kp~ - ams.org … · if y ¥= kp — 1, k a positive integer. in this paper we consider multipliers...

TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 275, Number 2, February 1983

L" MULTIPLIERS WITH WEIGHT | x \kp~ '

BY

BENJAMIN MUCKENHOUPT1 AND WO-SANG YOUNG2

Abstract. Let k be a positive integer and 1 < p < oo. It is shown that if T is a

multiplier operator on Lp of the line with weight | x \kp~\ then Tf equals a constant

times/almost everywhere. This does not extend to the periodic case since m(j) =

l/j,j ¥* 0, is a multiplier sequence for Lp of the circle with weight |x|*''_l. A

necessary and sufficient condition is derived for a sequence m(j) to be a multiplier

on L2 of the circle with weight | x |2/t~'.

1. Introduction. This is a continuation of our previous work with R. L. Wheeden

on multipliers for weighted Lp spaces [2, 3]. The underlying space we consider here

will either be the line R or the circle [-77,w]; / and / will denote the Fourier

transform and inverse Fourier transform of/. For 1 <p < 00, Lp is the space of

functions/with || / H^ = (/ \ff | x \ydx)l/p < 00. Let S^ be the set of all functions

whose Fourier transforms are infinitely differentiable, have compact support and

vanish near the origin. A function m, locally integrable in R\{0}, is said to be a

multiplier for LP(R) if

(1.1) f$mf)X\x\ydx<cf\f\P\x\ydx

for all / in S^. There is no analogue of S^ for the periodic case. Instead, for

k = 0,1,2,..., we let Sk be the set of trigonometric polynomials with f(j) — 0,

j — 0,1,... ,k. A sequence of numbers {m(j)}jL_g0 is a multiplier for Lp(\--n, w]) if

(1.1) holds for/in Sk, k = [(y + $/p] — 1. (As usual, [a] is the largest integer less

than or equal to a.)

In [2] we characterized multipliers for L2., y ^ 2k — 1, k a positive integer.

Results for the line and the circle are analogues of each other. In [3] we studied

weighted Lp results for various classes of multipliers. In particular, sufficient

conditions were obtained for a function m(x) to be a multiplier for Lp for all y > -1

if y ¥= kp — 1, k a positive integer. In this paper we consider multipliers for the

"missing indices", y = kp — 1, k a positive integer. Unlike [2 and 3], the results

obtained here are different for the line and the circle. We will show in §2 that

multipliers on L£ ,(R) are trivial by proving the following theorem.

Received by the editors December 9, 1981.

1980 Mathematics Subject Classification. Primary 42A45.

1 Research supported in part by N.S.F. grants.

2 Research supported in part by N.S.F. and NRC of Canada grants.

©1983 American Mathematical Society

0002-9947/82/0000-0361 /$04.75

623

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

624 BENJAMIN MUCKENHOUPT AND WO-SANG YOUNG

Theorem (1.2). Let k be a positive integer, 1 < p < oo, and m be locally integrable

in R\{0}. If for o/Z/wS«,,

(1.3) / \(mf)"\P\x\kp~'dx^cf \jf\x\kp~ldx

with c independent off, then m equals a constant a.e.

In contrast to this theorem we will now show that m(j) = l/j, j ¥" 0, is a

multiplier for Lg ß-ir, it]), 1 <p < oo, k a positive integer. Let/be a trigonomet-

ric polynomial in Sk_,. Then

i(mf)"(x) = fj(t) dt - ¿£/(0 *■

By Hardy's inequahty [5, p. 272] and the fact that | x Y < c if | x \ < it,

/TT fit_ \jfU)dtI \kp~X A <r\x\ dx «= cf 1/fW

¿p-ldx.

To estimate the integral of the second term, we observe that, since/ G S,fc-i»

(1.4)

I - I , / k-\fit fit

/ tf(t)dt\= f \t- 2'V

(1 - «-")/(/)A

The fact that 2*=,' zV/ = -log(l - z) + 0(z*) shows that the right side of (1.4) is

bounded by cfl„ 11 \k |/(í) | ¿ft. Hence, we have

f ¿|/_/(0<*f 1*1*'"'* < c( JT\Ak\f{t)\dt)P

=Sc/: i'i""'i/(.)f¿<,

by Holder's inequahty. Therefore m(j)= l/j is a multiplier for L£ ,([-vr, w]).

