Christ M., Kenig C.E., Sadosky C. Harmonic Analysis and Partial Differential Equations: Essays in Honor of Alberto P. Calderon

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Chapter 4: Transference Couples and Weighted Maximal Estimates

Put

(n3f)(x)

= >A,(k)f(x-k),

for allxER.

Since IA, (k) f (x - k)i IAv(k) f (x - k) 1, we can use dominated convergence

in fl (Z) to infer that as j -* oo,

(3i)

(x) -a

(iii)(x),

for almost all x E R. (4.5)

Using Theorems 4.4-(ii) and (4.4), we have for j > 0,

I17'jf IIPLP(W)

A'(m)f(x+k-m)

k=-oo m=-oo

Wk dx

PIIAjIIMD,w(T) f E If(x+k)IPwkdx

0 k=-oo

IIfIILD(W)'

From this estimate and (4.5) we infer with the aid of Fatou's Lemma that

II1 f II LP(w) <_ P21P IIAIIM

(T)

IIf II LP(W) ,

for all f E LP(w).

(4.6)

Next, observe that for g E S(R), and j > 0, we have 7tjg E S(R), and

(flag)

(Y) E Ai (k)e

2"il,g(y)

= Aj

(e2"iy)

g(y),

for all y E R.

k=-j

By Lebesgue's Theorem, Aj (e2"iy) -*

'(y), for almost all y E R.

Con-

sequently, 7tjg -+ ((Pg)v in L2(R). Combining this with (4.5), we see that

7tg = (4bg)v, for all g E S(R). In view of (4.6), the proof is now complete. 0

The following result sets up strong ties between M,,,,,) and MP,,,,(T).

Theorem 4.6. Suppose that w E Wp, %P E L°° (R), and the support of T is

a subset of [-1/2,1/21. Define

E LOO(T) by writing

2"ii

i(e

)=fi(t),

for -2<t<2'

(4.7)

Then in order that IF E Mp,W(R) it is necessary and sufficient that

b E

Mp,,.,(T). If this is the case, then

7?_1

IIV)IIM,,w(T) <_

W(T), (4.8)

where i

is a positive constant depending only on p and w.

Berkson, Paluszyiiski, and Weiss

Proof. For the sufficiency proof, let 4i(t) = .0 (e2x't), for all t E K By

Theorem 4.5, 41 E Mp,,(R), and II1DIIM,,w(R) < P2/P IIVGIIM,,,(T)

Since the

characteristic function Xi-1/2,1/2) E Mp,,,,(It), and 'I' ='tXi_1/2,1/2) a.e. on Ht,

we see that 'I' E Mp,,,,(R) with IIWIIM,,w(R) < Kp,, IIIGIIM,,w(T)'

Conversely, suppose that 'IQ E Mp,,(R). The reasoning leading up to [7,

Theorem (4.17)] readily adapts to the present circumstances so as to show

that

E Mp,,,,(T), with

KPM IIWIIM,,w(R)' We omit the details

for expository reasons.

Remark. In the unweighted setting, the necessity assertion of Theorem 4.6,

together with the left-hand inequality in (4.8), are contained in [3, Theorem

1] and [13, Theorem (2.3)].

It will now be convenient to introduce a further item of notation.

Definition 4.7. Suppose that 8 : R -+ C is compactly supported. Define

8# : R -+ C by writing

8(x - k), for all x E IL

(4.9)

k=-oo

Using Theorems 4.5 and 4.6, we arrive at the following result.

Theorem 4.8. Suppose that w E Wp, It E Mp,,,,(R), and the support of

'I' is a subset of [-1/2,1/2]. Then ty# E Mp,u,(R), and II'I'#IIM,,w(R)

Kp,,, IIWIIM,,w(R).

The next section will review the key features of transference couples.

Theorem 4.8 will play an instrumental role in setting up transference couples

needed for the applications in §4.4.

4.3 Transference Couples

In all that follows G will be a locally compact group with given left Haar

measure A. If X is a Banach space, and 1 < p < oo, we denote by LP(A, X) the

space of all X-valued, A-measurable functions g such that fo 119(u) IlP dA(u) <

co.

