on S0(Hn) from (5.8).

The last part is then obvious. �

Theorem 5.6. For a bounded operator Tf = f ∗ u on L2(H1), the following con-ditions are equivalent:

(i) T commutes with all the rotations of Hn, i.e. T (fU ) = (Tf)U for everyU ∈ U(n);

(ii) T maps radial functions into radial functions;(iii) u is radial;(iv) the Fourier multiplier M(λ) has the form

M(λ) =(δj,kν

(λ, |k|

))j,k

;

(v) T = µ(iT, L), for some bounded spectral multiplier µ(λ, ξ) on Fn.

If these conditions are satisfied, then the spectral multiplier µ(λ, ξ) and the diag-onal entries ν(λ, k) of M(λ) are related by the identity

ν(λ, k) = µ(λ, |λ|(2k + n)

).

HEISENBERG GROUP 89

Proof. The implication (i)⇒(ii) is trivial, because if f is radial, (Tf)U = T (fU ) =Tf for every U , hence Tf is radial. To prove that (ii)⇒(iii), take a radial functionϕ ∈ S(Hn) with

∫Hn

ϕ = 1. Then the functions

ϕε(z, t) =1

ε2n+2ϕ(zε,t

ε2

)

form an approximate identity as ε→ 0. In particular,

limε→0

T (ϕε) = limε→0

u ∗ ϕε = u

in S ′(Hn). But T (ϕε) is radial by assumption, hence u is also radial.

Assume now that u is radial. Then, for any U ∈ U(n),

(Tf)U = (u ∗ f)U = uU ∗ fU = u ∗ fU = T (fU ) ,

which give the implication (iii)⇒(i).

The implication (iv)⇒(ii) is obvious by Theorem 5.4. To prove that (i)⇒(iv),we fix ϕ ∈ D(R \ {0}) and k ∈ N, and define

ϕk(z, t) =1

(2π)n+1

∫

R

ϕ(λ)∑

|k|=k

Φλk,k(z, t) |λ|n dλ .

Then ϕk ∈ S0(Hn) and, by Theorem 5.4, it is radial. Hence, since we know that(i) implies (ii), T ϕk is also radial.

Since T commutes with diagonal rotations, we know from Theorem 5.2 that theFourier multiplier of T is diagonal. Let m(λ,k,k) be its entries on the diagonal.Then

T ϕk(λ,k,k) = ϕk(λ,k,k)m(λ,k,k) = δ|k|,kϕ(λ)m(λ,k,k) ,

must depend only on λ and |k|. By the arbitrarity of ϕ and k, m(λ,k,k) = ν(λ, |k|).Finally the equivalence between (iv) and (v) is proved as in Theorem 4.2. �

Define

(5.9) Ψλk(z, t) =

(k + n− 1

n− 1

)−1 ∑

|k|=k

Φλk,k(z, t) ,

for k ∈ N. It follows from this definition that

(5.10) Ψλk(z, t) = eiλtψk

(|λ||z|2

),

where ψk is a Schwartz function.

The relevance of the functions Ψλk in our context is clarified by the next state-

ment.

90 CHAPTER IV

Proposition 5.7. Up to scalar multiples, Ψλk is the unique radial function in

span {Φλk,k : |k| = k}. If |k| = k, then PΦλ

k,k = Ψλk , and, if f ∈ L1(Hn),

(5.11) P f(λ,k,k) =

∫

Hn

f(z, t)Ψλk(z, t) dz dt .

Proof. We start by proving (5.10). If f ∈ L1 ∩ L2(Hn), it follows directly from(5.8), and it extends to any integrable f by continuity, since Ψλ

k ∈ L∞(Hn).Take now

u =∑

|k|=k

ckΦλk,k ∈ L∞(Hn) .

Then, for f ∈ L1(Hn),

〈Pu, f〉 = 〈u, Pf〉=

∑

|k|=k

ckP f(λ,k,k)

=∑

|k|=k

ck〈Ψλk , f〉 ,

by (5.11). Hence Pu is a scalar multiple of Ψλk , and the rest of the statement follows

easily. �

6. Applications

Fourier analysis on Hn can be used to derive the regularity properties of thesub-Laplacian. We shall prove estimates showing that if f and Lkf are in L2(Hn),then all the derivatives of f up to a certain order are (at least locally) in L2. Thegeneral name for this type of results is sub-elliptic estimates, and they are typicalof hypoelliptic operators.

In the second part of this Section, we characterize Fourier transforms of radialSchwartz functions.

The most efficient way to state hypoelliptic estimates for L is in terms of left-invariant vector fields, because they can be stated in global form.

Consider “higher-order left-invariant derivatives” of f , meant as expressions likeT 2X1Y2Y1X

32f , or like Z2

2Z1Z1Tf . The order of the factors in these expressions isrelevant; we call them non-commutative monomials21. Clearly one can switch fromnon-commutative polynomials in the Xj, Yj, T to non-commutative polynomials inthe Zj , Zj, T by simple formal manipulations. The use of the Zj , Zj is preferablefor us, because formulas for the Fourier transform are simpler. Observe also that,due to the relations

[Xj, Yj] = T , [Zj, Zj] =i

2T ,

21We mention that it follows from the Poincare-Birckhoff-Witt theorem (see, e.g., the notes

Sub-Laplacians on nilpotent Lie groups). that every left-invariant differential operator on Hn canbe written as a non-commutative polynomial.

HEISENBERG GROUP 91

different non-commutative polynomials can give the same differential operator22.In particular, one can always replace a non-commutative polynomial by anotherone not containing T , without altering the differential operator.

Let P (Z, Z) be a non-commutative polynomial in the Zj , Zj only. The degreeof P is defined in the usual way. If T also appears in P , it must be countedas a factor of degree 2. With this convention, the non-isotropic order of a left-invariant diffeerential operator is well defined, as the degree of any non-commutativepolynomial representing it.

Theorem 6.1. Let N ∈ N, and assume that f and LN/2f are in L2(Hn). Thenalso P (Z, Z, T )f ∈ L2(Hn) for every P of non-isotropic degree at most N , and

‖Pf‖2 ≤ CN

(‖f‖2 + ‖LN/2f‖2

).

Moreover, all partial derivatives of f up to the order [N/2] are locally in L2.

Proof. It is sufficient to take a monomial P = P (Z, Z) in the Zj , Zj of degree

d ≤ 2N . For each λ 6= 0, P f(λ) is given by

P f(λ) = f(λ)P (Uλ, Lλ) , or P f(λ) = f(λ)P (Lλ, Uλ) ,

depending on the signum of λ, where Uλ, Lλ stand for the matrices Uλ,i, Lλ,i in(5.4). The diagonal matrix Dλ with

(Dλ)k,k = 1 +(|λ|(2|k| + n)

)N/2

is such that(

(I + LN/2)f)(λ) = f(λ)Dλ .

Therefore, if

Mλ =

{P (Uλ, Lλ)D−1

λ if λ > 0 ,

P (Lλ, Uλ)D−1λ if λ < 0 ,

we have that

P f(λ) = ((I + LN/2)f

)(λ)Mλ .

