Chol RI Zhenqiu ZHANG

Abstract Let(X,d,µ)be a metric measure space satisfying both the upper doubling and the geometrically doubling conditions in the sense of Hytönen.In this paper,the authors obtain the boundedness of the commutators of θ-type Calderón-Zygmund operators with RBMO functions from L∞(µ)into RBMO(µ)and from(µ)into(µ),respectively.As a consequence of these results,they establish the Lp(µ)boundedness of the commutators on the non-homogeneous metric spaces.

Keywords Non-homogeneous space,θ-Type Calderón-Zygmund operator,Commutator,RBMO(µ)space,(µ)space

1 Introduction and Preliminaries

The theory of Calderón-Zygmund operators and commutators plays an important role in harmonic analysis and partial differential equations.The theory of commutators were introduced in a general form by Calderón [2–3],in which Calderón showed that these kinds of operators are bounded on L2.In [4],the authors proved that given a singular integral T with standard Calderón-Zygmund kernel,the operator[b,T]=bT −T b is bounded in Lp,1

Recently,many mathematicians pay attention to the study of non-doubling measure spaces.One of the most general settings to which Calderón-Zygmund theory extends naturally is the spaces of homogeneous type in the sense of Coifman and Weiss [5],Many results from real and harmonic analysis on Euclidean spaces have their natural extensions on these space(see,for example,[5–6,8]).A metric space (X,d)equipped with a nonnegative Borel measureµ is called a space of homogeneous type if (X,d,µ)satisfies the following measure doubling condition that there exists a positive constant Cµ,depending on µ,such that for any ball B(x,r)={y ∈X :d(x,y)

This definition was introduced by Coifman and Weiss in [5].We point out that d may also be assumed to be a quasi-metric.However,for the simplicity,in this paper,we always assume that d is a metric; see also [9].The measure doubling condition (1.1)plays a key role in the classical theory of Calderón-Zygmund operators.However,many results on the classical Calderón-Zygmund theory have been proved still valid if the measure doubling condition is replaced by a weaker condition such as the polynomial growth condition; see,for example,[7,15,18].To be precise,let k ∈(0,∞),X be a metric space endowed with a metric d and a nonnegative“k-dimensional”Borel measureµ in the sense that there exists a positive constant C0such that for all x ∈X and r ∈(0,∞),

Such a measure need not satisfy the doubling condition (1.1).In [15],Tolsa established Calderón-Zygmund theory for non doubling measures.Because the measures satisfying (1.2)are only different form,not more general than,the measures satisfying (1.1),the theory with this kinds of non-doubling measures is not in all respects a generalization of the corresponding theory of spaces of homogeneous type.Hytönen in[9]introduced a new class of metric measure spaces satisfying the so-called upper doubling and the geometrically doubling conditions (see Definitions 1.1–1.2 below).This new class of metric measure spaces includes both spaces of homogeneous type and metric spaces with the measures satisfying(1.2)as special cases.Recently,many results on the Calderón-Zygmund theory have been built on the non-homogeneous metric measure spaces (see [1,10–11,13,18]).Let us mention that Bui and Duong [1]showed that if the Calderón-Zygmund operator is bounded on L2(µ),the commutator of this operator with a function RBMO(µ)is bounded on Lp(µ),1 < p < ∞on non-homogeneous metric measure space.

θ-type Calderón-Zygmund operator was introduced by Yabuta in [17],In [16],the authors studied the properties of θ-type Calderón-Zygmund operator and commutator.In this paper,we study the commutators of θ-type Calderón-Zygmund operator with RBMO function on nonhomogeneous metric measure space.We show that this commutators is bounded from L∞(µ)into RBMO(µ)and from(µ)into L1(µ),respectively.To state our main result,we recall some necessary notations which will be used in the proof of our main results.We start with the notion of the upper doubling and geometrically doubling metric measure space which introduced in [1,9].

Definition 1.1 Ameasure µ in the metric space (X,µ)is said to be an upper doubling measure if there exists a dominating function λ with the following properties:

(i)λ:X ×(0,∞)→(0,∞).

(ii)For any fixed x ∈X,r →λ(x,r)is increasing.

(iii)There exists a constant Cλ>0 such that λ(x,2r)≤Cλλ(x,r)for all x ∈X,r >0.

(iv)The inequality µ(x,r):=µ(B(x,r))≤λ(x,r)holds for all x ∈X,r >0.