If k E L' (A), g E LP(A, X), then the convolution f * g is defined for

A-a.a. x E G by

(k * 9) (x) =

J c

k(xy)g

(y-') dA(y)

k(y)9 (y-lx) dA(y)'

As is well known, convolution by k is a bounded linear mapping of LP(A, I)

into itself, since

Ilk * 9IIP <- IIk1i1 II91IP'

Chapter 4: Transference Couples and Weighted Maximal Estimates

We shall denote by Np,x(k) the norm of convolution by k E L' (.1) on LP(A, X).

In the special case when X = C, we shall write Np(k) in place of Np,c(k).

The proof of [5, Lemma (4.2)] (reproduced in [8, p. 77]) shows that if p

is an arbitrary measure, and X is a nonzero closed subspace of LP(µ), then

Np,x(k) = Np(k).

Obviously, in the case of the general Banach space X, we have NN,x(k) <

IIkIILI(A), and familiar classical examples show that in general Np,x(k) can

have a much smaller order of magnitude than IIkDILI(),).

For this reason,

transference methods for transplanting individual convolution operators must

aim at preserving convolution norms rather than L'-norms of convolution

kernels.

The remainder of this section will be devoted to a review of some essential

background items concerning the notion of transference couple. This notion

was introduced in [8], where further details can be found.

Definition 4.9. Let B(X) be the algebra of all bounded linear operators

mapping a Banach space X into itself. A transference couple defined on G

and acting in X is a pair (S, T) of strongly continuous mappings of G into

B(X) such that, after writing S =_ {Su}ueG, T = {Tu}UEG, we have:

<oo;

2. cTmsup{IITuII:uEG} <oo;

all uEG,veG.

In particular, if R is a strongly continuous, uniformly bounded represen-

tation of G in X, then (R, R) is a transference couple. Consequently, results

for transference couples generalize traditional transference methods, which

are based on representations of G.

Definition 4.10. Given k E L'(.), and a transference couple (S,T) defined

on G and acting in a Banach space X, we use X-valued Bochner integration

to define the transferred convolution operator Hk E B(X) by writing:

Hkx = fk(u)Txd)t(u),

for all x E X.

Notice that by elementary reasoning we have IIHkII 5 cT IIkIILI(A) When

the group G is amenable, this crude estimate can be considerably improved

as follows.

Theorem 4.11 ([8, Theorem (2.7)]). If G is an amenable group, k E

L '(A), and (S, T) is a transference couple defined on G and acting in a Ba-

nach space 1, then

IIHkII 5 cscTNp,x(k),

for 1 < p < oo.

Berkson, Paluszyriski, and Weiss

In the special case when X is a closed subspace of LP(p), where p is an

arbitrary measure and 1 < p < oo, the conclusion in Theorem 4.11 can be

replaced by the estimate IIHkII < cscTNP(k). In order to discuss the coun-

terpart of this result for maximal estimates, we first introduce the following

notation. Given a sequence {k1}'1 C L1(A), we denote by NP ({k}1)

(E [0, co]) the strong type (p, p) norm of the maximal convolution operator

on LP(A) defined by the sequence of convolution kernels {kk}9° 1. We shall

also need the following auxiliary notion.

Definition 4.12. Suppose that (Y, p) is an arbitrary measure space, 1 < p <

oo, and G is an amenable group. Let S = {Su}uEC be a strongly continuous

mapping of G into B (LP (p)). We say that S is a subpositive family provided

that there is a family of positive operators P = {Pu}uEC C B (LP(p)) such

that

1. for each u E G, and each f E LP(p), we have I Suf I < Pu(I f I) p-a.e. on

2. cP = SUNEG Pull < oo.

The transference by couples of the bounds for maximal convolution op-

erators has the following form.

Theorem 4.13. [8, Theorem (2.11)]. Suppose that p is an arbitrary

measure, 1 < p < oo, and G is an amenable group. Suppose further that

{k1 }J°

1 C

L' (A), and let (S, T) be a transference couple defined on G and

acting in X = LP(p). Then if S is a subpositive family, we have (in the

notation of (4.9)(ii) and (4.12)(ii)):

II0f II LP(µ) S cpcrrNP

({kj}, 1)

Iif IIL'(,.),

for all f E LP(p),

where fit f = Sup,EN I 11k, f I

Although in what follows we shall not need to transfer square function

estimates, it seems appropriate to mention here that for subspaces of LP(p),

Theorem 4.11 can be generalized to square functions under milder hypotheses

on transference couples than those employed in Theorem 4.13. Methods for

transferring the bounds associated with square functions defined by sequences

of multiplier transforms were initiated in [4] and [1]. The methods used to

establish [1, Theorems 2.2 and 2.8] are readily extended to the setting of

transference couples, where they furnish the following two results.

