Báo cáo hóa học: "WEIGHTED ESTIMATES FOR COMMUTATORS ON NONHOMOGENEOUS SPACES" pot

This paper is to establish the weighted norm inequality for commutators of Calder ´on-Zygmund operators with RBMOμ functions by an estimate for a variant of the sharp maximal function in

ON NONHOMOGENEOUS SPACES

WENGU CHEN AND BING ZHAO

Received 14 November 2005; Revised 8 March 2006; Accepted 11 March 2006

Letμ be a Borel measure onRd which may be nondoubling The only condition thatμ

must satisfy isμ(Q) ≤ c0l(Q) nfor any cubeQ ⊂ R d with sides parallel to the coordinate axes and for some fixedn with 0 < n ≤ d This paper is to establish the weighted norm

inequality for commutators of Calder ´on-Zygmund operators with RBMO(μ) functions

by an estimate for a variant of the sharp maximal function in the context of the nonho-mogeneous spaces

Copyright © 2006 W Chen and B Zhao This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

Letμ be some nonnegative Borel measure onRdsatisfying

for any cube Q ⊂ R d with sides parallel to the coordinate axes, wherel(Q) stands for

the side length ofQ and n is a fixed real number such that 0 < n ≤ d Throughout this

paper, all cubes we will consider will be those with sides parallel to the coordinate axes Forr > 0, rQ will denote the cube with the same center as Q and with l(rQ) = rl(Q).

Moreover,Q(x, r) will be the cube centered at x with side length r.

The classical theory of harmonic analysis for maximal functions and singular inte-grals on (Rd,μ) has been developed under the assumption that the underlying

mea-sure μ satisfies the doubling property, that is, there exists a constant c > 0 such that μ(B(x, 2r)) ≤ cμ(B(x, r)) for every x ∈ R d andr > 0 But recently, many classical results

have been proved still valid without the doubling condition; see [1–18] and their refer-ences

Orobitg and P´erez [11] have studied an analogue of the classical theory of A p(μ)

weights inRd without assuming that the underlying measureμ is doubling Then, they

Hindawi Publishing Corporation

Journal of Inequalities and Applications

Volume 2006, Article ID 89396, Pages 1 14

DOI 10.1155/JIA/2006/89396

obtained weighted norm inequalities for the (centered) Hardy-Littlewood maximal func-tion and corresponding weighted estimates for nonclassical Calder ´on-Zygmund oper-ators They also considered commutators of those Calder ´on-Zygmund operators with BMO(μ) functions The purpose of this paper is to establish weighted estimates for

com-mutators of those nonclassical Calder ´on-Zygmund operators with RBMO(μ) in this new

setting

Let us introduce some notations and definitions Given two cubesQ ⊂ R inRd, we set

K Q,R =1 +

NQ,R

k =1

μ

2k Q

l

whereN Q,Ris the first integerk such that l(2 k Q) ≥ l(R) K Q,Rwas introduced by Tolsa in [15]

Givenβ d(depending ond) big enough (e.g., β d > 2 n), we say that some cubeQ ⊂ R d

is doubling ifμ(2Q) ≤ β d μ(Q).

Given a cubeQ ⊂ R d, letN be the smallest integer ≥0 such that 2N Q is doubling We

denote this cube byQ.

Letη > 1 be some fixed constant We say that a function b(x) is in RBMO(μ) if there

exists some constantc1such that for any cubeQ,

1

μ(ηQ)



Q

b − m

Q bdμ ≤ c1,

m Q b − m R b ≤ c1K Q,R for any two doubling cubesQ ⊂ R,

(1.3)

wherem Q b =1/μ(Q)

Q b dμ The minimal constant c1is the RBMO(μ) norm of b, and it

will be denoted by b  ∗ The RBMO(μ) function space was introduced by Tolsa in [15] and shares more properties with the classical BMO function space than BMO(μ) space.

We say a kernelk(x, y) :Rd × R d \{(x, y) : x = y } → Cis ann-dimensional Calder

´on-Zygmund kernel in the new setting if

(1)| k(x, y) | ≤ A/ | x − y | nifx = y,

(2) there exists 0< γ ≤1 such that

k(x, y) − k(x,y)+k(y, x) − k(y, x) ≤ A | x − x | γ

| x − y | n+γ (1.4)

if| x − y | > 2 | x − x |

A bounded linear operatorT from L2(μ) to L2(μ) is said to be a Calder ´on-Zygmund

operator withn-dimensional kernel k if for every compacted supported function f ∈

L2(μ),

T f (x) =



Rd k(x, y) f (y)dμ(y) forx ∈suppf (1.5) Forr > 0, we define the truncated operators by

