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Estimates for fractional type Marcinkiewicz integrals with non-doubling measures
Journal of Inequalities and Applications volume 2014, Article number: 285 (2014)
Under the assumption that μ is a non-doubling measure on satisfying the growth condition, the authors prove that the fractional type Marcinkiewicz integral ℳ is bounded from the Hardy space to the Lebesgue space for with kernel satisfying a certain Hörmander-type condition. In addition, the authors show that for , ℳ is bounded from the Morrey space to the space and from the Lebesgue space to the space .
MSC:46A20, 42B25, 42B35.
Let μ be a nonnegative Radon measure on which satisfies the following growth condition: for all and all ,
where and n are positive constants and , is the open ball centered at x and having radius r. So μ is claimed to be non-doubling measure. If there exists a positive constant C such that for any and , , the μ is said to be doubling measure. It is well known that the doubling condition on underlying measures is a key assumption in the classical theory of harmonic analysis. Especially, in recent years, many classical results concerning the theory of Calderón-Zygmund operators and function spaces have been proved still valid if the underlying measure is a nonnegative Radon measure on which only satisfies (1.1) (see [1–8]). The motivation for developing the analysis with non-doubling measures and some examples of non-doubling measures can be found in . We only point out that the analysis with non-doubling measures played a striking role in solving the long-standing open Painlevé’s problem by Tolsa in .
Let be a μ-locally integrable function on . Assume that there exists a positive constant C such that for any with ,
and for any ,
The fractional type Marcinkiewicz integral ℳ associated to the above kernel and the measure μ as in (1.1) is defined by
If μ is the d-dimensional Lebesgue measure in , and
with Ω homogeneous of degree zero and for some , then K satisfies (1.2) and (1.3). Under these conditions, ℳ in (1.4) is introduced by Si et al. in . As a special case, by letting , we recapture the classical Marcinkiewicz integral operators that Stein introduced in 1958 (see ). Since then, many works have appeared about Marcinkiewicz type integral operators. A nice survey has been given by Lu in .
In 2007, the Hörmander-type condition was introduced by Hu et al. in , which was slightly stronger than (1.3) and was defined as follows:
However, in this paper, we discover that the kernel should satisfy some other kind of smoothness condition to replace (1.6).
Definition 1.1 Let , . The kernel K is said to satisfy a Hörmander-type condition if there exist and such that for any and ,
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).
The purpose of this paper is to get some estimates for the fractional type Marcinkiewicz integral ℳ with kernel K satisfying (1.2) and (1.7) on the Hardy-type space and the space. To be precise, we establish the boundedness of ℳ in for in Section 2. In Section 3, we prove that ℳ is bounded from the space to the Morrey space , from the space to the Lebesgue space for .
Before stating our results, 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 , ηQ 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.
Lemma 1.2 For a function , , conditions (i) and (ii) below are equivalent.
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, then(1.10)
and also for any cube R such that and ,
Remark 1.4 For , (1.9) is equivalent to
for any two cubes with (see Remark 2.7 in ).
Lemma 1.5 Let , , and . Then the fractional integral operator defined by
is bounded from to (see ).
Lemma 1.6 Let , , . Suppose that satisfies (1.2) and (1.3) and ℳ is as in (1.4). Then there exists a positive constant such that for all bounded functions f with compact support,
Proof of Lemma 1.6 By Minkowski’s inequality, we have
By Lemma 1.5 then
Throughout this paper, we use the constant C with subscripts to indicate its dependence on the parameters. For a μ-measurable set E, denotes its characteristic function. For any , we denote by its conjugate index, namely .
2 Boundedness of ℳ in Hardy spaces
This section is devoted to the behavior of ℳ in Hardy spaces. In order to define the Hardy space , Tolsa introduced the grand maximal operator in .
Definition 2.1 Given , is defined as
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, the norm of is defined 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
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 2.4 It was proved in  that for each , the atomic Hardy space is independent of the choice of ρ.
Applying the theory of Meda et al. in , we easily get the result as follows.
Theorem 2.5 Let , . Suppose that K satisfies (1.2) and the condition and . Then ℳ is bounded from the Hardy space into the Lebesgue space, namely there exists a positive constant C such that
Proof of Theorem 2.5 Without loss of generality, we may assume that and as a finite 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 cubes 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 ℳ (see Lemma 1.6), we have that
Denote simply by . Invoking the fact that , we thus get
Here we have used the fact that
see  for details.
The estimates for and give the desired estimate for . With a similar argument, we have
Combining the estimates for and yields the estimate for I.
For , , , we have , by Minkowski’s inequality, we get
For any , we have . It follows that
Here we have used the fact that .
Combining the estimates for I, II and III yields that
and this is the result of Theorem 2.5. □
3 Boundedness of ℳ in spaces
In this section, we discuss the boundedness for ℳ as in (1.4) in the space for and , respectively.
Firstly, we need to recall the definition of Morrey space with non-doubling measure denoted by , which was introduced by Sawano and Tanaka in .
Definition 3.1 Let and . The Morrey space is defined by
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 is a Banach space with its norms (see ). By using the Hölder inequality to (1.4), it is easy to see that for all , then
Theorem 3.2 Let , . Suppose that satisfies (1.2) and the condition, ℳ is defined as in (1.4). Then there exists a positive constant C such that for all ,
Theorem 3.3 Let and . Suppose that satisfies (1.2) and the condition, ℳ is defined as in (1.4). Then there exists a positive constant C such that for all bounded functions f with compact support,
Remark 3.4 As a special condition, we take , Theorem 3.3 can be deduced with a similar method of Theorem 3.2.
Proof of Theorem 3.2 For any cubes Q and R in such that satisfies , let
It is easy to see that and are real numbers. By Lemma 1.2, 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
Now let us estimate the term ,
In order to estimate , we write
It is easy to get that for any ,
For , since , , thus we get
By a similar argument, it follows that
Finally, by the condition , which the kernel conditions, applying Minkowski’s inequality, and the fact that , 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 , denote simply by N. Write
As in the estimate for the term , then
We conclude from , that
Taking mean over , we obtain
Analysis similar to that in the estimates for shows that
Finally, we get (3.2) and this is precisely the assertion of Theorem 3.2. □
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Jiang Zhou is supported by the National Science Foundation of China (Grant No. 11261055) and the National Natural Science Foundation of Xinjiang (Grant Nos. 2011211A005, BS120104).
The authors declare that they do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Lu, G., Zhou, J. Estimates for fractional type Marcinkiewicz integrals with non-doubling measures. J Inequal Appl 2014, 285 (2014). https://doi.org/10.1186/1029-242X-2014-285
- non-doubling measure
- fractional type Marcinkiewicz integral
- Hardy space