OF BOCHNER-RIESZ OPERATORS ON SOME WEIGHTED HARDY SPACES

OF BOCHNER-RIESZ OPERATORS ON SOME WEIGHTED HARDY SPACES

LIU LANZHE AND TONG QINGSHAN

Received 17 May 2004 and in revised form 3 November 2004

We show the boundedness for the commutator of Bochner-Riesz operator on some weightedH¹space.

1. Introduction

Let b be a locally integrable function. The maximal operatorB^δ_∗,b associated with the commutator generated by the Bochner-Riesz operator is defined by

B_∗^δ_,b(f)(x)=sup

r>0

B^δ_r,b(f)(x), (1.1)

where

B^δ_r,b(f)(x)=

RⁿB_r^δ(x−y)f(y)b(x)−b(y)d y (1.2) and (B_r^δ( ˆf))(ξ)=(1−r²|ξ|²)^δ₊fˆ(ξ). We also define that

B^δ_∗(f)(x)=sup

r>0

B_r^δ(f)(x), (1.3)

which is the Bochner-Riesz operator (see [8]). Let E be the space E= {h:h = sup_r>0|h(r)|<∞}, then, for each fixedx∈Rⁿ,B^δ_r(f)(x) may be viewed as a mapping from [0, +∞) to E, and it is clear that B_∗^δ(f)(x)= B^δ_r(f)(x) and B_∗^δ_,b(f)(x)= b(x)B_r^δ(f)(x)−B^δ_r(b f)(x).

As well known, a classical result of Coifman et al. [4] proved that the commutator [b,T] generated by BMO(Rⁿ) functions and the Calder ´on-Zygmund operator is bounded onL^p(Rⁿ) (1< p <∞). However, it was observed that [b,T] is not bounded, in general, fromH^p(Rⁿ) toL^p(Rⁿ) and fromL¹(Rⁿ) toL^1,∞(Rⁿ) forp≤1. But, ifH^p(Rⁿ) is replaced by some suitable atomic spaceH_b^p(Rⁿ) andH_B¹(Rⁿ) (see [1,6,7,9]), then [b,T] maps continuouslyH_b^p(Rⁿ) intoL^p(Rⁿ) andH_B¹(Rⁿ) into weakL¹(Rⁿ) forp∈(n/(n+ 1), 1]. The main purpose of this paper is to establish the weighted boundedness of the commutators

Copyright©2005 Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences 2005:2 (2005) 195–201 DOI:10.1155/IJMMS.2005.195

related to Bochner-Riesz operator and BMO(Rⁿ) function on some weightedH¹space.

We first introduce some definitions (see [1,6,7,9]).

Definition 1.1. Letb,wbe locally integrable functions andw∈A1(i.e.,Mw(x)≤cw(x) a.e.). A bounded measurable functionaonRⁿis said to be (w,b)-atom if

(i) suppa⊂B=B(x0,r), (ii)aL^∞≤w(B)⁻¹, (iii)a(y)d y=

a(y)b(y)d y=0.

A temperate distribution f is said to belong toH_b¹(w) if, in the Schwartz distributional sense, it can be written as

f(x)=^∞

j=1

λ_ja_j(x), (1.4)

whereaj’s are (w,b)-atoms,λj∈C, and^∞_j=1|λj|<∞. Moreover,fH_b¹(w)∼^∞_j=1|λj|. Definition 1.2. Letw∈A1. A function f is said to belong to weighted BlockH¹ space H_B¹(w) if f can be written as (1.4), where aj’s arew-atoms (i.e.,aj’s satisfy Definition 1.1(i), (ii), and (iii)a(y)d y=0) andλ_j∈Cwith

∞ j=1

λj

1 + log⁺ 1

λj <∞. (1.5)

Moreover,f_HB¹_(w)∼_∞

j=1|λj|(1 + log⁺((_i|λi|)/|λj|)).

Now, we formulate our results as follows.

Theorem 1.3. Let b∈BMO(Rⁿ) and w∈A1. Then the maximal commutator B^δ_∗,b is bounded fromH_b¹(w)toL¹_w(Rⁿ)whenδ >(n−1)/2.

Theorem 1.4. Let b∈BMO(Rⁿ) and w∈A1. Then the maximal commutator B^δ_∗,b is bounded fromH_B¹(w)toL^1,_w^∞(Rⁿ)whenδ >(n−1)/2.

Theorem 1.5. Let b∈BMO(Rⁿ) and w∈A1. Then the maximal commutator B^δ_∗,b is bounded fromH¹(w)toL^1,_w^∞(Rⁿ)whenδ >(n−1)/2.

2. Proof of theorems

Proof ofTheorem 1.3. It suffices to show that there exists a constantC >0 such that for every (w,b)-atoma,

B^δ_∗,b(a)_L1

w≤C. (2.1)

Letabe a (w,b)-atom supported on a ballB=B(x0,R). We write

Rⁿ

B_∗^δ_,b(a)(x)w(x)dx

=

|x−x0|≤2R

B_∗^δ_,b(a)(x)w(x)dx+

|x−x0|>2R

B^δ_∗,b(a)(x)w(x)dx≡I+II.