(v)And λ(x,r)≈λ(y,r)for all r >0,x,y ∈X and d(x,y)≤r.

Obviously,a space of homogeneous type is a special case of upper doubling spaces,if we take the dominating function λ(x,r)=µ(B(x,r)).On the other hand,a metric space (X,d,µ)satisfying the polynomial growth condition (1.2)is also an upper doubling measure space by taking λ(x,r)=C0rk.We now recall the notion of the geometrically doubling space introduced in [9].

Definition 1.2 Ametric space (X,d)is called geometrically doubling if there exists some N0∈N={1,2,···} such that for any ball B(x,r)⊂X ,there exists a finite ball coveringof B(x,r)such that the cardinality of this covering is at most N0.

Let (X,d)be a metric space.In [9],Hytönen showed that the following statements are mutually equivalent:

(i)(X,d)is geometrically doubling.

(ii)For any ε ∈(0,1)and any ball B(x,r)⊂X,there exists a finite ball covering{B(xi,εr)}iof B(x,r)such that the cardinality of this covering is at most N0ε−n,where and in what follows,N0is as in Definition 1.2 and n=log2N0.

(iii)For every ε ∈(0,1),any ball B(x,r)⊂X contains at most N0ε−ncenters {xi}iof disjoint balls {B(xi,εr)}i.

(iv)There exists M ∈N such that any ball B(x,r)⊂X contains at most M centers {xi}iof disjoint balls

We now recall the coefficients KB,Qand K′B,Qfor any two balls B ⊂Q which were introduced in [1].For any two balls B ⊂Q,let

where NB,Qis the smallest integer satisfying 6NB,QrB≥rQ.If λ(x,ar)=amλ(x,r)for all x ∈X and a,r>0,it is not difficult to show that KB,Q≈K′B,Q.However,in general,we only have KB,Q≤CK′B,Q.This definition is a variant of the definition in [15].Similar to Lemma 2.1 in [15],the authors in [1]showed the following property.

Lemma 1.1(i)If Q ⊂R ⊂S are balls in X,then

(ii)If Q ⊂R are of compatible size,then KQ,R≤C.

(iii)If αQ,··· ,αN−1Q are non (α,β)-doubling ballsthen KQ,αNQ≤C.

Throughout this paper,C always means a positive constant independent of the main parameters involved,but it may be different in different contents.

2 Main Results and Proofs

At first,based on the definition of θ-type Calderón-Zygmund operator in [17],we define the θ-type Calderón-Zygmund operator on non-homogeneous metric measure spaces as follows.

Definition 2.1Let θ be a non-negative,non-decreasing function on R+=(0,∞)satisfying

A kernel K(·,·)∈L1loc(X ×X{(x,y): x=y})is called a θ-type Calderón-Zygmund kernel if the following conditions hold

and

when d(x,y)≥2d(x,x′).

A linear operator T is called the θ-type Calderón-Zygmund operator with kernel K(·,·)satisfying (2.2)and (2.3)if for all f ∈L∞(µ)with bounded support and x /∈suppf,

The RBMO(µ)space for the general non-homogeneous space(X,µ)was introduced by Hytönen in [9],and studied systematically in [1].Both of the authors also gave some characterizations of RBMO(µ)space.

Definition 2.2Given a ball B ⊂X,let N be the smallest non-negative integer such that=6NB is doubling(such a ball B exists due to Lemma 1.1).Let ρ>1 be some fixed constant.We say that f ∈(µ)is in the RBMO(µ)if there exists some constant C >0 such that for any ball B,

and for any two doubling balls Q and R such that Q ⊂R,

where mBf is the mean value of f over the ball B.Then we set

We need the following equivalent property of Definition 2.2 (see[1]).

Lemma 2.1For f ∈(µ),the following three are equivalent:

(a)f ∈RBMO(µ).

(b)There exists some constant Cbsuch that for any ball B,

and for any two balls Q and R such that Q ⊂R,

(c)There exists some constant Ccsuch that for any doubling ball B,

and

for any two doubling balls Q ⊂R.

Definition 2.3Let the kernel K satisfy Definition 2.1.The commutator of the θ-type Calderón-Zygmund operator T with RBMO function is defined by

The main result of our paper is given as follows.