In §§3-5, we characterize all multipliers for L\k_x([-Ti, tt]), k a positive integer. In

the following A denotes the forward difference, i.e., A°m(j) — m{j), and for

5 = 0,1,2,..., Ai+1w(j) = tfm(j + 1) — Asm(j). Also, for any integer / and for

j = 0, l,2,...,/(i) is defined by /(î) = /(/ - 1)(/ - 2) ■••(/- * + 1) for s > 1 and

/(O) = i jjje characterization is given in Theorem (1.5).

Theorem (1.5). Let k be a positive integer and {m(j)}f=^x be a sequence of

numbers with m(j) = 0 for 0 <j < k — 1. Then

i. ,\ /'i/ ;\-i2, ,2/t-l , /•" . ,2 , .2*-l ,(1.6) / |(m/)| |x| rfx<c/ 1/1 |jc| dx

forallf G iS¿_| if and only if m G Ie",and,for h — \,2,...,k, m satisfies

\m{j + l)-2lzMs)/s^m(j)\\ c(1.7) 2 /2<*-"2

/^o log(/i + 1) '

ft=l,2,..

In this paper, c denotes a positive constant which depends on k and m and may

vary from line to line.


Lp MULTIPLIERS WITH WEIGHT | x\kp ' 625

2. The nonperiodic case. To prove Theorem (1.2) we first observe that (1.3) implies

that m G U°. This can be shown in exactly the same way as in the proof of Theorem

(2.2) in [2]. The rest of the proof is contained in the following lemmas.

Lemma (2.1). Let m G L00, Kp < oo, and k= 1,2,.... Suppose {/„}£=, is a

sequence of functions in S^, such that for sufficiently large n,

(2.2) f\(mfny\P\x\kp-ldx<clogn,JR

(2.3) f\ff\x\kp~ldx<clogn,JR

and

(2.4) there is a subset A C [1,2] with positive measure such that c \f„(x) \ > log n for

x G A. Then m = constant a.e.

Lemma (2.5). Let 1 <p < oo, and k = 1,2,_ Then there exists a sequence of

functions (/„)^=i in Sqq such that (2.3) and (2.4) hold for sufficiently large n.

Proof of Lemma (2.1). We first consider the case k—\. The inequality

(2.6) ( ( \f(x + h);Kx)\P \x\p-2dhdx^cf\jf\x\p-ldxJR JR \h\ JR

is proved in [5, pp. 139-140] for p — 2. The same proof, using a continuous version

of Paley's theorem [6, Volume II, p. 121] instead of Plancherel's formula, implies

(2.6) for 1 < /> < 2. Similarly,

(2.7) f f \f(* + h) -f(x)('dhdx<c(lAPlxr>dXi 2<p<oo.•'R-'R l/J •'R

We shall consider the case 1 < p < 2. The case 2 < p < oo can be done in the

same way. By (2.6) and Minkowski's inequality, we have

(2.8)

f \(mfnj\p\xr]dx>cf\f|m(*+ h);m{x)\PdhJa Ja Jo luf

p I \P~2|/„(x)| \x\" dx

M

_cf f \m(x + h)\P\Ux + h)-fn(x)\P\xrdhdx

Jr Jr \hf

The left side is less than c log n by (2.2). The first term on the right is larger than

cf f lm{x + h);m{x)lPdhdx(ioëny

JA JR \h\

by (2.4). Using the boundedness of m, (2.6) and (2.3), we see that the second term on

the right has absolute value less than c log n. Combining all these, we have

r c \m(x + h) — m(x)\ „ , ,. ,\-„\ \ J—*-'--^-^-dhdx^cilogn) pJA JR \h\


x dx

Ak-Dp-lx dx.


for all sufficiently large «. This implies that

f f \m{x + h);m{x)\Pdhdx = o,

JA JR \hf

which in turn implies that m(x) = constant a.e.

To prove Lemma (2.1) for the case k > 1, we shall show that there exists a

sequence {g„}¡?=1 in S^ satisfying (2.2)-(2.4) with k replaced by k — I. The lemma

will then follow by induction.

To do this, we define g„(x) = /» f„(t) dt. Then, since g„(x) = -/_*„ /„(i) dr, it is

easy to verify that gn G L1 and g„(x) = -(l//x)/„(x), x =^ 0. Hence g„ G §„<,, and

by (2.4),

|*.(*)l = T7l/-W|>clogii|x|

for x G /I. To show that (g„) satisfies (2.2) for A; — 1, we observe that

/R|(mg„)T|xr-,)/'-,iix=/J(^-m/1)

= f\r{mfjdtJR\Jx

Hardy's inequality [5, p. 272] implies that the last integral is less than

cf \{mf„)y | x f'"1*,•'R

which in turn is less than clog n by (2.2). The proof that (g„} satisfies (2.3) for k — 1

is similar.