Theorem 4.14 (Scholium). Suppose that p is an arbitrary measure, 1 <

p < oo, X is a closed subspace of LP(p), and G is an amenable group.

Chapter 4: Transference Couples and Weighted Maximal Estimates

Suppose further that {kj}j_1 C L'(\), and let (S, T) be a transference couple

defined on G and acting in X. If a is a constant such that

`Ejt

lki *

f,I2}1/2C all

`r.°1

If7I2}/2II11

LP(1) LP(A)

for all sequences {f,} J°

C LP(A),

then we have (in the notation of (4.9) and (4.10)):

Ilfvo

.I2}1'211

< a c

II{vo

.I2}1/211

(4.10)

j=1 99

,j=1 k; 9j -

LP(µ)

for all sequences {gj}'1 C X.

Theorem 4.15 (Scholium). Assume all the hypotheses of Scholium 4.14

except (4.10). If 0 is a constant such that

1/2

Ikj*fi2}

5 Q IIf IILP(A) ,

for all f E LP(A),

LP(A)

then we have (in the notation of (4.9) and (4.10)):

1/2

{

IHkj9l2

j=1

< ,B CS cT II9IILo(,,)

for all 9 E X.

LP(p)

4.4 Transference of Maximal Estimates from

LP(T) to LP(w), w E Wp

Throughout what follows we consider a weight function w E WP, where 1 <

p < oo.

For u E R, we denote by ry the corresponding character of R

specified by ryu (t) - e2"i". Hence by Theorem 4.4(iii), -y E MP,(1[t), with

T. Suppose now that IF E Mp, (R) and the support of T is a subset

of [-1/2,1/2]. For each u E IIt define F. E MP,u,(R) by writing r,, = y.T.

Invoking Theorem 4.8, we see that ru E MP,,,,(R), and

IIr#IIM,,W(ue)

:5 KP,W

IIINIIM,,,(R)

IIWIIM,,W(R),

(4.11)

where

ry_u(t}I'u (t) _ E e-2"tumq,(t - m),

for all t E R. (4.12)

m=-oo

Berkson, Paluszyriski, and Weiss

Notice, in particular, that.the mapping u E llS H -y-,,F

E Mp,u,(R) has pe-

riod 1 on IL Moreover, the strong continuity of {ru}uER asserted by Theorem

4.4(iii) shows by Banach-Steinhaus that SUPUEl_1,1J IIyuIIM,,W(R) < 00. Conse-

quently, we find with the aid of periodicity in u and (4.11) that

sup 11y-UM

IIM,,W(.)

usup) IIy_uru IIM,,,(R) <_ KP, ,

II"IIM,,,(R) .

(4.13)

UER

Now let G be the linear manifold consisting of all g E LP(w) fl L2(R) such

that g vanishes a.e. in the complement of some corresponding compact set.

Since w E Ap(IR), L is dense in LP(w). It is also clear that, for each u E P,

the translate of 'Ti by u, u), belongs to Mp,"(Bt) with corresponding

multiplier transform on LP(w) specified by f E LP(w) F- s y_uT

(yu f ). It

follows from these observations and (4.12) that for each g E L, the mapping

u E R

T (P'") *g E LP(w) is continuous. An application of uniform bound-

edness, furnished by (4.13), now shows that the mapping u E R * T (P'") #

7-uru

is continuous with respect to the strong operator topology of B (LP(w)). For

u E 1R, and z = e2x" `, we can rewrite (4.12) in the form

y_u(t)I'u(t) = E z-"P(t - m),

for all t E lit.

m=-oo

Assembling the foregoing facts, we arrive at the following theorem.

Theorem 4.16. Suppose that w E Wp, ' E Mp,"(R), and the support of

is a subset of [-1/2,1/2]. Then for each z E T, Mp,"(R) contains the

function q,(z) specified by

41(z) (t)

= E z-m'Y (t - m), for all t E lt.

m=-oo

Moreover,

slip I I"(z) I I

M,. (R)

< KPH" II T II

M,,w(R) ,

(4.14)

and the mapping z E T H T ( )

is continuous with respect to the strong

operator topology of B (LP(w)).

