- Open Access
Estimates for fractional type Marcinkiewicz integrals with non-doubling measures
© Lu and Zhou; licensee Springer 2014
Received: 26 November 2013
Accepted: 11 July 2014
Published: 18 August 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.
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 .
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 .
However, in this paper, we discover that the kernel should satisfy some other kind of smoothness condition to replace (1.6).
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 .
where is the smallest positive integer k such that . The concept was introduced in , where some useful properties of can be found.
- (i)There exist some constant and a collection of numbers such that these two properties hold: for any cube Q,(1.8)
- (ii)For any given p, , there is a constant such that for every cube Q, then(1.10)
for any two cubes with (see Remark 2.7 in ).
is bounded from to (see ).
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 .
for all ,
for all .
We recall the atomic Hardy space as follows.
there exists some cube R such that ,
- (3)for , there are functions supported on cubes and numbers such that , and
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.
see  for details.
Combining the estimates for and yields the estimate for I.
Here we have used the fact 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 .
Remark 3.4 As a special condition, we take , Theorem 3.3 can be deduced with a similar method of Theorem 3.2.
and so estimate (3.1) is proved.
Finally, we get (3.2) and this is precisely the assertion of Theorem 3.2. □
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).
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