This completes the proof of Lemma (2.1).

Proof of Lemma (2.5). We first show that there is a sequence {#„_*}"=, in L°°

with compact support such that for sufficiently large n,

(2.9) g„tk(o) = r„,M = ■■■ = Äftr^o) = o,

(2.10) ( \gn,k\P\x\ P dx<clogn,JR

and

(2.11) \gnM{x)\> clogn forxG[l,2],

The g„ k's are defined as follows. For n = 1,2,..., let

g„,i(*) = ^""'[X[i/„,i](^) - X[i,„](*)]•

For k > 1, gn>t is defined recursively by

(2-12) g„,,(x) = x-'[gn>fc_,(x) - g„,,_,(2x)].

We can also write gn k(x), k > 1, as

(2-13) &.*(*) =-¿T 2 ^,yg„,,(2^x),x 7=0


Lp MULTIPLIERS WITH WEIGHT | X \kp '

where ckj is given by

(2.14) 2ckjXj= n 1-$ •

627

j=0 1=0

That {g„ k) satisfies (2.9) and (2.10) follows readily from (2.12) and induction on

k. The proof of (2.11) is divided into two cases: k — 1 and k > 1. For k = 1, we

have

|g„,i(*)|;

•i 1 -i 1[ -dt\-\ -(e-x,-l)dt\-\ -e~ix'dtJ\/n l \J\/n { \J\ l

r" 1

The first term on the right is equal to log n. The second is majorized by \\/nxt/t dt,

which is less than 2 for x G [1,2]. By integration by parts, the third term is equal to

-ixn

e~ix r

-ix J\-dt

i ix f

This is less than 1/wx + 1/x + (l/x)(l — l/n) < 2, x G [1,2]. Hence, for suffi-

ciently large n, | gnl(x) \> clog n, x G [1,2].

For k > 1, we recall that g(n%(0) = 0, / = 0,1,... ,k - 2, and write

fc-2 /x'(

&.*(*) = ÈnAX) * 2 J]ên?k(°)1=0

*v2 (-«y.

/=0/!

dt.

Using (2.13) and a change of variables, we get

(2.15) g„,,(x) = 2lckJi**-*f -¿-g„,,(0<=n ■'r /* '7 = 0

A:-li 1 (-ix2^r)

¡x2-j( _ y (-¿x2 f)

/=o

k-\

/!í/í

y c 2><*-2>f _LIzí^L!_¿,

fc-l , [ k-\ ( ■ ».

y=o V»f*L /=o /!

{-Jx2-Jt)'

k~] t- 1

y=0 1 «

«2"-'i _ y (~?x2 ?)

/=0/!

dt

dt.

For x G [1,2], the modulus of the first term on the right is larger than

1

(k-l)l

k-\

2 ckJ2-J7 = 0

log«.

This, because of (2.14), is larger than clog n for some c > 0. To estimate the second

term on the right of (2.15) we observe that

eix2.Jt _ *"' (-ix2-Jt)'

1=0/! k\



Hence the modulus of the second term is majorized by c, xG[l,2]. Finally,

estimating the integral of the terms separately, we obtain that the modulus of the last

term of (2.15) is less than c, xG[l,2]. Hence, when n is sufficiently large,

I £„,*(•*) I^clog «,x S [1,2].We have thus obtained a sequence {gn¡k}™=l that satisfies (2.9)—(2.11). To finish

the proof of the lemma, all we have to do is approximate each gnk by a function

/„ G §oo in both the L2 norm and L^p_x norm. This is possible because of Theorem

(6.1) of [2]. This completes the proof of Lemma (2.5) and hence the proof of

Theorem (1.2).

3. Periodic case: preliminaries. The proof of Theorem (1.5) is more difficult, partly

because there are multipliers for the periodic case, and partly because of the

complications that result from replacing derivatives by differences. We shall need the

following lemmas.

Lemma (3.1). Let k = 1,2,..., let f be a trigonometric polynomial. Then

\fu+i)-^=äi's)/^-Wü)\2¡2k

(3.2) f\f(x)\2\x\2k-ldx~Z 2-* j I'¥=0

where, as usual, » means that each side is bounded above by c times the other side.