Theorem 4.16 will be used to generate transference couples defined on T

and acting in LP(w). To begin with, we specialize the assertions in Theorem

4.16 by taking 'Ti to be the Fourier transform of the de la Vallee Poussin

Chapter 4: Transference Couples and Weighted Maximal Estimates

kernel of order 7r/2-in other words, we consider the special case where ' is

the function b on R defined as follows:

(i) b is linear on each of the intervals [-1/2,-1/4],[1/4,1/2]; (4.15)

1, if -1/4 < t < 1/4;

(ii) b(t) =

if Its > 1/2.

Since BV(R) C Mp,,(1R), we obviously have b E Mp,,(R). For u E R, let

Nu =

-tub. From the relationship of b to the de la Vallee Poussin kernel, we

find by Fourier inversion that

u)2

{cos

-7r(t

- cos ((t + u)7r) } ,

for u E lit,

t E 1[t.

(Qu)" (t)

_ 72(t +

(4.16)

Next observe that for u E R, the restriction 6,,

I[_1/2,1/21

can be regarded as

an element of BV(T) C Mp,,,(T). It follows by Theorem 4.5 that for u E IR

we have for each f E 11(w),

f (x) _

(Qu)" (M) f (x - m), for almost all x E 1[t

(4.17)

m=-oo

We now proceed to use (4.16) and (4.17) in order to show that (in the

notation of Theorem 4.16) { T s t i 1 } Z 1. is a subpositive family of operators on

LP (w). Let U E (0, 1). It follows from (4.16) that there is a positive absolute

constant 5 such that

(0u)" (m) I < m2

for all m E Z \ {0}, and I (0u)" (0) I < b.

(4.18)

It is an elementary fact that there is a continuous function f E BV(T) such

that f"(0) = b, and fl(m) = 5/m2, for all m E Z \ {0}. In fact, f has the

following explicit description:

7r2 t2

(e`t)=b(1+

7rt+2 I,

for0<t<27r.

Specializing A in Theorem 4.5 to the function f, we infer that there is a

bounded positive operator T E B (IP(w)) such that for each f E 11(w), j3f

is specified by

(i) (x)

= b f (x) + m2

f (x - m),

for almost all x E lL

(4.19)

mEZ\{0}

Berkson, Paluszyziski, and Weiss

Comparing (4.17), (4.18), and (4.19), we find that for each u E [0, 1),

IT (P#") f

i (If () a.e. on R, for all f E L7(w).

(4.20)

Now let z E T, and put z = e2," `, where u E [0, 1). Then

b(z), and

hence by (4.20) we see that

ITb

)

l f l < f I) a.e. on IR,

for all f E LP(w). (4.21)

With the aid of (4.21), we can now state the following theorem.

Theorem 4.17. Suppose that w E Wp. Let b and

f be as in (4.15) and

(4.19), respectively. For z E T, define BZ E'.23 (LP(w)) by setting BZ = r_ q3,

where z = e2x=,. and u E [0,1). Then {Bz}ZET is a family of positive operators

on LP(w) such that

CB = SUP IIB=II < oo.

(4.22)

zET

For each z E T,

T(6p

)

BZ(If I) a.e. on R, for all f E LP(w). (4.23)

Consequently { T i) } is a subpositive family in the sense of Definition

b ZFT

(4.12).

Remark. Notice that the constant CB in (4.22) depends only on p and w.

For each z E T, let S,z E93 (L3(w)) be defined by

Szf = 'Y-1/4T

b()) (`Y,/4f) ,

for all f E LP(w).

(4.24)

It is clear that for each z E T, the operator SZ in (4.24) is the multiplier

transform on LP(w) corresponding to the element of MP,,,(R) specified by

t E R F4 E00 z-mb (t + 4 - m). It follows with the aid of Theorem 4.16

that z E T '-+ SZ is uniformly bounded and strongly continuous.

