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Estimates for Marcinkiewicz commutators with Lipschitz functions under nondoubling measures
Journal of Inequalities and Applications volume 2013, Article number: 388 (2013)
Under the assumption that μ is a nondoubling measure on satisfying the growth condition, the author proves that the commutator generated by the Marcinkiewicz integral operator and the Lipschitz function is bounded from the Hardy space into for with the kernel satisfying a certain Hörmander-type condition. Moreover, the author shows that for , is bounded from the Morrey space into , from into and from into , respectively.
MSC:42B25, 47B47, 42B20, 47A30.
In recent years, harmonic analysis on spaces with nondoubling measures has become a very active research topic. There has been significant progress in the study of boundedness for singular integrals on these spaces; see [1–8]. Among a long list of research papers, some of them [9–11] are on the Marcinkiewicz integral operators. The motivation for developing the analysis with nondoubling measures and some important examples of nondoubling measures can be found in .
We recall that a nonnegative Radon measure μ on is said to be a nondoubling measure if there is a positive constant such that for all and all it satisfies:
where n is a positive constant and , is the open ball centered at x and having radius r.
Let be a locally integrable function on . Assume that there exists a constant such that for any with ,
and for any ,
The Marcinkiewicz integral ℳ associated to the kernel and the measure μ as in (1.1) is defined by
Let , the Marcinkiewicz commutator is formally defined by
If μ is the d-dimensional Lebesgue measure in , and
with Ω homogeneous of degree zero and for some , then it is easy to verify that satisfies (1.2) and (1.3), and ℳ in (1.4) is just the higher dimensional Marcinkiewicz integral defined by Stein in , which is important in classical harmonic analysis and is a focus of active research; see [14–20]. Particularly, we should mention the work of Torchinsky and Wang , where they established the boundedness for the commutator generated by the Marcinkiewicz integral and function with . However, it is also worth to study the different behavior of another type commutator generated by the Marcinkiewicz integral and function, which was recently studied by Mo and Lu in  when Ω is homogeneous of degree zero and satisfies the cancellation condition. They obtained its boundedness from into for and .
When μ satisfies growth condition (1.1), ℳ as in (1.4) was first introduced by Hu et al. in , where the boundedness of such an operator in with and the Hardy space were established under the assumption that ℳ is bounded on with the kernel satisfying (1.2) and (1.3). Moreover, they got the same estimates for the commutator defined as (1.5) with when the kernel satisfies (1.2) and (1.6), which is slightly stronger than (1.3) and is defined as follows:
However, in our problem, we discover that the kernels should satisfy some other kind of smoothness to replace condition (1.6).
Definition 1.1 Let , . We say that the kernel K satisfies a Hörmander-type condition if there exist and such that for any and ,
Directly, one can see that condition (1.7) can be rewritten as
We note that this kind of smoothness was not new. Condition (1.7′) is similar to the Hörmander-type condition which allows that the integral operator can be controlled by a maximal operator in doubling measure spaces, and also useful in the research of Schrödinger operators; see [23–25] for details. We denote by the class of kernels satisfying this condition. It is clear that these classes are nested,
We should point out that is not condition (1.6).
In , by supposing that the kernel K satisfies (1.2) and (1.3), the authors studied the commutator in the case of and established that it is bounded from into for and . Furthermore, when condition (1.3) is replaced by (1.7), is bounded from into for some and , from into for some and , respectively.
The purpose of this paper is to get some estimates for the commutator with the kernel K satisfying (1.2) and (1.7) on the Hardy-type space and spaces. To be precise, we establish the boundedness of in for in Section 2. In Section 3, we prove that is bounded from to the Morrey space , from to for .
Before stating our result, we need to recall some necessary notation and definitions. For a cube , we mean a closed cube whose sides are parallel to the coordinate axes. We denote its center and its side length by and , respectively. Let , αQ denote the cube with the same center as Q and . Given two cubes in , set
where is the smallest positive integer k such that . The concept was introduced in , where some useful properties of can be found.