Proof. We first observe that for s = 0,1,...,

(3.3) A7(/)= 2 (-iy^(Sh)f{j + h)=f f(t)(e--l)se-'J'dt.h=o x ' J~«

Hence the right side of (3.2) is equal to

¿> «2i ¿¡t¥-0

Jit)k~\ ,(s)

r'"- 2 77-(e-"-l)J e-ij'dt

which, by Parseval's identity, is equal to

» l¥=0 /2 k

The lemma will be proved if we show that

(3.4) 2l¥>0

-illik=¿(i^/s\)(e-'- iy\ ,2/fc-l

2 k

To prove (3.4), we let $(z) = z'. Repeated integration by parts, as in the proof of

Taylor's theorem for a real variable, shows that for z = e~", 11 \< 77,

^■ïfi'-^iirliH)'-«,J=o s- Jr(k-l)\


Lp MULTIPLIERS WITH WEIGHT | X \kp ' 629

where T is the path on the unit circle joining 1 to z. From this it follows that for / an

integer,

(3.5) e-«<- 2 tt(*-"- i) t/ 7F--nTf/"^ - 0 *j = 0 (fc-1

•■th®^ -«•>*"'*•Hence

Therefore

-'"- 2 tt^""-1)'l/(/c)l i i* i,i*n*

C it! ^ <c|/| ''I '

20<|/|<1/|/|

On the other hand, we have

*-l ;(i)

e-,v,_2.-i(/WA,)(e-,_ir|-*-< c m

i2*

- 2 Tr(e-"-i);i = 0

*2#(<

j = 0S!

+(k~l)l]

,*-i, ,ft-i

i*-i

Thus

|g-'ft-S;rj(/W/5!)(e-ft-l)'| _ 2*-i2, J-—-L<c|r|

1»i/M/2*

We have proved the inequality " < " in (3.4).

To prove the reverse inequahty in (3.4), we consider two cases: \t\< \(2/Tr)k and

\t\>$(2/ir)k. For 111< \{2/ir)k, it is enough to restrict the sum in (3.4) to

- i(2/ir)krl < / < 0. For / < 0, | /<*> \/kl> (1/fc!)\ I \k. Also,

|/V"«(l -e'e)k-lde\^\f'(l-e'e)k-[eiedeKo I I •'o

_I /"'(«r«'+ ')»_i)(i -eie)k-'eiedeKo

The first term on the right is equal to | l — e" \k/k, which is larger than {2/ir)k \ t \k/k

for | /1< 7T. The second term on the right is less than (| t \k+l/(k + 1)) | / + 1 | ,

which is less than \(2/itf \ t \k/k for 0 > / > -\{2/mf \ 11"'. Using (3.5), we

obtain, for -1(2/77)* 11 \x < I < 0,

Ck-\ Us)

-"- 2 ^(e-u-ns=o A-

,¡k i ¡k

\t\ ,



and so

-ff-Q1(/(')A0(g-',-l),| 2*-1

-(l/4)(2A)*(l/|i|)</<02 A

For the case 11 \> \{2/ir)k, we only have to take the term /=-l. Since

| (-!)<*> |/fc! = Land

\f'e-W(l-eie)k-ldeKo

U -«"!'U) * ' í <77,

(3.5) imphes that the left side of (3.4) is bounded below by (2/7r)2*(| t \2k/k2). This

in turn is larger than C11 \~x for ^(2/77)fc <| 11< 77. This completes the proof of

Lemma (3.1).

The second lemma deals with products.

Lemma (3.6). Let k — 1,2, — For any two functions f, g defined on the integers,

and any integers j and I,

(/g)(/ + 0- 2 ^(/g)(/) =/(/ + /)s = 0

Si

k~\ ,(s)

g(j + i)- 2 hr^gU)s = 0

S\

k-\

1h=0

/(/ + /)- 2 ^—^-tifij + h)s = 0

S\

1(h)

Proof. This identity can be verified using induction and Leibnitz's rule for

products:

(3.7) **(/*)(/) = 2 (î)A*g(/)A*-*/(/ + A),h=0 V '

where/ is an integer, and A: is a nonnegative integer.

The last lemma relates the L|*-i n(>rrn of (mf) to the Fourier coefficients of/.

Lemma (3.8). Let k— 1,2,..., {m{j)}f=_<x a sequence of numbers, f be a trigono-

metric polynomial and A a set of integers. Then

(3.9)

f\{mf)V*~'¿x<c2 2 {m{j+/)|21/0 +1)~2^°ei')/s^'M

j l¥=0 I2k

h=0 j /#0 p


Lp MULTIPLIERS WITH WEIGHT I x\kp ' 631

and

(3.10)

/ 1/ ?\'l¿ i ,2a: — 1 , ^ ^-,\{mf)\ \x\ dx>c2 2-" jSA ¡¥=0

\m(j + /) - 2k=¿ (l^/s\)A°m(j)\ ^ 2

k~\

i 1 1 K^o-^y.fa(;)|Wj)fA=l /E/4 /=^0

-Hye/i /=¿=o

My + /)ri/(y + 0-s*rj(/(I)/J!)A'Ay)|,2^

where the c 's depend only on k.

