Xiangxing TAO(陶祥興)Yuan ZENG (曾媛)*Department of Mathematics， Zhejiang University of Science and Technolog，Hangzhou 310023,China.E-mail : xatau@163.com; zy 1347256531@163.com

Xiao YU (喻曉)Department of Mathematics，Shangrao Normal University，Shangrao 334001，China E-mail : ya2000s@163.com

Definition 1.1Let X be a non-empty set.A function d : X ×X →[0,∞) is called a quasi-distance on X if

(i) d(x,y)=0 if and only if x=y;

(ii) d(x,y)=d(y,x);

(iii) there exists a finite constant cd≥1 such that, for every x,y,z ∈X,

As usual,we denote by(X,d)the set X endowed with a quasi-distance d.Letting B(x,r)={y ∈X :d(x,y)0)form a base for a complete system of neighborhoods of X, and so X is a Hausd?rffspace.Note that the balls B(x,r) are not in general open sets; if they are, then they form a base for the topology of X (see [10], among others, for details).

Definition 1.2We say that (X,d,μ) is a homogeneous space if

(i) X is a set endowed with a quasi-distance d such that the balls are open sets in the topology induced by d;

(ii) μ is a positive Borel measure on X, satisfying the doubling condition

with the constant C0>1 independent of x ∈X and r >0;

(iii) if B(x,r1)∩B(y,r2) /= ?and r1≥r2, then there exists a constant C1> 1 such that B(y,r2)?B(x,C1r1).

To facilitate the study of certain classes of pseudo-differential operators, Yabuta [3] introduced the ω-type Calder′on-Zygmund operators on Euclidean space.Here we define the ω-type Calder′on-Zygmund operators on homogeneous space as follows:

Definition 1.3Let ω be a non-negative, non-decreasing function on (0,+∞) satisfying

A kernel K(·,·) ∈L1loc(X ×X{(x,y):x=y}) is called an ω-type Calder′on-Zygmund kernel if when d(x,y)>2d(x,x′), it holds that

A linear operator Tωis called the ω-type Calder′on-Zygmund operator with kernel K(·,·)satisfying the above conditions if Tωis bounded on L2(X) and, for all f with bounded support and x /∈suppf,

We note that when ω(t) = tδwith 0 < δ ≤1, the ω-type operator is just the classical Calder′on-Zygmund operator T with a standard kernel (see [5, 11]).Given a function b ∈BMO(X)or b ∈VMO(X) and the ω-type Calder′on-Zygmund operator Tω, the linear commutator[b,Tω]generated by b and Tωis defined initially for a smooth and compactly supported function f as

In 1976, Coifman-Rochberg-Weiss[12] obtained that if T is a Calder′on-Zygmund operator on Lp(Rn) and b ∈BMO(Rn), then commutator [b,T] is a linear bounded operator on Lp(Rn)for p ∈(1,∞).On the other hand, Uchiyama[13]proved a compactness characterization of the commutator[b,T]such that b ∈VMO(Rn)if and only if[b,T]is a compact operator on Lp(Rn)(see also [14]).Krantz and Li [15] discussed the compactness of [b,T] on Lp(X), where X is a space of homogenous type.

Recently, Dao and Krantz[9]proved the Lorentz boundedness of commutators[b,T]on the space of homogeneous type, and showed the compactness of commutators [b,T], where T is a singular integral associated to the Szeg? kernel, on strictly pseudo-convex domains or a convex domain of finite type in Cn.It is natural to ask whether the commutator [b,Tω] is bounded and compact on the Lorentz spaces Lp,r(X).

In this paper, we will overcome the difficulties brought by Lorentz space Lp,r(X) and the ω-type Calder′on-Zygmund operators Tω,and establish the boundedness and compactness of the commutators [b,Tω] through decomposition technology and the study of a truncated operator of Tω.Our main results are as follows:

Theorem 1.4Let 1

Remark 1.5Here we remark that, under the condition (1.1), the ω-type Calder′on-Zygmund operator Tωis well defined, and we can show the Lp,r(X)-boundedness for Tω; see Theorem 3.1 below.However,the commutator[b,Tω]has a greater degree of singularity than the corresponding ω-type Calder′on-Zygmund operator Tω, so we need a slightly stronger condition(1.2) in our theorems.

In the case X = Rnwith the standard Euclidean metric and μ as the Lebesgue measure,we obtain a compactness characterization of [b,Tω] on Lorentz space Lp,r(Rn).

Theorem 1.6Let 1 < p < ∞, r ∈[1,∞].Assume that b ∈VMO(Rn), and that Tωis an ω-type Calder′on-Zygmund operator with ω satisfying (1.2).Then, the commutator [b,Tω]is compact on Lp,r(Rn).

The rest of this paper is organized as follows: in the next section,we provide the definitions and fundamental functional inequalities involving the Lorentz space, the maximal function,the sharp maximal function, BMO and VMO space, the truncated operator and the maximal truncated operator.Section 3 is devoted to the proof of Theorem 1.4.In Section 4, we provide the proof of Theorem 1.6.

2 Definitions and Preliminary Results

2.1 Lorentz Space

First, we recall the definition of the decreasing rearrangement of a function.

Definition 2.1Given a measurable function f defined on (X,μ), the decreasing rearrangement of f, denoted by f?, is the function defined on [0,∞) by

One can see Grafakos [16] for more basic properties of the decreasing rearrangement of functions.Now we give the definition of the Lorentz space.