Observe that both P (Uλ, Lλ) and P (Lλ, Uλ) have non-zero entries only on onesingle diagonal, and the k-th entry along this diagonal is dominated by

(|λ|(|k| +

1))d/2

. Therefore the norms ‖Mλ‖L(`2) are uniformly bounded in λ. It follows fromTheorem 3.1 (which holds also on Hn, as already mentioned in Section 5) that

‖Pf‖2 ≤ C‖f + LN/2f‖2 ≤ C(‖f‖2 + ‖LN/2f‖2

).

The last part of the statement follows from the fact that, by the explicit ex-pression of the vector fields, L2-estimates for Zig, Zig, Tg imply local L2-estimatesfor ∂zi

g, ∂zig, ∂tg. An induction argument shows that, in our hypotheses, we can

control locally all partial derivatives of f up to order [N/2]. �

A similar argument, made simpler by the fact that all the matrices involved arediagonal, gives the following result.

22In more correct terms, a non-commutative polynomial is an element of the tensor algebraT over C generated by the Zj , Zj , T . The map assigning to each element of T the corresponding

composition of left-invariant vector fields on Hn has a kernel, equal to the ideal I generated by

the elements Zj ⊗ Zk − Zk ⊗ Zj − i2δj,kT . The quotient T /I identifies left-invariant differential

operators on Hn and is called the universal enveloping algebra of the Lie algebra hn.

92 CHAPTER IV

Theorem 6.2. . Assume that f and Lsf are in L2(Hn) for some s ∈ R+. ThenLs1(i−1T )s2−s1f ∈ L2(Hn), for s1, s2 ≥ 0, s2 ≤ s, and

‖Ls1(i−1T )s2−s1f‖2 ≤ Cs

(‖f‖2 + ‖Lsf‖2

).

A further refinement gives global Sobolev estimates for poliradial functions.

Corollary 6.3. Suppose that f is poliradial and that f, LN/2f ∈ L2(Hn). Then

∂αz ∂

βz ∂

mt f ∈ L2(Hn) for all multi-indices α, β and all m ∈ N such that |α| + |β| +

2m ≤ N , and

‖∂αz ∂

βz ∂

mt f‖2 ≤ CN

(‖f‖2 + ‖LN/2f‖2

).

Proof. Consider the right-invariant sub-Laplacian

L(r) = −n∑

i=1

((X

(r)i )2 + (Y

(r)i )2

).

If Lf(λ) = f(λ)M(λ), then L(r)f(λ) = M(λ)f(λ). Since M(λ) is diagonal, it

commutes with f(λ) if f is poliradial. Hence L(r)f = Lf . Since the analogue of

Theorem 6.1 also holds for right-invariant operators, we have that P (Z(r)i , Z

(r)i )f ∈

L2(Hn) for every non-commutative monomial P of degree at most 2N .Starting from the identities

Zi + Z(r)i = 2∂zi

, Zi + Z(r)i = 2∂zi

,

and expressing ∂t as a commutator, we reduce the problem of estimating ∂αz ∂

βz ∂

mt f

to proving that

‖Q(Z, Z, Z(r), Z(r))f‖2 ≤ C(‖f‖2 + ‖LN/2f‖2

),

if Q = Q(Z, Z, Z(r), Z(r)) is a non-commutative monomial of degree at most N .It is a general fact (and it can be easily verified from the explicit formulas or

from the Fourier transforms) that right-invariant vector fields commute with left-invariant ones. Hence they also commute with L, and, by Theorem 3.1 extendedto Hn, with its spectral projection, and ultimately with its fractional powers. Thesame can be said interchanging left and right.

We can therefore write

Q = Q1(Z, Z)Q2(Z(r)Z(r)) ,

where Q1 and Q2 are monomials of degrees d1 and d2 respectively, with d1+d2 ≤ N .By Theorems 6.1 and 6.2,

‖Qf‖2 ≤ C(‖Q2(Z

(r)Z(r))f‖2 + ‖Ld1/2Q2(Z(r)Z(r))f‖2

)

≤ C(‖f‖2 + ‖(L(r))d2/2f‖2 + ‖Q2(Z

(r)Z(r))Ld1/2f‖2

)

≤ C(‖f‖2 + ‖(L(r))d2/2f‖2 + ‖Ld1/2f‖2 + ‖(L(r))d2/2Ld1/2f‖2

)

= C(‖f‖2 + ‖Ld2/2f‖2 + ‖Ld1/2f‖2 + ‖L(d1+d2)/2f‖2

)

≤ C(‖f‖2 + ‖LN/2f‖2

). �

HEISENBERG GROUP 93

We conclude this section with the proof of some identities providing conditionson the Fourier transform side that correspond to decay at infinity of the function.

In Rn this is provided by formulas like F(xjf)(ξ) = i∂ξjf(ξ). In Hn, staying within

radial functions, we look for formulas relating f(λ) with |z|2f(λ) and with tf(λ).If f is radial, we set

f(λ, k) = f(λ,k)

if |k| = k.

Lemma 6.4. Assume that f ∈ S(Hn) is radial. Then

(|z|2f)(λ, k) =2

|λ|((2k + n)f(λ, k) − (k + n)f(λ, k + 1) − kf(λ, k − 1)

),

and

(tf)(λ, k) = −i∂λf(λ, k) − i

2λ

(nf(λ, k) − (k + n)f(λ, k + 1) + kf(λ, k − 1)

).

Proof. Take f ∈ S(Hn), not necessarily radial. For each i = 1, . . . , n,

Zi − Z(r)i = − i

2zi∂t , Zi − Z

(r)i =

i

2zi∂t .

Therefore

(zi∂tf)(λ, j,k) =

i√

2kiλf(λ, j,k− ei) − i√

2(ji + 1)λf(λ, j + ei,k) if λ > 0 ,

−i√

2(ki + 1)|λ|f(λ, j,k + ei) + i√

2ji|λ|f(λ, j− ei,k) if λ < 0 ,

and

(zi∂tf)(λ, j,k) =

i√

2(ki + 1)λf(λ, j,k + ei) − i√

2jiλf(λ, j − ei,k) if λ > 0 ,

−i√

2ki|λ|f(λ, j,k− ei) + i√

2(ji + 1)|λ|f(λ, j + ei,k) if λ < 0 .

Therefore, for every λ 6= 0,

(|zi|2∂2t f)(λ, j,k) = −2(ji + ki + 1)|λ|f(λ, j,k)

+ 2|λ|√

(ji + 1)(ki + 1)f(λ, j + ei,k + ei)

+ 2|λ|√jikif(λ, j− ei,k− ei) .

Summing over i and restricting to f radial and j = k, we obtain that

˜(|z|2∂2t f)(λ, k) = −2(2k + n)|λ|f(λ, k) + 2|λ|(k + n)f(λ, k + 1)

+ 2|λ|kf(λ, k − 1) .

94 CHAPTER IV

But ˜(|z|2∂2t f) = ˜(∂2

t |z|2f) = −λ2(|z|2f), and this proves the first formula.