Proof. In view of Lemmas (3.1) and (3.6), and the fact that | l(h)/h\\< c\l\h for

any integer / and h = 1,... ,/c — 1, it is sufficient to show that for h = l,...,k — 1

and any integers/ and /,

\m{j +0- S^*-1 ((/ - h)(s)/s\)tfmU + ft)|2

'3' > ^ /2A-2A/#0 '

<c2l¥=0

\m(j + l)-2kZtl(l(s)/s\)Asm(j)\

i2k-2h

To prove (3.11), define

(3.12) R(j,l,k,h)= 2 l .' A-m(/ + /»)--rA-m(/)..5=0

The proof of (3.11) will be completed by showing that

(3.13) 2 lRU±":h)l <c]¡2k-2h

m(j+l)-lk=t'(lis)/s\)Asm(h)\

22k-2hl¥=0 ' 1*0

To prove (3.13) we will first prove the identity

(3.14)k-h-\ ,. , ,(r) t r

*(;./.*.*) = 2 tt-iP-l (-»'-'('„)1=0 q=0

To do this use the expansion in (3.3) on the term Asm(j + h) in (3.12) to show that

(3.15)

(; + * + «)- 2 ^#-^0)

k-h-\

R(j,l,k,h)= 2s = 0

_/(*) (/-*)(i) Í

rA^m(/) + iL7f^2 (-ir,(i)w(7 + A + «)?=o



The difference of the right side of (3.14) and the right side of (3.15) is

k-h-i

2s=0

k-h-l /, .\(0 t¡(s)_ 2

(=0 q = 0

to pro\

ttm{j)

l ALïfi-i(-iMi)(*îf)

to show that this is 0 it is enough to prove that

fc-A-l (I-U\M t

(3.16)q = 0

for 0 < s < k — h — l.By induction on í or by (8) on p. 10 of [4], the inner sum

equals (sh_t). Therefore, the right side of (3.16) equals

Since 0=Ss<rC — A — 1, Vandermonde's identity shows that (3.17) equals s\(ls) =

l(s\ This completes the proof of (3.16) and establishes (3.14).

To prove (3.13), change the order of summation in (3.14) to show that for / ¥= 0,

fc-ft-i

R(j,l,k,h)<c 2q = 0

k-h-l /, , \(s)

s=05!

„*-A-l

Hence

2 |*(/,/,&,*)! fc-A-l

i2k-2h 2 2 ^<7=0 /=/=0 «

k — h— 1 /li \(f)

í = 05!

*-£-* h(/ + * + «)- S£Ô~' ((* + îfAij^O)!c 2

?=0,2k~2h

\h + ?rwhich is less than the right side of (3.13). This completes the proof of Lemma (3.8).

4. Periodic case: necessity. Suppose m(j) = 0 for 0 <j < k — 1 and (1.6) holds

for all / G Sk_x. We first observe that {m(j)} G /°°. This can be seen by taking

f(x) — e'jx, j < -1 and / > k, in (1.6). The rest of the proof is based on the

following lemmas.

Lemma (4.1). Let k = 1,2,..., and {m(j)}fl_x G l°°. Suppose there is a trigono-

metric polynomial f, a positive constant B — B{f), a constant D depending only on k,

and a nonempty set of nonzero integers A such that

(4.2) /(/) = o, y = o,i,...,*-i,

(4.3)

(4.4)

(4.5)

( |(»j/)v| |x|"" dx<DB,J -m

f \f\2\x\2k~'dx<DB,•'-77

\fU)\> B\j\k-\ je A,



and

(4.6)

Then, for h

k-h-l\Ahf(j)\< DB\j]

1,2,..., k, m satisfies

h =0,1,...,*- l,j GA.

h(/ + /)-2^(/(->A!)A-m(/)l/2* 5'(4.7) 2 j«"-» 2

7E/4 /^O

where c depends only on k and \\m\\x.

This will be used with functions given by the next two lemmas.

Lemma (4.8). Let k= 1,2,_ 77ien there exists an integer N > 2k, a number

1 < r < 2, and a sequence of trigonometric polynomials {fn}™= x such that for n> N,

(4.2), (4.4)-(4.6) are satisfied with A = (/: n < \j \ < nr) and B — clog n, where c is

positive and independent of n.