The following characterization of the Lipschitz space for in  plays a key role in the proof of theorems.
Lemma 1.1 For a function , conditions I, II and III below are equivalent.
There is a constant such that
for μ-almost every x and y in the support of μ.
There exist some constant and a collection of numbers such that these two properties hold: for any cube Q,(1.8)
and for any cube R such that and ,
For any given p, , there is a constant such that for every cube Q, we have(1.10)
where, and in the sequel,
and also for any cube R such that and ,
In addition, the quantities , and with a fixed p are equivalent and denoted by .
Remark 1.2 For , (1.9) is equivalent to
We also need the following lemma for the -boundedness of , which was proved in .
Lemma 1.2 Let , . Suppose that satisfies (1.2) and (1.3) and that is as in (1.5). If ℳ is bounded on , then there exists a positive constant such that for all bounded functions f with compact support,
where and .
Throughout this paper, we use the constant C with subscripts to indicate its dependence on the parameters. We denote simply by if there exists a constant such that ; and means that and . For a μ-measurable set E, denotes its characteristic function. For any , we denote by its conjugate index, namely, .
2 Boundedness of in Hardy spaces
This section is devoted to the behavior of the commutator in Hardy spaces. In order to define the Hardy space , Tolsa introduced the ‘grand’ maximal operator in .
Definition 2.1 Given , we define
where the notation means that and satisfies
for all ,
for all .
Definition 2.2 The Hardy space is the set of all functions satisfying that and . Moreover, we define the norm of by
We recall the atomic Hardy space as follows.
Definition 2.3 Let . A function is called an atomic block if
there exists some cube R such that ,
for , there are functions supported on cubes and numbers such that , and
Then we define
Define and as follows:
where the infimum is taken over all possible decompositions of f in atomic blocks, is the set of all finite linear combinations of -atoms.
Remark 2.1 It was proved in  that for each , the atomic Hardy space is independent of the choice of ρ.
To establish the boundedness of operators in Hardy-type spaces on , one usually appeals to the atomic decomposition characterization (see [28, 29]) of these spaces, which means that a function or distribution in Hardy-type spaces can be represented as a linear combination of atoms. Then the boundedness of linear operators in Hardy-type spaces can be deduced from their behavior on atoms in principle. However, Meyer  (see also ) gave an example of whose norm cannot be achieved by its finite atomic decompositions via -atoms. Based on this fact, Bownik  (Theorem 2) constructed a surprising example of a linear functional defined on a dense subspace of , which maps all -atoms into bounded scalars, but yet cannot extend to a bounded linear functional on the whole .
Recently, in , a boundedness criterion was established via Lusin function characterizations of Hardy spaces on as follows: a sublinear operator T extends to a bounded sublinear operator from Hardy spaces with to some quasi-Banach space B if and only if T maps all -atoms into uniformly bounded elements of B for some . Here and in what follows means the integer part of real t. This result shows the structural difference between atomic characterization of via -atoms and -atoms. On the other hand, Meda et al.  independently obtained some similar results by grand maximal function characterizations of Hardy spaces on . In fact, let , and integer , and let be the set of all finite linear combinations of -atoms. Denote by the set of all continuous functions. For any , when or when , Meda et al. in  proved that can be achieved by a finite atomic decomposition via -atom when or continuous -atom when ; from this, they further deduced that if T is a linear operator and maps all -atoms with or all continuous -atoms with into uniformly bounded elements of some Banach space B, then T uniquely extends to a bounded linear operator from to B which coincides with T on these -atoms.
According to the theory of Meda et al. , we get the result as follows.