(2.2)

ForI, takingq >1, by H¨older’s inequality and theL^q-boundedness ofB_∗^δ_,b(see [2]), we see that

I≤CB_∗^δ_,b(a)_L^q_w·w(2B)^1−1/q≤Ca_L^q_ww(B)^1−1/q≤C. (2.3) ForII, letb0= |B(x0,R)|⁻¹

B(x0,R)b(y)d y, then

II≤ ∞ k=1

2^k+1R≥|x−x0|>2^kR

b(x)−b0B_∗^δ(a)(x)w(x)dx

+ ∞ k=1

2^k+1R≥|x−x0|>2^kRB_∗^δb−b0

a(x)w(x)dx=II1+II2.

(2.4)

ForII1, we chooseδ0such that n−1

2 < δ0<min

δ,n+ 1

2 (2.5)

and consider the following two cases.

Case 1(0< r≤R). In this case, note that (see [8])

B^δ(z)≤C1 +|z|₋(δ+(n+1)/2)

, (2.6)

we have, for|x−x0|>2|y−x0|, B^δ_r(a)(x)≤Cr⁻ⁿ

B(x0,R)

a(y)

1 +|x−y|/r^δ+(n+1)/2d y

≤C|B|^(δ0+(n+1)/2)/n2^k+1B^−(δ0+(n+1)/2)/n

w(B)⁻¹.

(2.7)

Case 2(r > R). In this case, note that

∇^βB^δ(z)≤C1 +|z|₋(δ+(n+1)/2)

(2.8) for anyβ=(β1,. . .,βn)∈(N∪ {0})ⁿand|x−x0|>2|y−x0|, where

∇^β= ∂

∂x1 β1

···

∂

∂x_n

βn

, (2.9)

by the vanishing condition ofa, we gain B^δ_r(a)(x)≤Cr⁻⁽ⁿ⁺¹⁾

B(x0,R)

a(y)y−x0 1 +x−x0/r^δ+(n+1)/2d y

≤C|B|^(δ0+(n+1)/2)/n2^k+1B^−(δ0+(n+1)/2)/n

w(B)⁻¹.

(2.10)

CombiningCase 1withCase 2, we obtain II1≤C

∞ k=1

2^k+1R≥|x−x0|>2^kR

b(x)−b0|B|^(δ0+(n+1)/2)/n

×2^k+1B^−(δ0+(n+1)/2)/n

w(B)⁻¹w(x)dx

≤C ∞ k=1

2^{−k(δ0+(n+1)/2)}w(B)⁻¹

2^k+1R≥|x−x0|>2^kR

b(x)−b0w(x)dx.

(2.11)

Sincew∈A1,wsatisfies the reverse of H¨older’s inequality as follows:

1

|B|

Bw(x)^pdx

1/ p

≤ C

|B|

Bw(x)dx (2.12)

for any ballBand some 1< p <∞(see[10]). Using the properties of BMO(Rⁿ) functions (see [10]), and notingw∈A1, then

wB2

B2 · B1

wB1

≤C (2.13)

for all ballsB1,B2withB1⊂B2. Thus, by H¨older’s and reverse of H¨older’s inequalities for w∈A1, we get, for 1/ p+ 1/ p=1,

II1≤C ∞ k=1

2^{−k(δ0+(n+1)/2)}w(B)⁻¹2^k+1B 1

2^k+1B

2^k+1B

b(x)₋b0^pdx

1/ p

× 1

2^k+1B

2^k+1Bw(x)^pdx

1/ p

≤CbBMO

∞ k=1

k2^{−k(δ0−(n−1)/2)}

w2^kB 2^kB

|B_| w(B) ^≤C.

(2.14)

ForII2, similar to the estimate ofII1, we obtain B_r^δb−b0

a(x)≤CbBMOw(B)⁻¹|B|^(δ0+(n+1)/2)/nx−x0^{−(δ0+(n+1)/2)}, (2.15) thus

II2≤CbBMO

∞ k=1

w(B)⁻¹|B|^(δ0+(n+1)/2)/n2^kB^−(δ0+(n+1)/2)/n

w2^kB

≤CbBMO

∞ k=1

2^{−k(δ0−(n−1)/2)}

w2^kB 2^kB

|B| w(B) ^≤C.

(2.16)

This finishes the proof ofTheorem 1.3.

To proveTheorem 1.4, we recall the following lemma (see [5,10]).