T r f (x) =



and define the maximal operator associated withT as follows:

T ∗ f (x) =sup

r>0

2 Sharp maximal function estimates for commutators

In [15], Tolsa defined a sharp maximal operatorM#f (x) such that

f ∈RBMO(μ) ⇐⇒ M#f ∈ L ∞(μ), (2.1) where

M#f (x) =sup

x ∈ Q

1

μ (3/2)Q

Q

f − m

Q fdμ + sup

x ∈ Q ⊂ R

Q, R doubling

m Q f − m R f

We also consider the noncentered doubling maximal operatorN:

N f (x) = sup

x ∈ Q

Q doubling

1

μ(Q)



By [15, Remark 2.3], forμ-almost all x ∈ R done can find a sequence of doubling cubes

{ Q k}kcentered atx with l(Q k)→0 ask → ∞such that

lim

k →∞

1

μ

Q k

Q k

So,| f (x) | ≤ N f (x) for μ-a.e x ∈ R d Moreover, it is easy to show thatN is of weak type

(1,1) and bounded onL p(μ), p ∈(1,∞]

In order to obtain the estimate for a variant of the sharp maximal function for the commutators of Calder ´on-Zygmund operators defined as above with RBMO(μ)

func-tions, we need the following definition

A functionB : [0, ∞)→[0,∞) is called a Young function if it is continuous, convex, increasing, and satisfyingB(0) =0 andB(t) → ∞ast → ∞ We define theB-average of a

function f over a cube Q by means of the following Luxemburg norm:

 f B,Q,(ρ) =inf

λ > 0 : 1 μ(ρQ)



Q B f (y)

λ

dμ ≤1

The generalized H¨older’s inequality

1

μ(ρQ)



Q

f (y)g(y)dμ(y) ≤  f B,Q,(ρ) g B,Q,(ρ) (2.6)

holds, whereB is the complementary Young function associated to B For every locally

integrable functionf , define its maximal operator M B,(ρ)by

M B,(ρ) f (x) =sup

x ∈ Q

Theorem 2.1 Let b ∈RBMO(μ), let 0 < δ <  < 1, there exists C = C δ,  such that

M#

δ



[b, T] f

(x) ≤ C  b  ∗

M ,(3/2)(T f )(x) + M2

(9/8) f (x) + T ∗ f (x)

where M#

δ f (x) = M#(| f | δ)1/δ , M p,(ρ) f (x) =supx ∈ Q((1/μ(ρQ))

Q | f | p dμ)1/ p , 0 < p < ∞ Set M(ρ) f (x) = M1,(ρ) f (x).

Before proving the theorem, another equivalent norm for RBMO(μ) is needed

Sup-pose that for a given functionb ∈ L1loc(μ) there exist some c2and a collection of numbers

{ b Q}Q(i.e., for each cubeQ, there exists b Q ∈ R) such that

sup

Q

1

μ(ηQ)



Q

b − b Qdμ ≤ c2,

b Q − b R  ≤ c2K Q,R for any two cubesQ ⊂ R.

(2.9)

Then, set b  ∗∗ =infc2, where the infimum is taken over all the constantsc2and all the numbers{ b Q}satisfying (2.9) By [15, Lemma 2.8, page 99], for a fixedη > 1, the norms

 ·  ∗and ·  ∗∗are equivalent

Proof of Theorem 2.1 We follow the argument from [15, proof of Theorem 9.1] LetQ =

Q(x, r) be a cube with center x and side length r For 0 < δ < 1 and α, β ∈ R, we have

|| α | δ − | β | δ | ≤ | α − β | δ Let{ b Q}Qbe a sequence of numbers satisfying



Q

b − b Qdμ ≤2μ(2Q)  b  ∗∗, (2.10) for all cubesQ and

b Q − b R  ≤2K Q,R b  ∗∗ (2.11) for all cubesQ, R with Q ⊂ R For any cube Q, we denote h Q:= − m Q(T((b − b Q)f χRd \

(4/3)Q)) We will show that for all x, Q with x ∈ Q,

1

μ

(3/2)Q

Q

[b, T] f − h Qδ

dμ

 1/δ

≤ C  b  ∗∗

M ,(3/2)(T f )(x) + M2

(9/8) f (x)

; (2.12) and for all cubesQ, R with Q ⊂ R, x ∈ Q,

h Q − h R  ≤ C  b  ∗∗

M(92/8) f (x) + T ∗ f (x)

K Q,R2 . (2.13)

To obtain (2.12) for some fixed cubeQ and x with x ∈ Q, we rewrite [b, T] f :