Suppose next that 0 E MP,(JR), and the support of 0 is a subset of

[-1/2,0]. Applying Theorem 4.16 to A, we define the family {TI-ET S

B (17(w)) by putting TZ = T

l, for each z E T. Thus, z E T '-+ T,, is

strongly continuous, and

sup IIT=II <- KP,, IIoiIMp. ,(R)

zET

Chapter 4: Transference Couples and Weighted Maximal Estimates

In view of our hypothesis that the support of A is contained in [-1/2, 01, and

the definition of b in (4.15), it is easy to see that for t; E T, z E T, we have

(t + -

0(t),

for all t E IL

{mooe_ml

This shows that SST,, = Tt,z f and we have established that the pair of operator

families (S, T) just defined is a transference couple defined on T and acting in

L1(w). Comparing (4.23) and (4.24), we also see that {Sz}:ET is a subpositive

family.

At this juncture, it will be convenient to exhibit the following result, a

corollary of Theorem 4.16. Here and henceforth a will denote normalized

Haar measure on T.

Theorem 4.18. Assume the hypotheses of Theorem 4.16. Suppose that k E

L' (or), and define the function t E L' (R) by putting

P(t) = 1] k(m)T(t - m),

for all t E lit

M=_00

Then P E Mp,4,(JR), and

tr,W)

f = fk(z)T'fdcr(z), for all f E L'(w). (4.25)

Proof. For f E LP(w), we can use the results in Theorem 4.16 to define

55k f by the Bochner integral on the right-hand side of (4.25). Then 55k E

B (L7(w)). Let g belong to the linear manifold G in L1(w) which was de-

scribed right after (4.13). Since W has support contained in [-1/2,1/2], easy

calculations using the definition of the multipliers t(1), z E T, along with

the compact support of g, show that

$k9 = (t W.

(4.26)

For f E S(R), we obtain a sequence

C G such that Ilgn - f II

Ln(W)

and 119- - f 16

0 (for example, we can take gn =

f )V ) Since

55k E B (L1(w)), 5jkgn -4 f kf in L1(w). Applying (4.26) to the sequence

{g,,}',, we see that 1 k9n -+ (t f )v in LZ(R). Consequently, bkf = (fi f )v,

for all f E S(R), and this suffices to complete the proof.

In order to facilitate further discussion, we now introduce two more items

of notation. Given a sequence {Q1}j,1 C Mp,W(R), we shall symbolize by

Tr,W

({Q,}j>1)

E [0, oo] the strong type (p, p) norm of the maximal operator

Berkson, Paluszyriski, and Weiss

on L1(w) defined b y the sequence {T (,`'') } .

If 0 E 1-(Z), and 8: R -4 C

is bounded, measurable, and compactly supported, we define the bounded

measurable function WW,e : R -3 C by writing

E ¢(m)8(t

- m),

for all t E R. (4.27)

"I--- - 00

The stage is now set for the application of Theorem 4.13 to the transference

couple (S, T) described after the second remark. In view of Theorem 4.18,

this immediately furnishes the following maximal theorem.

Theorem 4.19. Suppose that w E Wp, 0 E Mp,u,(R), and the support of 0

is contained in [-1/2,0]. Then for each sequence {kj},>1 C L'(T), we have

(in the notation of (4.27)) {N °}j>1 C Mp,,,(R), and

I {k,}f>1)

"P-

.>1)

Kp,. IIOIIMM.u,(R) Np

where 'JTp, ({W

(respectively, Np ({k,}j>1)) denotes the strong

type (p, p) norm of the maximal operator on L1(w) (respectively, L1(T)) de-

fined by the multiplier transforms (respectively, convolution operators) corre-

sponding to (respectively, to the convolution kernels {k3}j>1).

Suppose next that 0 E Mpm (R), and the support of ,& is contained in

[0,1/2]. We modify the preceding transference couple (S, T) as follows. For

each z E T, define Sz E 93 (L1(w)) by putting

Szf = 71/4T

(')

(7-114f)

for all f E L"(w).

Define the family {7':}zET C M (LP (w)), by putting TZ = T

f, for each

z E T. Considerations analogous to those used in establishing Theorem

4.19, now show that (S, T) is a transference couple defined on T and acting

in L1(w), with {S-}"ET a subpositive family, and that the conclusions of

Theorem 4.19 remain valid for ,& in place of A. Combining this fact with

Theorem 4.19, we arrive at the following result regarding the transference of

maximal estimates.

Theorem 4.20. Suppose that w E Wp, %F E Mp,4,(R), and the support of 1P

is contained in [-1/2,1/2]. Then for each sequence {k;},>, 9 L'(T), we