In order to prove the second formula, we start from the derivative of f(λ, k) in λ.By (5.10) and (5.11), if f is radial,

f(λ, k) =

∫

Hn

f(z, t)eiλtψk

(|λ||z|2

)dz dt ,

so that

∂λf(λ, k) = i

∫

Hn

tf(z, t)eiλtψk

(|λ||z|2

)dz dt

+ sgn λ

∫

Hn

|z|2f(z, t)eiλtψ′k

(|λ||z|2

)dz dt

= i(tf)(λ, k) + sgnλ

∫

Hn

|z|2f(z, t)eiλtψ′k

(|λ||z|2

)dz dt .

Observing that

n∑

i=1

zi∂ziψk

(|λ||z|2

)= |λ||z|2ψ′

k

(|λ||z|2

),

we obtain

(6.1)

∂λf(λ, k) = i(tf)(λ, k)

− 1

λ

∫

Hn

n∑

i=1

∂zi

(zif(z, t)

)Ψλ

k(z, t) dz dt

= i(tf)(λ, k) − 1

λ

˜( n∑

i=1

∂zi(zif)

)(λ, k)

(observe that∑∂zi

(zif) is also radial).

Using again the fact that Zi − Z(r)i = i

2zi∂t, and that Zi +Z

(r)i = 2∂zi

, we have

i∂t∂zi(zif) = (Zi + Z

(r)i )(Zi − Z

(r)i )f .

With computations similar to the previous ones, we find that, for λ 6= 0,

λ ∂zi(zif)(λ,k,k) =

λ

2

(f(λ,k,k)+kif(λ,k−ei,k−ei)−(ki+1)f(λ,k+ei,k+ei)

).

Summing over i,

˜( n∑

i=1

∂zi(zif)

)(λ, k) =

1

2

(nf(λ, k) + kf(λ, k − 1) − (k + n)f(λ, k + 1)

).

Inserting this identity in (6.1), we find the stated formula. �

HEISENBERG GROUP 95

The formula indicated in Lemma 6.4 are rather complicated, since they involvestrange second-order differences in k. A considerable simplification occurs if we

combine the two formulas to express the Fourier transform of( |z|2

4± it

)f . Setting

w±(z, t) = |z|2

4 ± it, we have the following identities:

(6.2) (w+f)(λ, k) =

∂λf(λ, k) − k + n

λ

(f(λ, k + 1) − f(λ, k)

)if λ > 0 ,

∂λf(λ, k) − k

λ

(f(λ, k) − f(λ, k − 1)

)if λ < 0 ,

and

(6.3) (w−f)(λ, k) =

−∂λf(λ, k) +k

λ

(f(λ, k) − f(λ, k − 1)

)if λ > 0 ,

−∂λf(λ, k) +k + n

λ

(f(λ, k + 1) − f(λ, k)

)if λ < 0 .

96 CHAPTER IV

CHAPTER V

SPECTRAL MULTIPLIERS OF THE SUB-LAPLACIAN

1. The heat kernel on Hn

Our discussion of spectral Lp-multipliers of L on Hn begins with the study ofthe heat kernel. This is defined as the kernel ps(z, t), for s > 0, such that

f ∗ ps = e−sLf .

We recall some general facts about semigroups and evolution equations on Liegroups.

The semigroup property e−(s1+s2)L = e−s1Le−s2L implies that

ps1+s2= ps1

∗ ps2

for every s1, s2 > 0. This identity extends to s = 0 if we set p0 = δ0, consistentlywith the fact that e0L = I. Moreover, the map s 7−→ e−sL is continuous from[0,∞) to L(L2) with the strong topology.

The identity ddse

−sLf = Le−sLf holds for every s > 0 and f ∈ L2(Hn). There-

fore, the function u(s, z, t) = e−sLf(z, t) satisfies the homogeneous heat equation

(∂s + L)u = 0 ,

which implies that(∂s + L)ps = 0

in the sense of distributions.By the already cited Hormander’s theorem23, the operator ∂s +L is hypoelliptic,

which implies that ps(z, t) is smooth in (s, z, t).Finally, it follows from Hunt’s theorem24 that the ps define probability measures,

i.e.

(1.1) ps(z, t) ≥ 0 ,

∫

Hn

ps(z, t) dz dt = 1 .

We shall prove now further properties of the ps, using the Fourier analysis onHn.

23See L. Hormander, Hypoelliptic second-order differential equations, Acta Math. vol.119(1967), p.147-171.

24See A. Hunt, Semigroups of measures on Lie groups, Trans. Amer. Math. Soc. vol.81

(1956), p.264-293.

Typeset by AMS-TEX

SPECTRAL MULTIPLIERS 97

By Theorem 5.6 in Chapter IV,

ps(λ, k) = e−s|λ|(2k+n) ,

so that, by the inversion formula in Theorem 5.1 of Chapter IV,

(1.2) ps(z, t) =1

(2π)n+1

∫

R

∑

k∈N

(k + n− 1

n− 1

)e−s|λ|(2k+n)Ψλ

k(z, t) |λ|n dλ .

We show that ps is a Schwartz function, and, more precisely that it decaysexponentially at infinity.

Proposition 1.1. For λ 6= 0,

∑

k∈N

(k + n− 1

n− 1

)e−s|λ|(2k+n)Ψλ

k(z, t) =2neiλt

sinhn(|λ|s)e−

|λ||z|2

tanh(|λ|s) .

The proof is based on the following lemma.

Lemma 1.2. For 0 ≤ r < 1 and z ∈ C,

∑

k∈N

rkϕk,k(z) =1

1 − re−

1+r4(1−r)

|z|2 .

Proof. By (3.3) and (3.4) in Chapter I,

hj,k(z) = (−2)je|z|2

4 ∂kz ∂

jze

−|z|2

2 ,

and, by (2.2) in Chapter IV,

ϕk,k(z) =(−2)k

k!e

|z|2

4 ∂kz ∂

kz e

−|z|2

2 =1

2kk!e

|z|2

4 ∆ke−|z|2

2 .

Then ∑

k∈N

rkϕk,k(z) = e|z|2

4

∑

k∈N

rk

2kk!∆ke−

|z|2

2 = e|z|2

4 F (r, z) ,

where the series converges for r < 1 because |ϕj,k(z)| ≤ 1. Recalling that, for s > 0,

(1.3) F(1

2πse−

|z|2

4s ) = e−s|ζ|2 ,

we have

F(∆ke−|z|2

2 ) = π|ζ|2ke−|ζ|2

2 .

Therefore

F (r, z) = πF−1( ∑

k∈N

rk

2kk!|ζ|2ke−

|ζ|2

2

)

= πF−1(e−1−r2 |ζ|2)

=1

1 − re−

|z|2

2(1−r) .

98 CHAPTER V

Multiplying by e|z|2

4 , the proof is completed. �

Proof of Proposition 1.1. Assume first that λ = 1. By (5.9) and (5.10) in Chap-ter IV, and by Lemma 1.2,

∑

k∈N

(k + n− 1

n− 1

)e−s(2k+n)Ψ1

k(z, t) = eit∑

k∈Nn

e−s(2|k|+n)ϕk,k(z)

= eitn∏

i=1

( ∑

k∈N

e−s(2k+1)ϕk,k(zi)

)

= eitn∏

i=1

e−s

1 − e−2se− 1+e−2s

1−e−2s |zi|2

=eit

(es − e−s)ne− es+e−s

es−e−s |z|2

=2neit

sinhn se−

|z|2

tanh s .