[b, T] f =b − b Q

T f − T

b − b Q

f1



− T

b − b Q

f2



where f1= f χ(4/3)Q, f2= f − f1 Let us estimate the term (b − b Q)T f first Take 1 < r < ε/δ By H¨older’s inequality, we have

1

μ

(3/2)Q



Q

b(y) − b Q

T f (y)δ

dμ(y)

1/δ

μ

(3/2)Q

Q

b(y) − b Qδr

dμ(y)

1/δr

1

μ (3/2)Q

Q

T f (y)δr

dμ(y)

1/δr

≤ C  b  ∗∗ M δr,(3/2)(T f )(x) ≤ C  b  ∗∗ M ε,(3/2)(T f )(x).

(2.15)

SinceT : L1(μ) → L1,∞(μ) (see [9]) and 0< δ < 1, Kolmogorov’s inequality and

gener-alized H¨older’s inequality yield

1

μ (3/2)Q

Q

T

b − b Q

f1(y)δ

dμ(y)

1/δ

μ (3/2)Q

(4/3)Q

b(y) − b Q

f (y)dμ(y)

≤ Cb − b Q

expL,(4/3)Q,(9/8)  f L LogL,(4/3)Q,(9/8),

(2.16)

while John-Nirenberg inequality implies that

1

μ (3/2)Q

 (4/3)Qexp b(y) − b Q

C  b  ∗

So there exists a positive constantC such that for all cubes Q,

b − b Q

Therefore

1

μ((3/2)Q)



Q

T

b − b Q

f1(y)δ

dμ(y)

1/δ

≤ C  b  ∗ M L LogL,(9/8) f (x). (2.19)

In order to prove (2.12), we only need to estimate| T((b − b Q)f2)− h Q| δ Note that

K Q,2 k(4/3)Q =1 +

k+1

j =1

μ

2j Q

l

2j Qn ≤1 + (k + 1)C0≤ Ck. (2.20)

Forx, y ∈ Q, we have

T

b − b Q

f2



(x) −T

b − b Q

f2



(y)



Rd \(4/3)Q

| y − x | γ

| z − x | n+γb(z) − b Qf (z)dμ(z)

∞



k =1



2k(4/3)Q \2k −1 (4/3)Q

l(Q) γ

| z − x | n+γb(z) − b2k(4/3)Q+b Q − b2k(4/3)Qf (z)dμ(z)

∞



k =1

2− kγ 1

l

2k Qn



2k(4/3)Q

b(z) − b2k(4/3)Qf (z)dμ(z) +C

∞



k =1

k2 − kγ  b  ∗ 1

l

2k Qn



2k(4/3)Q

f (z)dμ(z)

∞



k =1

μ (9/8)2 k(4/3)Q

2k(4/3)Q

b(z) − b2k(4/3)Qf (z)dμ(z) +C

∞



k =1

k2 − kγ  b  ∗ M(9/8) f (x)

∞



k =1

2− kγb − b2k(4/3)Q

expL,2 k(4/3)Q,(9/8)  f L LogL,2 k(4/3)Q,(9/8)+C  b  ∗ M(9/8) f (x)

≤ C  b  ∗ M L LogL,(9/8) f (x) + C  b  ∗ M(9/8) f (x).

(2.21)

Forρ > 1, it is easy to see M(ρ) f (x) ≤ M L LogL,(ρ) f (x) Thus

T

b − b Q

f2 

(x) −T

b − b Q

f2 

(y) ≤ C  b  ∗ M L LogL,(9/8) f (x). (2.22) According to Jensen’s inequality, we obtain

1

μ

(3/2)Q

Q

T

b − b Q

f2

 (y) − m Q

T

b − b Q

f2 δ

dμ(y)

 1/δ

μ

(3/2)Q

Q

T

b − b Q



f2

 (y) − m Q



T

b − b Q



f2 dμ(y)

≤ C  b  ∗ M L LogL,(9/8) f (x).

(2.23)

Note that forρ > 1, M(2ρ) f (x) ≈ M L LogL,(ρ) f (x) By (2.15), (2.16), and (2.23) we obtain (2.12)

For{ h Q}Q, we want to prove (2.13) Consider two cubesQ ⊂ R and x ∈ Q We denote

N = N Q,R+ 1 We writeh Q − h Rin the following way:

m Q

T

b − b Q

f χRd \(4/3)Q



− m R

T

b − b R

f χRd \(4/3)R

≤m Q

T

b − b Q

f χ2Q \(4/3)Q+m Q

T

b Q − b R

f χRd \2Q

+m Q

T

b − b R

f χ2N Q \2Q

+m Q

T

b − b R

f χRd \2N Q



− m R

T

b − b R

f χRd \2N Q

+m R

T

b − b R

f χ2N Q \(4/3)R

= M1+M2+M3+M4+M5.