Lemma (4.9). Let k= 1,2,..., and N>2k be an integer. Then there exists a

trigonometric polynomial f such that (4.2), (4.4)-(4.6) are satisfied with B > 0 and

A = {j:K\j\<N).

Suppose we have these three lemmas, m(j) = 0 for 0 </ < k — 1 and m satisfies

(1.6) for / in SA_|. Let N and n be as in Lemma (4.8). Given n> N, let / be a

nonnegative integer and let n = [pr']. Because of (1.6), the polynomials/, of Lemma

(4.8) satisfy (4.2)-(4.6) with B = clog« and D independent of n. Therefore, by

Lemma (4.1),

J2(h-l) \m(j + l)-2hs=x(l^/s\)A°m(j)\

/#o2* /•'log ft

Adding these for t = 0,1,2,... proves (1.7) for n> N. Similarly, from Lemma (4.9),

(1.6) and Lemma (4.1) we have

,2(A ,J2 \m(j + l)-2hszW>/s\)ù?m(j)\

l«í[/|</V l¥=02 h

<C

Combining this with the previous case proves (1.7) for all ja 3= 1.

Proof of Lemma (4.1). The proof for the case k = 1 uses Lemma (3.1) to prove a

discrete version of (2.8). The reasoning parallels that used in the proof of Lemma

(2.1) except for the conclusion.

For k > 1, the proof is by induction on k and is given in two steps. First we show

that there is a trigonometric polynomial g that satisfies (4.2)-(4.6) with k replaced by

k — 1, with B(g) — cB(f) and A(g) = A(f). From this it follows that m satisfies

(4.7) forh = l,...,k- 1.

We define g as follows:

1g(x)=ff(t)dt-—ftf(t)dt.



Then g G L1, g(0) = 0, and, for/ ¥= 0,

êU) = -/(/)/'/•Therefore g satisfies (4.2). To prove (4.3), we observe that

(4.10) r\(mg)"\2\x\2(k~l)~ldx

f[- mf,2(A— 1)-1

/TT pVT v/ (mf) dt, ,2(A:-1)-1x dx

. ,2(A;-1)-1x dx./IT f>7T A v

-77 K -77

By Hardy's inequality [5], the first term on the right is less than

\{mf)\ \x\ dx<cB(f).-77

The argument used to estimate the left side of (1.4) and its integral shows that the

second term on the right of (4.10) is also less than (4.11). Hence g satisfies (4.3). A

similar argument shows that it also satisfies (4.4).

That g satisfies (4.5) is obvious. To show that it satisfies (4.6), we observe that for

h = 0,1,...,*- 2,

A*g(/) = (A*[/0)//*],

where/* =/ if/ ¥= 0 and 0* = 1. By (3.7),

A*g(/) = -J2 (*)*Ay)5 = 0

\h-s_ I

U + 'Y

hence, for/ G A(f),

.A-l-A-l

|AAg(/)|<ç5(/)l/f

The second step is to show that m satisfies (4.7) for h = k. To do this we use (3.10)

with A = A(f). By (4.3) the left side is less than B(f). Using the fact that m G l°°,

Lemma (3.1) and (4.4), the last term on the right is also majorized by cB(f). For

h = l,...,k- 1,

2 i H^o-g^-y>/..^(»r mjí <J(/)[/£/4 1=^0 '

by (4.6) and (4.7) for A *s A: - 1. All of these and (4.5) yield (4.7) with A = *. This

completes the proof of Lemma (4.1).

Proof of Lemma (4.8). First we observe that it is sufficient to obtain a sequence

of L°° functions, {g„,*}"=i, with the same properties as {/„}. Then we can take/, to

be a partial sum of the Fourier series of gn k such that /"„ |/„ |21 x \2k~l dx < clog n,

and/„(/) = gn>kU) for |/1< (n + k)r.

The g„¡ks are defined as follows. Let

gn,\(x) = *~I[X[l/«2.l/«](^) - X[l/„,l](x)].


—Ill

Lp MULTIPLIERS WITH WEIGHT | X \kp~' 635

For k = 2,3,..., let

(4.12) gHtk(x) = (e-* - ir'[g„,,_,(x) - e*gHik_i(2x)].

Note that g„k is bounded and is supported in [2l~kn~2,1].

We show that {g„ k) satisfies (4.2) by induction on *. It is obvious that g„ ,(0) = 0.

