Journal of Function Spaces

Volume 2014 (2014), Article ID 765984, 8 pages

http://dx.doi.org/10.1155/2014/765984

Research Article

Marcinkiewicz Integrals on Weighted Weak Hardy Spaces

Yue Hu¹ and Yueshan Wang²

¹College of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo 454003, China

²Department of Mathematics, Jiaozuo University, Jiaozuo 454003, China

Received 13 January 2014; Accepted 26 February 2014; Published 3 April 2014

Academic Editor: Yongqiang Fu

Copyright © 2014 Yue Hu and Yueshan Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We prove that, under the condition , Marcinkiewicz integral is bounded from weighted weak Hardy space to weighted weak Lebesgue space for , where belongs to the Muckenhoupt weight class. We also give weaker smoothness condition assumed on Ω to imply the boundedness of from to .

1. Introduction and Results

Suppose that is the unit sphere in equipped with Lebesgue measure . Let be homogeneous of degree zero on satisfying and

where for any . Then, the Marcinkiewicz integral operator is defined by

where

The operator was originally introduced by Marcinkiewicz [1] in 1938 for and sign . In 1958, Stein [2] defined the Marcinkiewicz integral of higher dimensions and proved that if , then is of strong type for and of weak type . We say that if

In 1962, Benedek et al. [3] showed that if is continuously differentiable on , then is of strong type for all . In 1990, Torchinsky and Wang [4] considered the weighted case and proved that if , then, for all (Muckenhoupt weight class), is bounded on .

In 1999, Ding et al. [5] improved Torchinsky and Wang's result, by ridding of the smoothness condition, as follows.

Theorem A. Let , and satisfy (1). If , then there is independent of such that

where is the dual of such that .

In 2003, Ding et al. [6] discussed the boundedness of Marcinkiewicz integral on weighted Hardy space, getting the following results.

Theorem B. Let , and . Let satisfy (1). If , then there exists a constant independent of such that .

Theorem C. Let satisfy (1) and condition (. If , then there is a independent of such that .

In 2007, C. C. Lin and Y. C. Lin [7] showed that under weaker condition assumed on , which is called -Dini type condition of order , Marcinkiewicz integral operator is bounded from to .

We say that satisfies the -Dini condition if , is homogenous of degree zero on , and

where denotes the integral modulus of continuity of defined by

and is a rotation in with . If , we let .

In 2002, Ding et al. [8] considered the boundedness of Marcinkiewicz integral on weak Hardy space; they obtained the following result.

Theorem D. Suppose that satisfies (1) and

Then, there exists a constant such that

In this paper, we will investigate the boundedness of Marcinkiewicz integral on weighted weak Hardy space. We have the following conclusions.

Theorem 1. Suppose that  , satisfies (1) and . If and , then there exists a constant independent of such that .

Theorem 2. Let , satisfy (1) and

If , then there is a independent of such that .

Remark 3. From [8, Remark 2], Condition (10) is weaker than Condition (4).

2. Some Preliminaries and Notations

We begin this section with some properties of weights which play an important role in the proofs of our main results.

A weight is a nonnegative, locally integrable function on . Let denote the ball with the center and radius . For any ball B and denotes the ball concentric with whose radius is times as long. For a given weight function and a measurable set , we also denote the Lebesgue measure of by and set weighted measure .

Definition 4. A weight is said to belong to for , if there exists a constant such that for every ball, ,

The class is defined by replacing the above inequality with

The classical weight theory was first introduced by Muckenhoupt in the study of weighted boundedness of Hardy-Littlewood maximal functions in [9].

We know that if with , then for all and for some . Therefore, we will use the notation to denote the critical index of . Obviously, if , , then we have .

We state the following results about class that will be used later on.

Lemma 5 (see [10]). Let , . Then, for any ball and any , there exists an absolute constant such that

where depends neither on nor on .

Lemma 6 (see [10]). Let with . Then, for any , there exists an absolute independent of such that

We also need the following result about -Dini condition.

Lemma 7 (see [11]). Suppose that and satisfies -Dini condition. Then, for ,

Given a Muckenhoupt weight function on , for , we denote by the space of all functions satisfying

When , will be taken to mean , and we set . And also the weighted weak Lebesgue spaces is defined by

Let us now turn to the weighted weak Hardy spaces. The (unweighted) weak spaces have first appeared in the work of Fefferman et al. [12]. The atomic decomposition theory of weak space on was given by Fefferman and Soria in [13]. In 2000, Quek and Yang [14] introduced the weighted weak Hardy spaces and established their atomic decompositions. Moreover, by using the atomic decomposition theory of , Quek and Yang also obtained the boundedness of Calderón-Zygmund type operators on these weighted spaces.

We write to denote the Schwartz space of all rapidly decreasing smooth functions and to denote the space of all tempered distributions, that is, the topological dual of . Let , and . Define

where , and .

For the grand maximal function of is defined by

Then, we can define the weighted weak Hardy space by

Moreover, we set .