For a generic λ, it is sufficient to replace t by λt, s by |λ|s, and z by λ12 z or

|λ| 12 z, depending on the signum of λ. �

By (1.2), we then have

(1.4)

ps(z, t) =1

2πn+1

∫

R

eiλt

sinhn(|λ|s)e−

|λ||z|2

tanh(|λ|s) |λ|n dλ

=1

2πn+1

∫

R

eiλt

sinhn(λs)e−

λ|z|2

tanh(λs) λn dλ .

Observe that

(1.5) ps(z, t) = s−(n+1)p1

(s−

12 z, s−1t

),

which is the analogue of the scaling property for the Gauss-Weierstrass heat kernelfor the Laplacian in Rn.

Corollary 1.3. For s > 0, ps ∈ S(Hn).

Proof. We can take s = 1 and prove that Ftp1 is a Schwartz function. By (1.4),

Ftp1(z, λ) =λn

πn sinhn λe−

λ|z|2

tanh λ .

This function is analytic in (z, λ) and all of its derivatives decay rapidly atinfinity. �

We introduce on Hn the homogeneous norm

(1.6)∣∣(z, t)

∣∣ =(|z|4 + 16t2

) 14 =

∣∣|z|2 + 4it∣∣ 12 = 2|w+|

12 .

Proposition 1.4. The homogeneous norm (1.6) satisfies the inequality∣∣(z, t)(w, u)

∣∣ ≤∣∣(z, t)

∣∣ +∣∣(w, u)

∣∣ .

Proof. We have

∣∣(z, t)(w, u)∣∣2 =

∣∣∣|z + w|2 + 4i(t+ u− 1

2=m〈z|w〉

)∣∣∣=

∣∣|z|2 + |w|2 + 2〈w|z〉 + 4i(t+ u)∣∣

≤∣∣|z|2 + 4it

∣∣ +∣∣|w|2 + 4iu

∣∣ + 2|z||w|≤

∣∣(z, t)∣∣2 +

∣∣(w, u)∣∣2 + 2

∣∣(z, t)∣∣∣∣(w, u)

∣∣

=(∣∣(z, t)

∣∣ +∣∣(w, u)

∣∣)2

. �

The following estimate can be seen as a Gaussian estimate w.r. to the homoge-neous norm (1.6).

Corollary 1.5. There is a > 0 such that

p1(z, t) ≤ Ce−a∣∣(z,t)

∣∣2.

Proof. From (1.4) we obtain that

p1(z, t) ≤1

πn+1

∫ +∞

0

λn

sinhn λe−

λ|z|2

tanh λ dλ .

Since the function λ/ tanhλ is bounded from below by 1, we have

(1.7) p1(z, t) ≤e−b|z|2

πn+1

∫ +∞

0

λn

sinhn λdλ = Ce−b|z|2 .

Consider now

Fp1(ζ, λ) =1

coshn λe−

tanh λ4λ |ζ|2 .

It can be extended analytically in λ to the strip{λ+iτ : |τ | < π

2

}. For |τ | ≤ π

2−δ,Fp1(ζ, λ+ iτ) is integrable in (ζ, λ) and rapidly decreasing in λ, uniformly in ζ andτ . Therefore, a change of contour integration in the plane λ+ iτ gives

p1(z, t) =1

(2π)2n+1

∫

Cn×R

Fp1(ζ, λ+ iτ)ei(t(λ+iτ)+<e〈z|ζ〉

)dζ dλ

=e−τt

(2π)2n+1

∫

Cn×R

Fp1(ζ, λ+ iτ)ei(tλ+<e〈z|ζ〉

)dζ dλ

≤ Ce−τt

∫

Cn×R

∣∣Fp1(ζ, λ+ iτ)∣∣ dζ dλ

≤ Cτe−τt .

Combining this with (1.7), we obtain that, for some α > 0,

p1(z, t) ≤ Ce−α(|z|2+|t|

).

This gives the conclusion, since∣∣(z, t)

∣∣2 ∼ |z|2 + |t|. �

100 CHAPTER V

2. Smooth multipliers and Schwartz kernels

In his Section, we use the estimates obtained for the heat kernel to prove thatcertain spectral multipliers of L correspond to convolution with a Schwartz kernel.

Theorem 2.1. Let m be a smooth function supported on the interval[

14 , 4

]. Then

m(L)f = f ∗ k, where k ∈ S(Hn).

This theorem will be proved in a few steps. The multiplier must be first decom-posed appropriately, in order to take advantage of the heat kernel estimates.

Let µ(τ) = m(− log τ). Then µ is supported on [e−4, e−14 ] ⊂ [−π, π]. Extending

µ as a periodic function of period 2π, it can be expanded into a Fourier series

µ(τ) =∑

j∈Z

ajeijτ ,

with rapidly decreasing coefficients. Since∑

j aj = µ(0) = 0, we can write

µ(τ) =∑

j∈Z\{0}

aj(eijτ − 1) ,

hence, for ξ > 0,

(2.1) m(ξ) =∑

j∈Z\{0}

aj(eije−ξ − 1) .

We set

mj(ξ) = eije−ξ − 1 =

∞∑

`=1

(ij)`

`!e−`ξ ,

so that

mj(L) =∞∑

`=1

(ij)`

`!e−`L .

This makes sense, because the series converges in the operator norm in L(L2).Therefore, the convolution kernel kj of mj(L) equals

(2.2) kj(z, t) =

∞∑

`=1

(ij)`

`!p`(z, t) .

Lemma 2.2. We have‖kj‖2 ≤ C|j| .

Proof. Let

mj(ξ) =eije−ξ − 1

e−ξ.

Observe that mj is bounded on R+, and

‖mj‖∞ ≤ |j| .

Then, if kj is the convolution kernel of mj(L) and f ∈ L2(Hn),

f ∗ kj = mj(L)f = mj(L)e−Lf = f ∗ p1 ∗ kj .

Therefore,kj = p1 ∗ kj = mj(L)p1 .

It follows from Proposition 1.4 in Chapter I that

‖kj‖2 ≤ ‖mj(L)‖L(L2)‖p1‖2 = ‖mj‖∞‖p1‖2 ≤ C|j| . �

Lemma 2.3. There is a constant ν > 0 such that

(2.3)

∫

Hn

e

∣∣(z,t)∣∣|kj(z, t)| dz dt ≤ eν|j| .

Moreover, for every N ∈ N, there is CN > 0 such that

(2.4)

∫

Hn

∣∣(z, t)∣∣N |kj(z, t)| dz dt ≤ CN |j|N+n+2 .

Proof. By (2.2),

(2.5)

∫

Hn

e

∣∣(z,t)∣∣|kj(z, t)| dz dt ≤

∞∑

`=1

|j|``!

∫

Hn

e

∣∣(z,t)∣∣p`(z, t) dz dt .

By Proposition 1.4,

∫

Hn

e

∣∣(z,t)∣∣∣∣f ∗ g(z, t)

∣∣ dz dt ≤

≤∫

Hn

∫

Hn

e

∣∣(z,t)∣∣∣∣f

((z, t)(w, u)−1

)∣∣∣∣g(w, u)∣∣dw du dz dt

=

∫

Hn

∫

Hn

e

∣∣(z′,t′)(w,u)∣∣∣∣f(z′, t′)

∣∣∣∣g(w, u)∣∣dw du dz′ dt′

≤( ∫

Hn

e

∣∣(z′,t′)∣∣∣∣f(z′, t′)

∣∣ dz′ dt′)( ∫

Hn

e

∣∣(w,u)∣∣∣∣g(w, u)

∣∣dw du).