(2.24)

Let us estimateM1 Fory ∈ Q we have

T

b − b Q

f χ2Q \(4/3)Q

 (y)  ≤ C l(2Q) n



2Q

b − b Q | f | dμ

≤ Cb − b Q

expL,2Q,(9/8)  f L LogL,2Q,(9/8)

≤ C  b  ∗ M L LogL,(9/8) f (x) ≤ C  b  ∗ M2

(9/8) f (x).

(2.25)

So we deriveM1≤ C  b  ∗ M2

9/8) f (x) Let us consider M2 Forx, y ∈ Q,

T f

χRd \2Q



(y) =

Rd \2Q f (z)k(y, z)dμ(z)





≤







Rd \2Q f (z)

k(y, z) − k(x, z)

dμ(z)



+









Rd \2Q k(x, z) f (z)dμ(z)





≤







Rd \2Q

| y − z | γ

| y − z | n+γf (z)dμ(z)

+T ∗ f (x)

≤ C sup

Q0 x

1

l

Q0

n



Q0

| f | dμ + T ∗ f (x) ≤ CM(9/8) f (x) + T ∗ f (x).

(2.26) Thus

M2=b R − b Q

T f

χRd \2Q  ≤ CK Q,R b  ∗

T ∗ f (x) + CM2

(9/8) f (x)

For the termM4, we execute the process as in (2.21) For anyy, z ∈ R d, we get

T

b − b R

f χRd \2Q

 (y) − T

b − b R

f χRd \2Q

 (z)

≤ C  b  ∗ M L LogL,(9/8) f (x) ≤ C  b  ∗ M2

(9/8) f (x). (2.28)

The termM5can be estimated asM1 We can obtain

M5≤ C  b  ∗ M2

Finally we have to deal withM3 Fory ∈ Q, we have

b2k+1 Q − b R  ≤ CK2k+1 Q,R b  ∗ ≤ CK Q,R b  ∗ (2.30) Then,

T

b − b R

f χ2N \2Q

 (y)

N−1

k =1

1

l

2k Qn



2k+1 Q \2k Q

b − b R | f | dμ

N−1

k =1

1

l

2k Qn



2k+1 Q

b − b2k+1 Q | f | dμ + C

N−1

k =1

l

2k Qn



2k+1 Q

b2k+1 Q − b R | f | dμ

N−1

k =1

b − b2k+1 Q

expL,2 k+1 Q,(9/8)  f L LogL,2 k+1 Q,(9/8)

+C

N−1

k =1

K Q,R b  ∗ μ

2k+1 Q

l

2k Qn 1

μ

2k+1 Q

2k+1 Q | f | dμ

≤ C  b  ∗ M L LogL,(9/8) f (x) + CK Q,R b  ∗

N−1

k =1

μ

2k+1 Q

l

2k Qn M(9/8) f (x)

≤ C  b  ∗ M L LogL,(9/8) f (x) + CK2

Q,R  b  ∗ M(9/8) f (x)

≤ C  b  ∗ M(92/8) f (x)K Q,R2 .

(2.31)

Taking the mean overQ, we get

M3≤ C  b  ∗ M(92/8) f (x)K Q,R2 . (2.32)

By the estimates onM1,M2,M3,M4,M5, we can get (2.13)

Let us see how from (2.12) and (2.13) one obtains (2.8) IfQ is a doubling cube and

x ∈ Q, then we have by (2.12)

m Q[b, T] fδ

−h δ

Q 1/δ

μ(Q)



Q

[b, T] fδ

− h δ Qdμ1/δ

≤ C  b  ∗

M ,(3/2)(T f )(x) + M2

(9/8) f (x) + T ∗ f (x)

.

(2.33)

Also, for any cubeQ  x, K Q, Q≤ C, and then by (2.12) and (2.13) we get

1

μ((3/2)Q)



Q

[b, T] fδ

− m Q[b, T] fδdμ1/δ

μ

(3/2)Q

Q

[b, T] fδ

−h Qδdμ1/δ+h Qδ

−h

Qδ 1/δ

+h

Qδ

− m Q[b, T] fδ 1/δ

μ

(3/2)Q

Q

[b, T] f − h Qδ

dμ

1/δ

+h Q − h

Q+h δ



Q − m Q[b, T] fδ 1/δ

≤ C  b  ∗

M ,(3/2)(T f )(x) + M2

(9/8) f (x) + T ∗ f (x)

.