Lemma 8 (see [14]). Let . For every in , there exists a sequence of bounded measurable functions such that

(i) in the sense of distributions.(ii)Each can be further decomposed into , where satisfies the following:

(a)each is supported in a ball with and  . Here, denotes the characteristic function of the set and ;(b), where is independent of and ;(c) for every multi-index with .

Conversely, if has a decomposition satisfying (i) and (ii), then . Moreover, we have .

3. Proof of Theorem 1

It suffices to show that there exists a constant , such that, for any ,

For any given , we choose such that . Then, by Lemma 8, for every , we can write

where

and satisfies (a)–(c) in Lemma 8. Then,

We claim that the following inequality holds:

In fact, it follows from Minkowski's integral inequality and the bounded overlapping property of the cubes that

Notice that ; then from Chebyshev's inequality and the boundedness of on we have

We set

where . Then, we get

Since , then, by Lemmas 5 and 8 and the fact , we get

An application of Chebyshev's inequality gives us that

Below, we will give the estimate of integral . We have

For and , we have . Thus,

By Minkowski's inequality for integral,

From the fact that and Lemma 6, we have

Since , then there exists a such that . Thus, we have

Hence,

For and with , we can see that . Since , we have . By the vanishing moment condition of ,

But

Note that for any and . Thus,

From the fact that and is homogeneous of degree zero, we get

Then,

For , we have . Then,

Hence,

Since , then and for some . From Lemma 6,

Analogously, we have

Then,

Hence,

Combining the estimate of and , we get . This completes the proof of the Theorem 1.

4. Proof of Theorem 2

We follow the strategy of the proof of Theorem 1. For any given , we may choose such that . For every , and then by Lemma 8, we can write

where , and satisfies (a)–(c) in Lemma 8. Then,

Similar to the estimate of in the proof of Theorem 1, we obtain

We set

where . Then, we get

Since , and then by Lemmas 5 and 8 and the fact , similar to the estimate in the proof of Theorem 1, we get

An application of Chebyshev's inequality gives us that

Below, we give the estimate of integral . We have

For and , we have . Thus,

Then, Minkowski's inequality for integral gives us that

Since , then . From Lemmas 6 and 5, we have

Hence, for , Hölder's inequality and imply that

Then,

We now estimate . Since , we have and hence

Denote . Minkowski's inequality for integrals and the fact for give us

Notice that

By Hölder's inequality and Lemma 7, we have

Then,

But

from which we get

Thus,

This completes the proof of Theorem 2.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

L. Hörmander, “Estimates for translation invariant operators in spaces,” Acta Mathematica, vol. 104, pp. 93–140, 1960.

E. M. Stein, “On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz,” Transactions of the American Mathematical Society, vol. 88, pp. 430–466, 1958.

A. Benedek, A. P. Calderón, and R. Panzone, “Convolution operators on Banach space valued functions,” Proceedings of the National Academy of Sciences of the United States of America, vol. 48, pp. 356–365, 1962.

A. Torchinsky and S. L. Wang, “A note on the Marcinkiewicz integral,” Colloquium Mathematicum, vol. 60-61, no. 1, pp. 235–243, 1990.

Y. Ding, D. Fan, and Y. Pan, “Weighted boundedness for a class of rough Marcinkiewicz integrals,” Indiana University Mathematics Journal, vol. 48, no. 3, pp. 1037–1055, 1999.

Y. Ding, M. Y. Lee, and C. C. Lin, “Marcinkiewicz integral on weighted Hardy spaces,” Archiv der Mathematik, vol. 80, no. 6, pp. 620–629, 2003.

C. C. Lin and Y. C. Lin, “ boundedness of Marcinkiewicz integral,” Integral Equations and Operator Theory, vol. 58, no. 1, pp. 87–98, 2007.

Y. Ding, S. Lu, and Q. Xue, “Marcinkiewicz integral on Hardy spaces,” Integral Equations and Operator Theory, vol. 42, no. 2, pp. 174–182, 2002.

B. Muckenhoupt, “Weighted norm inequalities for the Hardy maximal function,” Transactions of the American Mathematical Society, vol. 165, pp. 207–226, 1972.

J. García-Cuerva and J. L. R. de Francia, Weighted Norm Inequalities and Related Topics, vol. 116 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherlands, 1985.

Y. Ding and S. Lu, “Homogeneous fractional integrals on Hardy spaces,” The Tohoku Mathematical Journal, Second Series, vol. 52, no. 1, pp. 153–162, 2000.

C. Fefferman, N. M. Rivière, and Y. Sagher, “Interpolation between spaces: the real method,” Transactions of the American Mathematical Society, vol. 191, pp. 75–81, 1974.

R. Fefferman and F. Soria, “The space weak ,” Studia Mathematica, vol. 85, no. 1, pp. 1–16, 1975.

T. Quek and D. Yang, “Calderón-Zygmund-type operators on weighted weak Hardy spaces over ,” Acta Mathematica Sinica, vol. 16, no. 1, pp. 141–160, 2000.