Therefore, if

ν =

∫

Hn

e

∣∣(z,t)∣∣p1(z, t) dz dt ,

we deduce from the fact that p` = p1 ∗ · · · p1 (` times) that

∫

Hn

e

∣∣(z,t)∣∣p`(z, t) dz dt ≤ ν` .

From (2.5) we obtain the first claimed estimate. We then pass to the second.Given r > 0, we split the integral into the sum of the two integrals, extended to

the set where∣∣(z, t)

∣∣ > r and∣∣(z, t)

∣∣ < r respectively.

Lemma 2.4. Let m and k be as in Theorem 2.1. Then, for every N ∈ N,∫

Hn

(1 + |(z, t)|

)N |k(z, t)| dz dt ≤ CN‖m‖CN+n+4 .

Proof. By (2.1),

k(z, t) =∑

j∈Z\{0}

ajkj(z, t) ,

so that, by (2.4),∫

Hn

∣∣(z, t)∣∣N |k(z, t)| dz dt ≤ CN

∑

j∈Z\{0}

|aj ||j|N+n+2 .

The conclusion follows from the inequality

|aj | = |µ(j)| ≤ |j|−M‖µ‖CM ≤ |j|−M‖m‖CM ,

valid for every M . �

The control by the CN+n+4-norm is not optimal, and it will be improved later.At this stage we do not need to be more precise than this.

End of the proof of Theorem 2.1. Let m(ξ) = eξm(ξ). Then m satisfies the same

assumptions of m, so that the convolution kernel k of m(L) satisfies, by Lemma 2.4,∫

Hn

∣∣(z, t)∣∣N |k(z, t)| dz dt ≤ CN‖m‖CN+n+4 ≤ C ′

N‖m‖CN+n+4 .

From the identity m(L) = e−Lm(L) = m(L)e−L, it follows that

k = k ∗ p1 = p1 ∗ k .

In particular, k is smooth. To prove that k ∈ S(Hn) is equivalent to provingthat for every non-commutative polynomial P (Z, Z) in the left-invariant vectorfields Zj , Zj, P (Z, Z)k is rapidly decreasing at infinity. We then have

P (Z, Z)k = k ∗ P (Z, Z)p1 ,

where P (Z, Z)p1 = g is rapidly decreasing at infinity.Fix the polynomial P and N ∈ N. For

∣∣(z, t)∣∣ > 1, we split the convolution

integral in two parts. If∣∣(w, u)| < 1

2

∣∣(z, t)∣∣, then

∣∣(w, u)−1(z, t)∣∣ ≥

∣∣(z, t)∣∣ −

∣∣(w, u)∣∣ > 1

2

∣∣(z, t)∣∣ ,

so that

(2.8)

∫∣∣(w,u)|< 1

2

∣∣(z,t)∣∣∣∣k(w, u)

∣∣∣∣g((w, u)−1(z, t)

)∣∣ dw du ≤

≤ C

∫∣∣(w,u)|< 1

2

∣∣(z,t)∣∣∣∣k(w, u)

∣∣∣∣(w, u)−1(z, t)∣∣−N

dw du

≤ C ′∣∣(z, t)

∣∣−N‖k‖1

≤ C ′′‖m‖Cn+4

∣∣(z, t)∣∣−N

.

104 CHAPTER V

Moreover,

(2.9)

∫∣∣(w,u)|> 1

2

∣∣(z,t)∣∣∣∣k(w, u)

∣∣∣∣g((w, u)−1(z, t)

)∣∣ dw du ≤

≤ 2N

∫∣∣(w,u)|> 1

2

∣∣(z,t)∣∣

∣∣(w, u)∣∣N

∣∣(z, t)∣∣N

∣∣k(w, u)∣∣∣∣g

((w, u)−1(z, t)

)∣∣ dw du

≤ C‖m‖CN+n+4

∣∣(z, t)∣∣−N

.

Putting together (2.8) and (2.9), we find that

(2.10)∣∣P (Z, Z)k(z, t)

∣∣ ≤ CP,N‖m‖CN+n+4

(1 +

∣∣(z, t)∣∣)N

,

and this concludes the proof. �

3. Mihlin-Hormander multipliers of L

We have the tools now to prove the sharp Mihlin-Hormander theorem for mul-tipliers of L.

We need however to make some preliminary digression on the realization ofHn asa space of homogeneous type, and on the corresponding Calderon-Zygmund theory.

There are two natural homogeneous-type structures on Hn, a left-invariant and aright-invariant one, that we denote here as (Hn,m, d`) and (Hn,m, dr) respectively.In both cases the measure m is the Lebesgue measure, and they differ in the choiceof the distance, which in one case is the left-invariant distance

d`

((z, t), (w, u)

)=

∣∣(w, u)−1(z, t)∣∣ ,

and in the other case the right-invariant distance

dr

((z, t), (w, u)

)=

∣∣(z, t)(w, u)−1∣∣ .

The terminology corresponds to the different invariance properties of d` and dr:while

d`

((ζ, τ)(z, t), (ζ, τ)(w, u)

)= d`

((z, t), (w, u)

),

for every (ζ, τ) ∈ Hn, we have instead

dr

((z, t)(ζ, τ), (w, u)(ζ, τ)

)= dr

((z, t), (w, u)

).

We then have two different notions of Calderon-Zygmund kernel.

Definition. A distribution u ∈ S ′(Hn) is a left Calderon-Zygmund kernel on Hn

if

(i) the operator Tf = f ∗ u extends to a bounded operator on L2(Hn);(ii) away from the origin, u coincides with a locally integrable function u(z, t),

and there is a constant C > 0 such that, for every (w, u) 6= (0, 0),

(3.1)

∫∣∣(z,t)

∣∣>4∣∣(w,u)

∣∣∣∣u

((w, u)(z, t)

)− u(z, t)

∣∣ dz dt ≤ C .

A distribution u ∈ S ′(Hn) is a right Calderon-Zygmund kernel on Hn if

(i) the operator Tf = u ∗ f extends to a bounded operator on L2(Hn);(ii) away from the origin, u coincides with a locally integrable function u(z, t),

and there is a constant C > 0 such that, for every (w, u) 6= (0, 0),

(3.2)

∫∣∣(z,t)

∣∣>4∣∣(w,u)

∣∣∣∣u

((z, t)(w, u)

)− u(z, t)

∣∣ dz dt ≤ C .

The following statement belongs to the general Calderon-Zygmund theory (seeCorollary 3.4 in Chapter II).

Proposition 3.1. If u is a left Calderon-Zygmund kernel on Hn, the operatorTf = f ∗ u is weak-type (1,1) and bounded on Lp(Hn) for 1 < p ≤ 2.

If u is a right Calderon-Zygmund kernel on Hn, the operator T ′f = u ∗ f isweak-type (1,1) and bounded on Lp(Hn) for 1 2.