Theorem 2.1Let T be θ-type Calderón-Zygmund operator defined by (2.4)as above and T is bounded on L2(µ).If b ∈RBMO(µ),then the commutator [b,T]is bounded from L∞(µ)into RBMO(µ)on non-homogeneous space.

ProofBy characterization of RBMO(µ)given by(2.9)–(2.10)in Lemma 2.1,We only need to prove the following two estimations:

(i)There exists some constant C such that for any doubling ball B,

(ii)For any two doubling balls Q and R with Q ⊂R,one has

We first check (2.11).Let {bB} be a family of numbers,satisfying

for balls B,and

for balls Q ⊂R.Denote hB=mB(T((b −bB)fχX6B)),f1=fχ6Band f2=fχX6B.

For any doubling ball B,we can write

By Hölder’s inequality,L2(µ)-boundedness of T and corollary 6.3 in [9],it follows

Using the doubling property of ball B and the coefficient KB,6B≤C,we have

In order to get the estimation of I3,we need to estimate |T((b −bB)f2)−hB|.

For x,y ∈B,by the definition (2.1),we get

Here we have used the following inequality that

From the above estimate and the choice of hB,we obtain and

Therefore we have (2.11).

Next we prove (2.12).For any x ∈Q,y ∈R,we write

With the argument similar to the estimate for I1it follows that

and

Therefore

We now estimate J3,Let N be the first integer k such that R ⊂6kQ.We denote QR=6N+1Q.Then we write

Similar to (2.13),we can show that

Here we have used the following inequality

and the fact rQR≈rR.

For J32,we have

The estimation of J33is similar.Hence J3≤C||f||L∞(µ)||b||∗KQ,R.We proved (2.12)and complete the proof of Theorem 2.1.

Now we are going to show that if a θ-type Calderón-Zygmund operator is bounded on L2(µ),then the commutators of θ-type Calderón-Zygmund operator with RBMO function is bounded from(µ)into L1(µ).Before stating our results,we first recall some definitions.

Definition 2.4Let ρ>1.A function a ∈(µ)is called a atomic block if

(1)t here exists some balls B such that supp(a)⊂B;

(3)there are functions ajsupported on balls Bj⊂B and numbers λj∈R such that

where the sum converges in L1(µ),andand the constant KBj,Bbeing given in the paragraph before Lemma 1.1.

where the infimum is taken over all the possible decompositions of f in atomic blocks.It was proved in [1]that the atomic Hardy space(µ)is independent of the choice of ρ.Here we choose ρ=6.

Theorem 2.2Assume that θ-type Calderón-Zygmund operator defined by (2.4)as above and T is bounded on L2(µ),then the commutators of T with RBMO function is bounded from(µ)into L1(µ)on non-homogeneous space.

ProofIt is enough to show that

for any atomic block a with supp(a)⊂B,a=where the ajis function satisfying property of the definition of atomic block.

For a atomic block a(x),we can write

First we will estimate I1.

In fact

Since

we write

By using Hölder’s inequality and the boundedness of T on L2(µ),we obtain

Similarly

Thus

For I11,we write

Let xjbe the center of ball Bj.According to the definition of atomic block,we obtain

Since (2.6)–(2.7)and Lemma 1.1,

Thus

and

Hence

Now,for I2,we can write

Thus

Let xBbe the center of ball B and r be the radius of ball B.Then

Similarly estimation of I111,we can show that

For I22,we get

Remark 2.1In [1],Bui and Duong showed that the space RBMO(µ)is embedded in the dual space ofThat is RBMO(µ)⊂()∗.So we can not obtain the above result by duality.

In [1],Bui and Duong established the following interpolation theorem.

Theorem 2.3Let T be a linear operator which is bounded from(µ)into L1(µ)and from L∞(µ)into RBMO(µ).Then T extends boundedly to Lp(µ)for all 1

By using Theorem 2.3,as a consequence of Theorem 2.1 and Theorem 2.2,we immediately obtain the following theorem.

Theorem 2.4Let T be θ-type Calderón-Zygmund operator defined by (2.4)as above and b ∈RBMO(µ).Then the commutator [b,T]can be extended to a bounded operator on Lp(µ)for all (1

Remark 2.2The classical theorem of the boundedness of the commutator on Lp(µ)space was obtained by using the pointwise estimate for commutator,see the papers [4,14].However,our method is different.We derive the Lp(µ)boundedness from the endpoint estimates on the non-homogeneous metric spaces.