Lemma2.1. Letw≥0and{g_k}be a sequence of measurable functions satisfying gk

L^1,w^∞≤1. (2.17)

Then, for every numerical sequence{λ_k},

k

λ_kg_k

L^1,w^∞

≤C

k

λ_k

+ log

j

λ_jλ_k

. (2.18)

Proof ofTheorem 1.4. ByLemma 2.1, it is enough to show that there exists a constantC such that

B_∗^δ_,b(a)_L^1,_w^∞≤C for eachw-atoma. (2.19) Letabe aw-atom supported on a ballB=B(x0,r). We write

wx∈Rⁿ:B_∗^δ_,b(a)(x)> λ

≤wx∈2B:B^δ_∗,b(a)(x)> λ+wx∈(2B)^c:B_∗^δ_,b(a)(x)> λ=I+II. (2.20) ForI, by theL^q-boundedness ofB_∗^δ_,bforq >1, we gain

I≤λ⁻¹B^δ_∗,b(a)χ_2BL1

w≤Cλ⁻¹B^δ_∗,b(a)_L^q_w·w(B)^1−1/q

≤Cλ⁻¹a_L^q_w·w(B)^1−1/q≤Cλ⁻¹. (2.21) ForII, letb0= |B|⁻¹

Bb(x)dx, notice that

B_∗^δ_,b(a)(x)=b(x)B^δ_r(a)(x)−B^δ_r(ba)(x)

=b(x)−b0

B_r^δ(a)(x)−B^δ_rb−b0

a(x)

≤b(x)−b0

B_r^δ(a)(x)+B_r^δb−b0

a(x)

≤b(x)−b0B^δ_∗(a)(x) +B^δ_∗b−b0

a(x),

(2.22)

we have

II≤w

x∈(2B)^c:b(x)−b0g_µ^∗(a)(x)>λ 2 +w

x∈(2B)^c:g_µ^∗b−b0

a(x)>λ

2 ⁼II1+II2.

(2.23)

Similar to the proof ofTheorem 1.3, we get II1≤Cλ⁻¹

(2B)^c

b(x)−b0B^δ_∗(a)(x)w(x)dx

=Cλ⁻¹ ∞ k=1

2^k+1B\2^kB

b(x)−b0B^δ_∗(a)(x)w(x)dx≤Cλ⁻¹bBMO,

II2≤Cλ⁻¹

(2B)^cB_∗^δb−b0

a(x)w(x)dx≤Cλ⁻¹bBMO.

(2.24)

Combining the estimate ofI,II1, andII2, we gain

wx∈Rⁿ:B_∗^δ_,b(a)(x)> λ≤Cλ⁻¹bBMO. (2.25)

This completes the proof ofTheorem 1.4.

Proof ofTheorem 1.5. . Given f ∈H¹(w), let f =

jλjajbe the atomic decomposition for f. By a limiting argument, it suffices to showTheorem 1.5for a finite sum of f =

Qλ_Qa_Qwith_Q|λ_Q| ≤CfH¹(w). We may assume that eachQ(the supporting cube ofaQ) is dyadic. Forλ >0 by [3, Lemma 4.1], there exists a collection of pairwise disjoint dyadic cubes{S}such that

Q⊂S

λQ≤Cλ|S|, ∀S,

S

|S| ≤λ⁻¹

Q

λ_Q,

Q⊂S

λ_Q|Q|⁻¹χ_Q

L^∞

≤Cλ.

(2.26)

Let E=

SS, where for a fixed cube Q, Q denotes the cube with the same center as Qbut with the side-length 4^√n times that ofQ. Then,|E| ≤Cλ⁻¹fH¹. SetM(x)=

S

Q⊂SλQaQ,N(x)=f(x)−M(x). By theL²boundedness ofB^δ_∗,band the well-known argument, it suffices to show that

wx∈E^c:B^δ_∗,b(M)(x)> λ≤Cλ⁻¹fH¹(w). (2.27)

BecauseB_∗^δ_,b(M)(x)≤

S

Q⊂S|λQ|B^δ_∗,b(aQ)(x), we have wx∈E^c:B^δ_∗,b(M)(x)> λ

≤Cλ⁻¹

E^cB^δ_∗,b(M)(x)w(x)dx

≤Cλ⁻¹

S

Q⊂S

λ_Q ∞ k=1

2^k+1Q\2^kQB^δ_∗,ba_Q(x)w(x)dx,

(2.28)

similar to the estimate ofTheorem 1.3, we get, whenx∈E^c,

B^δ_∗,baQ

(x)≤CbBMOw(B)⁻¹|Q|^(δ0+(n+1)/2)/nx−x0^{−(δ0+(n+1)/2)}

+Cb(x)−b0w(B)⁻¹2^{−k(δ0+(n+1)/2)}, (2.29)

thus, by H¨older’s and reverse of H¨older’s inequalities forw∈A1, we obtain wx∈E^c:B_∗^δ_,b(M)(x)> λ

≤Cλ⁻¹w(B)⁻¹

S

Q⊂S

λQ^∞

k=1

k2^{−k(δ0+(n+1)/2)}w2^kQ

≤Cλ⁻¹

S

Q⊂S

λQ^∞

k=1

k2^{−k(δ0−(n−1)/2)}

≤Cλ⁻¹

S

Q⊂S

λQ≤Cλ⁻¹fH¹(w).

(2.30)

This finishes the proof ofTheorem 1.5.

Acknowledgment

The author would like to express his gratitude to the referee for his very valuable com- ments and suggestions.

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Liu Lanzhe: College of Mathematics and Computer, Changsha University of Science and Technol- ogy, Changsha 410077, China

E-mail address:[email protected]

Tong Qingshan: College of Mathematics and Computer, Changsha University of Science and Tech- nology, Changsha 410077, China