In contrast with what we have seen in Rn, the fact that a convolution operator isbounded on Lp(Hn) for some p ∈ (1,∞), does not imply25 that the same operator

is also bounded on Lp′

(Hn). This is related to the non-commutative structure ofHn. The correct duality result is as follows.

Proposition 3.2. Let u ∈ S ′(Hn) and p ∈ (1,∞). The operator Tf = f ∗ u is

bounded on Lp(Hn) if and only if T ′g = u ∗ g is bounded on Lp′

(Hn). In this casethe two operator norms coincide.

Proof. Take f, g ∈ S(Hn). If f(z, t) = f(−z,−t), an explicit computation showsthat

〈f ∗ u, g〉 =

∫

Hn

f ∗ u(z, t)g(z, t) dz dt

= 〈u, f ∗ g〉= 〈u ∗ g, f〉 .

Therefore

‖T‖L(Lp) = sup‖f‖p≤1,‖g‖p′≤1

∣∣〈f ∗ u, g〉∣∣

= sup‖f‖p≤1,‖g‖p′≤1

∣∣〈u ∗ g, f〉∣∣

= ‖T ′‖L(Lp′) . �

We shall use the following consequence of Propositions 3.1 and 3.2.

25In fact this is false in general. This phenomenon is known as “asymmetry” of convolutionoperators, and it occurs on many non-commutative groups. It is not known if all infinite locally

compact non-commutative groups exhibit asymmetry. For a large class of l.c. groups, called

amenable and including the Heisenberg group, it is true however that if a convolution operator isbounded on some Lp, then it is also bounded on L2.

106 CHAPTER V

Corollary 3.3. Let u be a two-sided Calderon-Zygmund kernel (i.e. it is both leftand right C-Z kernel). Then Tf = f ∗u is bounded on Lp(Hn) for 1 12 on R+. Then m

is continuous and bounded, by (6.1) in Chapter II, so that m(L) is bounded onL2(Hn). Hence there exists u ∈ S ′(Hn) such that m(L)f = f ∗ u.

We shall prove the following result26.

Theorem 3.4. Assume that m(ξ) is a Mihlin-Hormander multiplier on R+ oforder s > 2n+1

2. If m(L)f = f ∗u, then u is a two-sided Calderon-Zygmund kernel.

Hence m(L) is weak-type (1,1) and bounded on Lp(Hn) for 1 0. We define

(3.3) mj(ξ) = m(2−jξ)η(ξ) ,

so that each mj is supported in[12 , 2

], and

(3.4) m(ξ) =∑

j∈Z

mj(2jξ) .

We call uj the distribution on Hn such that mj(L)f = f ∗ uj . Our aim is toprove that the uj are in fact integrable unctions, and that they satisfy conditionsanalogous to those imposed on the ϕj of Theorem 5.3 in Chapter II.

To begin with, we prove that the uj are integrable functions and that also(1 +

|(z, t)|)εuj(z, t) is integrable for some ε > 0. The following statement is not good

enough, but it is one of the starting points for an interpolation argument, whichwill lead to the desired estimates.

Lemma 3.5. Assume that m ∈ HN (R), with N ≥ n+5, and is supported in[

14 , 4

].

Then m(L)f = f ∗ u, where u is integrable on Hn and

∫

Hn

(1 + |(z, t)|

)2(N−n−5)|u(z, t)|2 dz dt ≤ CN‖m‖2HN .

Proof. It follows from Lemma 2.4, by a limiting argument, that u is integrable and∫

Hn

(1 + |(z, t)|

)N−n−5|u(z, t)| dz dt ≤ CN‖m‖CN−1 .

26This result has been proved independently by D. Muller and E.M. Stein, On spectral mul-

tipliers for Heisenberg and related groups, J. Math. Pures Appl. 73 (1994), 413-440, and by W.

Hebisch, Multiplier theorem on generalized Heisenberg groups, Coll. Math. 65 (1993), 231-239.The two papers contain different extensions to other nilpotent groups.


On the other hand, (2.10) implies that

(1 + |(z, t)|

)N−n−5|u(z, t)| ≤ CN‖m‖CN−1 .

By (6.1) in Chapter II, ‖m(j)‖∞ ≤ C‖m(j)‖H1 , so that ‖m‖CN−1 ≤ ‖m‖HN .Putting these inequalities together, we conclude the proof. �

The other starting point for the interpolation is a sharper estimate for s = 32.

Lemma 3.6. Assume that m ∈ H32 (R) and is supported in

[14 , 4

]. Then m(L)f =

f ∗ u, where u is square-integrable on Hn and

∫

Hn

(1 + |(z, t)|

)4|u(z, t)|2 dz dt ≤ C‖m‖2

H32.

Proof. The left-hand side is equivalent to the L2-norm of u plus the L2-normof w+u, by (1.6). We then use the Plancherel formula, recalling that u(λ, k) =m

(|λ|(2k + n)

):

‖u‖22 =

∞∑

k=0

(n+ k − 1

n− 1

)∫ +∞

−∞

∣∣u(λ, k)∣∣2 |λ|n dλ

∼∞∑

k=0

(k + 1)n−1

∫ +∞

−∞

∣∣m(|λ|(2k + n)

)∣∣2 |λ|n dλ

=

∞∑

k=0

(k + 1)n−1

(2k + n)n+1‖m‖2

2

≤ C‖m‖2

H32.

Moreover,

‖w+u‖22 =

∞∑

k=0

(n+ k − 1

n− 1

) ∫ +∞

−∞

∣∣(w+u)(λ, k)∣∣2 |λ|n dλ

∼∞∑

k=0

(k + 1)n−1

∫ +∞

−∞

∣∣(w+u)(λ, k)∣∣2 |λ|n dλ .

For λ > 0 we have

(w+u)(λ, k) = ∂λm(λ(2k + n)

)

− k + n

λ

(m

(λ((2k + n+ 2)

)−m

(λ(2k + n)

))

= (2k + n)m′(λ(2k + n)

)− (k + n)

∫ 2

0

m′(λ(2k + n+ s)

)ds

= −nm′(λ(2k + n)

)

− (k + n)

∫ 2

0

(m′

(λ(2k + n+ s)

)−m′

(λ(2k + n)

))ds

= A(λ, k) +B(λ, k) .

108 CHAPTER V

We make separate estimates of the contributions of A(λ, k) and B(λ, k) to theL2-integral. Clearly,

∫ +∞

0

∣∣A(λ, k)∣∣2 λn dλ = n2

∫ +∞

0

∣∣m′(λ(2k + n)

)∣∣2 λn dλ

=n2

(2k + n)n+1

∫ 4

14

∣∣m′(λ)∣∣2 λn dλ ,

so that

∞∑

k=0

(k + 1)n−1

∫ +∞

0

∣∣A(λ, k)∣∣2 λn dλ

≤ C∞∑

k=0

(k + 1)−2

∫ 4

14

∣∣m′(λ)∣∣2 λn dλ

≤ C‖m′‖22

≤ C‖m‖H

32.