Theorem 2.1 Let , and . Suppose that K satisfies (1.2) and condition. If , then is bounded from the Hardy space into the Lebesgue space, namely, there exists a positive constant C such that
Proof of Theorem 2.1 Via Remark 2.1, without loss of generality, we may assume that and as a finite sum of atomic blocks defined in Definition 2.3. It is easy to see that we only need to prove the theorem for one atomic block h. Let R be a cube such that , , and
where for , is a real number, , for , is a bounded function supported on some cube and it satisfies
By (2.1), we have
To estimate , we write
Choose and such that , and . By the Hölder inequality, the fact that and the -boundedness of (Lemma 1.2), we have that
Denote simply by . Invoking the fact that , we thus get
here we have used the fact that
The estimates for and give the desired estimate for . A similar argument tells us that
Combining the estimates for and yields the desired estimate for I.
For , , , we have . By the Minkowski inequality, we get
For any , we have . It follows that
For , by the Minkowski inequality, we have
here we used the fact that and .
We now turn to estimate . Note that for any , , we have , so by the Minkowski inequality,
Combining the estimates for I, II and III yields that
and this is the result of Theorem 2.1. □
3 Boundedness of in space
In this section, we investigate the boundedness for the commutator as in (1.5) in the space for and , respectively.
Definition 3.1 Let and . We define the Morrey space as
where the norm is given by
We should note that the parameter appearing in the definition does not affect the definition of the space , and the space is a Banach space with its norm; see . By using the Hölder inequality to (1.5), it is easy to see that for all , we have
Theorem 3.1 Let , , . Suppose that K satisfies (1.2) and condition, ℳ is bounded on and is defined as in (1.5). Then there exists a positive constant C such that for all ,
Theorem 3.2 Let , and . Suppose that K satisfies (1.2) and condition. If ℳ is bounded on and is defined as in (1.5), then there is a constant such that for all bounded functions f with compact support,
Theorem 3.3 Let , , and . Suppose that K satisfies (1.2) and condition, ℳ is bounded on and is defined as in (1.5). Then there exists a positive constant C such that for all ,
Remark 3.1 By the Minkowski inequality and the kernel condition, we get that
where is the fractional integral operator. Then .
Remark 3.2 Theorem 3.2 can be deduced as a conclusion of Theorem 3.1 in the case of .
Remark 3.3 Applying Lemma 1.1, a slight change in the proof of Theorem 3.1 actually shows Theorem 3.3 and we leave the details to the reader.
Proof of Theorem 3.1 For any cubes Q and R in such that satisfies , let
It is easy to see that and are real numbers. By Lemma 1.1, we need to show that for some fixed there exists a constant such that
Let us first prove estimate (3.1). For a fixed cube Q and , decompose , where and . Write that
For and , it follows that
In order to estimate the term , set
It is easy to get that for any ,
For , since , , we thus get
here we used the Minkowski inequality, , (1.10) of Lemma 1.1 and the fact that
By a similar argument, it follows that
Finally, by the condition , which the kernel K satisfies, and the fact that , applying the Minkowski inequality, we have
Combining these estimates, we conclude that
and so estimate (3.1) is proved. □
We proceed to show (3.2). For any cubes with , where Q is arbitrary and R is a doubling cube with , denote simply by N. Write
As in the estimate for the term , we have
We conclude from , that
Taking mean over , we obtain
Analysis similar to that in the estimate for shows that
Finally, we get (3.2) and this is precisely the assertion of Theorem 3.1.
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The author would like to thank the referee for his very careful reading and valuable remarks which made this article more readable. This work was supported by NSFC of China (No. 11261055), NSFC of Xinjiang (No. 2012211B28), Graduate Innovative Research Program Foundation of Jiangsu Province (2012) and Science Research Foundation of Yili Normal University (No. 2011YNZD009).
The author declares that they have no competing interests.
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Li, L. Estimates for Marcinkiewicz commutators with Lipschitz functions under nondoubling measures. J Inequal Appl 2013, 388 (2013). https://doi.org/10.1186/1029-242X-2013-388
- nondoubling measure
- Marcinkiewicz integral