Definition 2.2Given f,a measurable function on a measure space(X,μ)and 0

The set of all f satisfying ‖f‖Lp,r(X)< ∞is denoted by Lp,r(X), and is called the Lorentz space.

Remark 2.3For all 0 < p,r ≤∞, Lp,r(X) is a quasi-Banach space.These spaces are normable when p, r are greater than 1; see Theorem 1.4.11 in [16].

Moreover, if p=r, then we have that

Remark 2.4For all 0

In addition, we have H?lder’s inequality on the Lorentz space.

Proposition 2.5([9]) Let 0 < p1, p2< ∞, and r1,r2> 0.Suppose that f ∈Lp1,r1(X)and g ∈Lp2,r2(X).Then it holds that

2.2 Hardy-Littlewood Maximal Function and Sharp Maximal Function

ClearlyMf(x) ≤Mf(x).Moreover, applying the covering lemma, we can know that there exists a constant C =C(X)>0 such that

As a result, the boundedness properties ofMare identical to those of M.Therefore, in this paper we useMf whenever we refer to the maximal function.

Next, let us recall the definition of the sharp maximal function, introduced by Fefferman and Stein,

where the supremum is taken over all balls B ?X containing x.

For the Hardy-Littlewood maximal function and the sharp maximal function, we will use the following results of [9]:

Lemma 2.6Let p>1, and 1 ≤r ≤∞.ThenMis bounded on Lp,r(X).

Lemma 2.7For any p > 1 with r ∈[1,∞] there is a positive constant C = C(X,p,r)such that, for every f ∈Lp,r(X),

(i) if μ(X)=∞, then ‖f‖Lp,r(X)≤C‖f?‖Lp,r(X);

(ii) if μ(X)<∞, then ‖f -fX‖Lp,r(X)≤C‖f?‖Lp,r(X).

2.3 BMO and VMO Spaces

Remark 2.8In the case X = Rn, we have that VMO(Rn) = CMO(Rn), where the CMO(Rn) space denotes the closure of C∞c(Rn) in the BMO(Rn) topology.

Next, let us recall a well-known result obtained by John-Nirenberg[17], which will be used in the proofs of our results.

Lemma 2.9For every p ∈[1,∞), there exists a constant C = C(X,p) such that, for every f ∈BMO(X) and every ball B,

Moreover, the following lemma about BMO functions holds:

Lemma 2.10([7]) Let b ∈BMO and let M > 1.Then, for every ball Br(x) = B(x,r)and every positive integer j, we have that

2.4 Truncated Operator and Maximal Truncated Operator

In this subsection, we consider the case X = Rn.Letting Tωbe the ω-type Calder′on-Zygmund operator, we define the truncated operator Tεωas

To prove this we need the following result, which is due to Kolmogorov:

Lemma 2.12Given a weak(1,1)operator S, 0<δ <1, and a finite measure set E, then there exists a constant C depending only on δ such that

and this completes the proof.□

Using the boundedness ofMand Tωon Lp,r(Rn), p > 1, r ∈[1,∞] (see Lemma 2.6 and Theorem 3.1 in this paper),we know that inequality(2.1)with δ =1 implies that T?ωis bounded on Lp,r(Rn).

3 Proof of Theorem 1.4

Theorem 3.1Let p>1, r ∈[1,∞].If Tωis an ω-type Calder′on-Zygmund operator with ω satisfying (1.1), then Tωis bounded on Lp,r(X).Furthermore, we have that

ProofWe divide our proof into the following two cases.

Case 1r >1.We first consider the caseμ(X)=∞.Let x ∈X and let B =B(ν,δ)?X be a ball containing x.For any y ∈X, we write

On the other hand, for every y ∈B and z ∈Ej, it is obvious that d(x,z) > 2j+1cdδ for each j ≥1 and d(x,z)>2d(x,y).Then it follows from the smoothness condition of K that

It follows from the triangle inequality and inequality (3.2) that

It remains to prove that

Our desired conclusion then follows directly.□

Lemma 3.2For every q ∈(1,∞), Tωis an ω-type Calder′on-Zygmund operator with ω satisfying (1.2) and b ∈BMO(X).Then there exists a constant C =C(q,Tω,X) such that

Combining estimates (3.5)–(3.9) leads to our desired conclusion.□

It is easy to see that the condition(1.2) implies the condition(1.1).Thus, if ω satisfies the condition (1.2), we have that Theorem 3.1 holds.Now we give the proof of Theorem 1.4.

Proof of Theorem 1.4We divide our proof into two cases.

Case 1r >1.We first consider the caseμ(X)=∞.By applying Lemma 2.7 and Lemma 3.2, we can obtain that

This completes the proof of Theorem 1.4.□

4 Proof of Theorem 1.6

In order to prove this theorem, we use a compactness criterion for Lorentz spaces provided by Brudnyi [21].

Lemma 4.1Let 1

Then we say that G is relatively compact in Lp,r(Rn).

Since the proof of the rest lemma is the same as Lemma 7 in [22], which does not rely on the kernel smoothness condition, we omit the details.

Finally, we show the decay at infinity of the elements of G.The fact bε∈(Rn) implies that there exists a real number Rε> 0 such that supp bε?B(0,Rε).If we provide that R satisfies R>2Rεand is sufficiently large, then

Conflict of InterestThe authors declare no conflict of interest.