The estimate for the contribution of B(λ, k) is more delicate. We have

∫ +∞

0

∣∣B(λ, k)∣∣2 λn dλ =

= (k + n)2∫ +∞

0

∣∣∣∣∫ 2

0

(m′

(λ(2k + n+ s)

)−m′

(λ(2k + n)

))ds

∣∣∣∣2

λn dλ

≤ 2(k + n)2∫ 2

0

∫ +∞

0

∣∣∣m′(λ(2k + n+ s)

)−m′

(λ(2k + n)

)∣∣∣2

λn dλ ds

=2(k + n)2

(2k + n)n+1

∫ 2

0

∫ 4

0

∣∣∣m′(λ+

λs

2k + n

)−m′(λ)

∣∣∣2

λn dλ ds

≤ C

(k + 1)n−1

∫ 2

0

∫ 4

0

∣∣∣m′(λ+

λs

2k + n

)−m′(λ)

∣∣∣2

λ dλ ds

≤ C

(k + 1)n−2

∫ 4

0

∫ 2λ2k+n

0

∣∣m′(λ+ h) −m′(λ)∣∣2 dh dλ

≤ C

(k + 1)n−2

∫ 4

0

∫ 82k+n

0

∣∣m′(λ+ h) −m′(λ)∣∣2 dh dλ .

Summing over k and using the the Plancherel formula in R, we obtain that

∞∑

k=0

(k + 1)n−1

∫ +∞

0

∣∣B(λ, k)∣∣2 λn dλ

≤ C

∫ 4

0

∞∑

k=0

(k + 1)

∫ 82k+n

0

∣∣m′(λ+ h) −m′(λ)∣∣2 dh dλ

= C

∫ 4

0

∫ 8n

0

( ∑

k< 4h

(k + 1))∣∣m′(λ+ h) −m′(λ)

∣∣2 dh dλ

≤ C

∫ 4

0

∫ 8n

0

∣∣m′(λ+ h) −m′(λ)∣∣2 dhh2

dλ

≤ C

∫ +∞

0

∫

R

∣∣m′(λ+ h) −m′(λ)∣∣2 dλ dh

h2

= C

∫ +∞

0

∫

R

τ2|m(τ)|2|eihτ − 1|2 dτ dhh2

= C

∫

R

τ2|m(τ)|2∫ +∞

0

|eihτ − 1|2h2

dh dτ

= C

∫

R

|τ |3|m(τ)|2∫ +∞

0

|eih − 1|2h2

dh dτ

= C‖m‖2

H32.

We have so proved that

∞∑

k=0

(k + 1)n−1

∫ +∞

0

∣∣(w+u)(λ, k)∣∣2 |λ|n dλ ≤ C‖m‖2

H32.

Similar computations allow to obtain the same estimate for the integral overnegative values of λ. �

Corollary 3.7. Assume that m ∈ Hs(R), with s > 2n+12

, and is supported in[12, 2

]. If m(L)f = f ∗ u, then, for 0 < ε < s− 2n+1

2,

∫

Hn

(1 + |(z, t)|

)ε|u(z, t)| dz dt ≤ Cε‖m‖Hs .

Proof. Let ψ(ξ) be a smooth function supported in[14 , 4

]and identically equal to

1 on[12 , 2

]. Given a bounded function m(ξ) on R, let u be the convolution kernel

of (mψ)(L), and let S be the linear operator given by Sm = u. If, in particular, mis supported in

[12 , 2

], then mψ = m, so that Sm is the kernel u in the statement.

Denote by L2α(Hn) the space of functions f onHn such that

(1+|(z, t)|

)αf(z, t) ∈

L2(Hn). Then Lemma 3.6 states that

S : H32 (R) −→ L2

2(Hn) ,

and Lemma 3.5 states that if N ≥ n+ 5,

S : HN (R) −→ L2N−n−5(Hn) .

110 CHAPTER V

A simple modification to the proof of the interpolation Lemma 7.2 in Chapter IIshows that if

s =3

2θ + (1 − θ)N , α = 2θ + (1 − θ)(N − n− 5) ,

with 0 < θ < 1, thenS : Hs(R) −→ L2

α(Hn) ,

For fixed s > 32

and N > s, we find θ = N−sN− 3

2

, hence

α = αN = 2N − s

N − 32

+ (s− 3

2)N − n− 5

N − 32

.

As N → +∞, αN tends monotonically to s+ 12. Therefore

(3.5)

∫

Hn

(1 + |(z, t)|

)2(s+ 12−δ)|u(z, t)|2 dz dt ≤ Cδ‖m‖2

Hs ,

for every δ > 0.Assume now that s > 2n+1

2and 0 < ε < s− 2n+1

2. Then, if δ < s− 2n+1

2− ε,

∫

Hn

(1 + |(z, t)|

)ε|u(z, t)| dz dt ≤( ∫

Hn

(1 + |(z, t)|

)2(s+ 12−δ)|u(z, t)|2 dz dt

) 12

×( ∫

Hn

(1 + |(z, t)|

)−2(s+ 12−δ−ε)

dz dt

) 12

.

The last integral is convergent, because the exponent is strictly smaller than thenegative of the homogeneous dimension Q = 2n+ 2 of Hn. The conclusion followsfrom (3.5). �

We look now for a substitute of condition (c) of Theorem 5.3 in Chapter II onthe uj . We want to replace the ordinary L1-Lipschitz condition, which concern the“abelian” differences uj(z + w, t + u) − uj(z, t), with similar conditions, involving“non-abelian” differences of the form uj

((z, t)(w, u)

)−uj(z, t) and uj

((w, u)(z, t)

)−

uj(z, t).

Lemma 3.8. Assume that m ∈ Hs(R), with s > 2n+12 , and is supported in

[12 , 2

].

If m(L)f = f ∗ u, then

(3.6)

∫

Hn

∣∣u((z, t)(w, u)

)− u(z, t)

∣∣ dz dt ≤ C‖m‖Hs

∣∣(w, u)| ,

and

(3.7)

∫

Hn

∣∣u((w, u)(z, t)

)− u(z, t)

∣∣ dz dt ≤ C‖m‖Hs

∣∣(w, u)| .

Proof. The generic increment (w, u) can be written as

(w, u) = (w, 0)(√|u|e1, 0)(±i

√|u|e1, 0)(−

√|u|e1, 0)(∓i

√|u|e1, 0) ,


where the ± sign depends on the signum of u. This implies that it is sufficient toprove (3.6) and (3.7) for “horizontal” increments (i.e. with a zero t-component).In fact, restricting our attention to (3.6) and assuming u > 0 for simplicity, we canthen write

∫

Hn

∣∣u((z, t)(w, u)

)− u(z, t)

∣∣ dz dt ≤

≤∫

Hn

∣∣u((z, t)(w, u)

)− u

((z, t)(w, 0)(

√ue1, 0)(i

√ue1, 0)(−

√ue1, 0)

)∣∣ dz dt

+ · · ·+∫

Hn

∣∣u((z, t)(w, 0)

)− u(z, t)

∣∣ dz dt

=

∫

Hn

∣∣u((z, t)(−i√ue1, 0)

)− u(z, t)

∣∣ dz dt

+ · · ·+∫

Hn

∣∣u((z, t)(w, 0)

)− u(z, t)

∣∣ dz dt

≤ C‖m‖Hs

(|w| + 4

√u)

≤ C ′‖m‖Hs

∣∣(w, u)| .