(2.34)

On the other hand, for all doubling cubesQ ⊂ R with x ∈ Q such that K Q,R ≤ P0, where

P0is the constant in [15, Lemma 9.3, page 143] By (2.13) we have

h Q − h R  ≤ C  b  ∗

M(92/8) f (x) + T ∗ f (x)

K Q,R P0. (2.35)

So by [15, Lemma 9.3, page 143], we get

h Q − h R  ≤ C  b  ∗

M(92/8) f (x) + T ∗ f (x)

for all doubling cubesQ ⊂ R with x ∈ Q, using (2.13) again, we get



mQ



[b, T] fδ

− m R

 [b, T] fδ

≤m

Q

 [b, T] fδ

− h δ Q+h δ

Q − h δ R+hδ

R − m R

 [b, T] fδ

≤ C

 b  ∗

M ,(3/2)(T f )(x) + M2

(9/8) f (x) + T ∗ f (x)

K Q,Rδ

.

(2.37)

From the above estimates, we can obtain

M#

δ



[b, T] f

(x) ≤ C  b  ∗

M ,(3/2)(T f )(x) + M2

(9/8) f (x) + T ∗ f (x)

 Now we are in the position to give the definition of weights we will consider Here we will consider theA p(μ) weights introduced by Orobitg and P´erez in [11] So we need the assumption thatμ(∂Q) =0 for any cubeQ with sides parallel to the coordinates axes.

Let 1< p < ∞and letp = p/(p −1) We say that a weightw satisfies the A p(μ)

condi-tion if there exists a constantK such that for all cubes Q

1

μ(Q)



μ(Q)



Q w1− p dμ

p −1

And we define theA ∞(μ) class as A ∞(μ) =p>1 A p(μ).

Theorem 2.2 Let 0 < p < ∞ , let ρ > 1, w(x) ∈ A ∞(μ) defined above, then



Rd

T f (x)p

w(x)dμ(x) ≤ C



Rd



M(ρ) f (x)p

holds for every function f for which the left-hand side is finite.

Proof For each  > 0 we define the maximal operator

T  ∗ f (x) =sup

δ> 

We only need to prove that forw ∈ A ∞(μ), there exist suitable constants α, β, ε such that

w

x : T  ∗ f (x) > (1 + β)t, M(ρ) f (x) ≤ εt

≤ αw

x : T  ∗ f (x) > t

, t > 0, (2.42) for allα p < (1 + β) −1 We may assume f is nonnegative and locally integrable Follow the

idea of [11], we first consider the special case whenw =1, then (2.42) turns to

μ

x : T  ∗ f (x) > (1 + β)t, M(ρ) f (x) ≤ εt

≤ αμ

x : T  ∗ f (x) > t

SinceΩ= { x ∈ R d:T  ∗ f (x) > t }is open, we decompose it into disjoint Whitney cubes

Ω=j Q j, whereQ jare disjoint and 2ρ diam(Q j)≤dist(Q j,Ωc)≤8ρ diam(Q j), and ev-ery point ofRd at most lies in 4ρQ j cubes Obviously 4ρQ j ⊂Ω We will show that for givenβ > 0, 0 < α < 1, there exists c = c(β, α, n) such that for all j,

μ

x ∈ Q j:T  ∗ f (x) > (1 + β)t, M(ρ) f (x) ≤ εt

≤ αμ

4Q j



Summing over allj, we have

μ

x ∈ R d:T  ∗ f (x) > (1 + β)t, M(ρ) f (x) ≤ εt

≤ α4 n μ( Ω). (2.45) Chooseα such that α4 n < 1, then we can obtain (2.42) in the special case For the general casew, recall that if w ∈ A ∞(μ), then by [11, Lemma 2.3, page 2017], there exist positive constantsc, δ such that for all cubes Q and all E ⊂ Q,

w(E) w(Q) ≤ c μ(E)

μ(Q)

δ

Looking back at (2.44), we get

w

x ∈ Q j:T  ∗ f (x) > (1 + β)t, M(ρ) f (x) ≤ εt

≤ cα δ w

4Q j

Summing again overj, we obtain

w

x : T  ∗ f (x) > (1 + β)t, M(ρ) f (x) ≤ εt

≤ cα δ4n w( Ω). (2.48) Choosingα such that cα δ4n < (1 + β) −1, we can get (2.42)

Before proving the theorem, another equivalent norm for RBMO(μ) is needed

Sup-pose that for a given functionb ∈ L1loc(μ)... that a weightw satisfies the A p(μ)

condi-tion if there exists a constantK such that for all cubes Q

1

μ(Q)