Suppose {gn¡k-\}, k > 2, satisfies (4.2). Let

(4.13) ^ {x) = ^ä=M= | $(jyj*.e — 1 ■—

Then

^!A.-,(2x) = (1+e,,) | . U)eajxel —

Therefore, (4.12) implies

(4.14) en,kU) = kkü)-kJ[j/2]).It thus follows that g„k(0) = 0. Also, from (4.13), we have

g„,*-i(/) = k.kiJ + u - kAJ)-Sinceg„,A_i(/) = 0,y = 0,l,...,fc- 2, we have

ÍW.*(0) = *..*(!) = ••=*,,,*(*-1),

so (4.14) implies gnk(j) = 0,/ = 1,2,...,* — 1. This proves (4.2). Inequality (4.4)

with B = c log n can be verified by induction.

The proof of (4.5) and (4.6) is divided into two cases: * = 1 and * > 1. We first

consider the case * = 1. Let r be a number between 1 and 2 to be chosen later. We

write

+ /"""' -e~ij,dt- C -e~iJ'dt.V' t )n-i t

The first term is equal to (2 — r)log n. The modulus of the second term is less than

|/1 n~r, which is less than 1 for \j\< nr. The modulus of the third term is less than

(r — l)log«. Upon integrating by parts, the modulus of the last term is less than

2n/\j| , which is less than 2 for \j\> n. By picking r = f and N sufficiently large,

we obtain | gnl(j) |«* clog«, n <|/|< nr, n > N.

In order to simplify the proof of (4.5) and (4.6) for the case * > 2, we approximate

{gn,ki by {A„,a}> which is defined as follows. Let hn ,(x) = g„ ,(x). For * = 2,3,...,

let '

h„,k(x) = (-ix)~l[nn,k-i(x) - A„>A_,(2x)].

Again, hnk is bounded and is supported in [21_/c«"2,1]. As in the proof of Lemma

(2.5), we can, for * > 1, write

1 k~]

(4.15) A„,,(x) = 2 ck,jgnA(2Jx),

(-ÍX) y = 0



where ck] is given by

(4.16) l>;*y=í[(1_f)'It thus follows that

A:-l

j = 0

Kk{x)[k-l

X I 7 = 02 |g„,,(2>x)|, * = 2,3,...

For this and an induction on *, using the estimates

1 1

e-ix _ j -ix< c

1

we obtain

e-<x _ , -ix

k-l

C, for |x| < 77,

(4.17) |g„,*(x)-A„,,(x)|<-—— 2 |&,,,(2/x)|, * = 2,3,...I* I 7 = 0

To prove (4.5) for g„ t for * > 2, it is sufficient to show that

(4.18)k~2 /('> -

KM - 2 J-ñ*K,Mi=o '•

,k-l .> c\j\ log n, n <j <nr,n> N,

where r, N are numbers to be chosen later. The reason is as follows. Since

gB>*(7) = 0,7 = 0,l,...,*-2,

|gn>*(/)

k~2 /(') .

KM - 2 jv^'KA0)1=0 ' ■

k-l •(/)

(gn,,-An,,)(/)- 2 7rA/^-.*"A"-.*)(°)/=0

The second term on the right is equal to

(4.19)

k-l ,-(/)

/77 ™ - ,l'> ,[&,.*(')-*,,*(')] *'7'- 2 7r(e-"-i)' dt

Using the fact that

fe-2 .•</)■°'- 2 TT^"""1)

/=o< 4/1 M ,

and (4.17), it is seen that (4.19) is majorized by c |/1* '. Thus, if N is large enough

and (4.18) is assumed, we have

i*-ign.k(j)\> C\J\ " !og". h <l/'l <«',«>#•


Lp MULTIPLIERS WITH WEIGHT I x\kp ' 637

In order to prove (4.18) we use (4.15) and a change of variables to obtain

k~2 /(O „

KM - 2 ^-A'AB>fc(0)/=o •

k-l1

= 2 ck,s2«k-vf —W.Wi=0

A

• (-*y

Ac —2 •(/)

-72"''- 2 V(e-'2-'<-l)/=o

J/

2 cM2^-2>/"1 1 i<*-1> t i

i=0

fc-i

ir» (_/,)*-' t (*-l)!