Suppose therefore that u = 0 in (3.6). By composing if necessary, u with aunitary transformation of Cn, we can assume that w = re1, with r > 0. We claimthat we can transform the difference into an integral by the fundamental theoremof calculus.

Consider in fact the multiplier m(ξ) = eξm(ξ) is also in Hs(Hn) and is sup-ported in

[12, 2

]. If u is the corresponding convolution kernel, then u ∈ L1(Hn), by

Corollary 3.7, and u = u ∗ p1 = p1 ∗ u. Hence u is C∞. Therefore

(3.8)

u((z, t)(re1, 0)

)− u(z, t) =

∫ r

0

d

dsu((z, t)(se1, 0)

)ds

=

∫ r

0

X1u((z, t)(se1, 0)

)ds .

Now, X1u = u ∗X1p1 ∈ L1(Hn), and

‖X1u‖1 ≤ C‖u‖1 ≤ C ′‖m‖Hs ≤ C ′′‖m‖Hs .

Hence∫

Hn

∣∣u((z, t)(re1, 0)

)− u(z, t)

∣∣ dz dt =

∫

Hn

∣∣∣∣∫ r

0

X1u((z, t)(se1, 0)

)ds

∣∣∣∣ dz dt

≤∫ r

0

∫

Hn

∣∣X1u((z, t)(se1, 0)

)∣∣ dz dt ds

= r‖X1u‖1 .

In (3.7) the increments are on the left, hence (3.8) must be replaced by

u((re1, 0)(z, t)

)− u(z, t) =

∫ r

0

d

dsu((se1, 0)(z, t)

)ds

=

∫ r

0

X(r)1 u

((z, t)(se1, 0)

)ds ,

112 CHAPTER V

where the right-invariant vector field X(r)1 appears. The identity

X(r)1 u = X

(r)1 (p1 ∗ u) = (X

(r)1 p1) ∗ u

then leads us to the conclusion. �

We consider next condition (b) of Theorem 5.3 in Chapter II.

Lemma 3.9. Assume that m ∈ Hs(R), with s > 2n+12

, and is supported in[

12, 2

].

If m(L)f = f ∗ u, then∫

Hnu(z, t) dz dt = 0.

Proof. Let kj be the kernel in (2.2). By (1.1),

∫

Hn

kj(z, t) dz dt =∞∑

`=1

(ij)`

`!

∫

Hn

p`(z, t) dz dt =∞∑

`=1

(ij)`

`!= eij − 1 .

Since s > 32 , m is C1, and so is µ. It follows that the coefficients aj in (2.1) are

summable, because, by the Parseval formula,

∑

j∈Z\{0}

|aj| ≤( ∑

j∈Z\{0}

|jaj|2) 1

2

= ‖µ′‖2 .

Since u =∑

j∈Z\{0} ajkj , we have

∫

Hn

u(z, t) dz dt =∑

j∈Z\{0}

aj(eij − 1) = µ(1) = m(0) = 0 . �

Finally, some further remarks are needed, concerning the decomposition (3.4)of m.

Let mj , with j ∈ Z, be the multiplier in (3.3), with uj such that mj(L)f = f ∗uj .By Proposition 1.4 in Chapter I, if f, g ∈ L2(Hn), then

(3.9)

〈m(L)f |g〉 =

∫ ∞

0

m(λ) dνf,g(λ)

=∑

j∈Z

∫ ∞

0

mj(2jλ) dνf,g(λ)

=∑

j∈Z

〈mj(2jL)f |g〉 .

Hence m(L) =∑

j∈Zmj(2

jL) in the weak topology.

Lemma 3.10. Let

u(j)j (z, t) = 2−(n+1)juj(2

− j2 z, 2−jt) .

Then mj(2jL)f = f ∗ u(j)

j , and

u =∑

j∈Z

u(j)j ,

in the sense of distributions.

Proof. By Theorem 5.6 and (5.10) in Chapter IV,

uj(λ, k) =

∫

Hn

uj(z, t)eiλtψk

(|λ||z|2

)dz dt = mj

(|λ|(2k + n)

).

Therefore,

u(j)j (λ, k) =

∫

Hn

uj(z, t)eiλ2j tψk

(|λ|2j |z|2

)dz dt = mj

(2j|λ|(2k + n)

).

Applying Theorem 5.6 in Chapter IV again, we conclude that u(j)j is the convo-

lution kernel of mj(2jL). By (3.9), if f, g ∈ L2(Hn) and g∗(z, t) = g(−z,−t),

〈u|f ∗ g〉 = 〈u ∗ g∗|f〉= 〈m(L)g∗|f〉=

∑

j∈Z

〈mj(2jL)g∗|f〉

=∑

j∈Z

〈u(j)j |f ∗ g〉 .

Take now ϕ ∈ S(Hn). We want to prove that

〈u|ϕ〉 =∑

j∈Z

〈u(j)j |ϕ〉 .

Since u is radial, 〈u|ϕ〉 = 〈u|Pϕ〉 (and the same holds for u(j)j ), where P is the

orthogonal projection from L2(Hn) onto L2rad(Hn) (see (5.7) in Chapter IV). By

Lemma 5.5 in Chapter IV, Pϕ ∈ Srad(Hn).It is then sufficient to prove that every ϕ ∈ Srad(Hn) can be written as ϕ = f ∗g,

with f, g ∈ L2(Hn).By Proposition 2.7 in Chapter IV and its extension to Hn,

ϕ(λ, k) ≤ CN(1 + |λ|(k + 1)

)N,

for every N ∈ N. Therefore, taking N > n+ 1,

∫

R

∑

k∈N

(k + n− 1

n− 1

)∣∣ϕ(λ, k)∣∣ |λ|n dλ ≤ CN

∑

k∈N

∫

R

(k + 1)n−1

(1 + |λ|(k + 1)

)N|λ|n dλ

< +∞ .

By the Plancherel formula, if we impose that

f(λ, k) =∣∣ϕ(λ, k)

∣∣ 12 , g(λ, k) =

∣∣ϕ(λ, k)∣∣ 12 arg

(ϕ(λ, k)

),

then f, g ∈ L2(Hn) and f ∗ g = ϕ. �

114 CHAPTER V

We have now all the ingredients for the proof of Theorem 3.4.

Proof of Theorem 3.4. By Lemma 3.10, u =∑

j∈Zu

(j)j . By Lemmas 3.7, 3.8, 3.9,

the uj satisfy the following conditions with the same constant C:

(i) For some ε > 0,∫

Hn

(1 + |(z, t)|

)ε|uj(z, t)| dz dt ≤ Cε;

(ii)∫

Hnuj(z, t) dz dt = 0;

(iii) for every (w, u) ∈ Hn,

∫

Hn

∣∣uj

((z, t)(w, u)

)− uj(z, t)

∣∣ dz dt ≤ C∣∣(w, u)| ,

∫

Hn

∣∣uj

((w, u)(z, t)

)− uj(z, t)

∣∣ dz dt ≤ C∣∣(w, u)| .

A straightforward adaptation of the proof of Theorem 5.3 in Chapter II showsthat u is a two-sided Calderon-Zygmund kernel. Corollary 3.3 can then be ap-plied. �

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