+ 2 hkjf^»ri i

•v2 (-¿0

X

,-A*-1 f

*-2 .(/)J2"'- 2 /7r(e-"2"s-l)

/=o

/ y(*-i)

(*-l) (-¿2-0Ar-l

dt

k-l

+ 2 ck,2«k-»f-5 = 0

„-'_1__ 1

»- (-a)*" '

i i ivAc-l /

-,72"''- 2 J-ñ{e-»2¡-iy1=0

s=o J»-' (-it)

By (4.16) the modulus of the first term is equal to

k-l

k~2 ;<" /'

,j2 '- 2 ^{e-1'2" - l)'1=0 '•

<ft

J/.

n,(>-?)|n)T(l08")(2-r)-

Using the estimate

iji-k-l .(/)

'-2 V^"'2"-1) , /*-o

/=o (*-l)!(-/2"s0 A;-l

4/('2-*)\

the modulus of the second term is less than c \j \kn r, which is less than c \j \k ' for

|/1 < nr. A similar estimate shows that the modulus of the third term is less than

Ar-l

2' \ck,,\2->£k~\{\ogn){r-\).(*-l)!

By estimating the integral of each term separately we obtain that the modulus of the

last term is less than c\j\k~] for \j\> n. By choosing r close enough to 1 and N

large enough we get (4.18).

A similar argument shows that in order to prove (4.6) for k > 2, it is sufficient to

prove

(4.20)

k-h-2 .(/)

*KM~ 2 ^t-aa+/a„,a(o)/=0

c\j\ log«,

« <|/|< nr, n> N.



To prove (4.20) we observe that its left side is equal to

k-h-i .</)

Chn¿t\e-"-l)h e-"<- 2 V(e""_1)' dt.

This is less than

Ar-l

5 = 0

LA:,J

f" i i1"*! /», mi ,Ä. .Ar — /i — I. .k-h-l , , .k-h-l ,j \t\ \gn.i(2st)\\t\ \j\ \t\ dt<c\j\ log«,

by changing the variable and performing the integration. This concludes the proof of

Lemma (4.8).

Proof of Lemma (4.9). Define /(x) = 2k^y\s;Ne'Jx. Then/has all the required

properties.

We have thus completed the necessity part of the proof of Theorem (1.5).

5. Periodic case: sufficiency. Suppose m G I00, and, for A = 1,. ..,*, it satisfies

(1.7). We use (3.9) and have

(5.1) f$mfy\2\x\2k'ldx

.2 i ft,

<2 2j ¡¥=0

\m(j + Of |/(7 + /)- %=¿ (l^/s$AJ(j)\¡2k

A=0 j 1*0 r-

By the fact that m G lx and Lemma (3.1), the first term is bounded by

,2, ,2A—1/77 nl/l2 dx.

To estimate the second term, we let

/ .x _ v h(/+/)-2^1(/WA!)A^(7)l^,aUJ - L ,2k-2h

1*0 I2A = 0,1,...,*- 1.

Since/(/) = 0,/= 0,1,...,*- 1,

|A"/(/)| f f(t)(e-" - l)h

f 1/(01 \tJ\'\< 1/1/1

+ / lAOIM'l/f

fc-A-1 ,-(i)

■(e-«-iys = 0

/i , ,A—A . ,A:-/¡ ,/ \t\ dt

h . .k-h-l, ,k-h-l

dt

dt.

Therefore the last term of (5.1) is majorized by a sum of terms of the form

(5.2) c2l/l>i

w, ,„(/)(/ W)\\Akdtri«l< vi/1



and

(5.3) c2 \j\2k~2h~2»kÂJ)[j WOIM*"*),l/l>i ri/l/HI«* /

A = 0,1,...,* - 1. By Hardy's inequality with weights [1], (5.2) is bounded by

/"„ |/(x) |21 x \lk~ ' dx provided we have

2 l/l wk.h(J)—<c-i<|/|<« w

This inequality is true since the left side is less than 2,s;1/|S:„|/|2(*~'!~1)H'Â: A(/),

which is less than c, by (1.7) with A replaced by * — A. Similarly (5.3) is majorized by

J-*\f(x) \21 x |2*~' í/x provided we have

2 l/l ">A,A(./)log77«<C.1/1=-«

This is also true from (1.7) with A replaced by * — A. This concludes the proof of

Theorem (1.5).

References

1. B. Muckenhoupt, Hardy's inequality with weights, Studia Math. 44 (1972), 31-38.

2. B. Muckenhoupt, R. L. Wheeden and W.-S. Young, L2 multipliers with power weights. Adv. in Math,

(to appear).

3._, Weighted Lp multipliers (to appear).

4. J. Riordan, Combinatorial identities, Wiley, New York, 1968.

5. E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press,

Princeton, N. J., 1970.

6. A. Zygmund, Trigonometric series, 2nd ed., Cambridge Univ. Press, New York, 1959.

Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903

Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada


kp~ - ams.org … · if y ¥= kp — 1, k a positive integer. in this paper we consider multipliers...

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