Abstract and Applied Analysis

Volume 2013 (2013), Article ID 651941, 15 pages

http://dx.doi.org/10.1155/2013/651941

Research Article

Boundedness for Parametrized Littlewood-Paley Operators with Rough Kernels on Weighted Weak Hardy Spaces

Ximei Wei and Shuangping Tao

College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China

Received 27 April 2013; Accepted 18 June 2013

Academic Editor: Dashan Fan

Copyright © 2013 Ximei Wei and Shuangping Tao. 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.

The authors prove that the parametrized area integral and function are bounded from the weighted weak Hardy space to the weighted weak Lebesgue space as satisfies a class of the integral Dini condition, respectively.

1. Introduction and Main Results

Suppose that is homogeneous of degree zero on and satisfies

where denotes the unit sphere of    equipped with normalized Lebesgue measure , , . The parametrized area integral and function are defined by

respectively, where

It is well known that Littlewood-Paley functions are very important tools in harmonic analysis and PDE (see [1–3]). Some well-known results related to the classical Littlewood-Paley operators can be seen in [4–8]. In 1999, inspired by Hörmander’s work [9], when satisfies the Lipschitz condition of , Sakamoto and Yabuta [10] established the    boundedness of the parametrized area integral and the parametrized function and gave the boundedness on spaces and Campanato spaces. For any , , it is easy to see that the inclusion relationship

holds. In 2002, Ding et al. [11] extended the previous -boundedness to the case as belongs to . In 2007, Ding et al. [12, 13] gave the boundedness of the parametrized area integral and function on the Hardy space and weak Hardy space when satisfies a class of the integral Dini conditions. Recently, Wang and Liu [14] obtained the boundedness on the weighted Hardy space for the parametrized Littlewood-Paley operators with satisfying the logarithmic type Lipschitz conditions. On the other hand, the boundedness properties of the intrinsic square functions on weighted weak Hardy spaces were studied by Wang in [15]. Inspired by the results mentioned previously, in this paper, we will study the boundedness of the parametrized area integral and function on the weighted weak Hardy spaces.

Before stating our main results, let us recall some definitions. Firstly, let , . Then, the integral modulus of continuity of order of is defined by

where, denotes a rotation on and . The function is said to satisfy the -Dini condition, if

Secondly, given a weight function on , for , the weighted Lebesgue spaces is defined by

And also, the weighted weak Lebesgue spaces is defined by

Let us now turn to recall the definition of the weighted weak Hardy spaces. The weak Hardy spaces were first introduced in [16]. The atomic decomposition theory of weak spaces on was given by Fefferman and Soria in [17]. Later, Liu established the weak spaces on homogeneous groups in [18]. In 2000, Quek and Yang introduced the weighted weak Hardy spaces in [19] and established their atomic decompositions. Moreover, by using the atomic decomposition theory of , Quek and Yang also obtained the boundedness of operators on these weighted spaces in [19]. Let , , and . Define

where, , , .

For , the grand maximal function of is defined by

Then, weighted weak Hardy space is defined by . Moreover, we set .

Our main results are stated as follows.

Theorem 1. Let satisfying (1) and the following condition

Then, for , , there exists a constant such that

The relationship between condition (11) and condition is not clear up to now. We point that the conclusion of Theorem 1 still holds if we replace the condition (11) by the condition. In other words, we have the following result.

Theorem 2. Let , , satisfying (1). Then, for , , there exists a constant such that

Theorem 3. Let satisfying (1) and the following condition

Then, for , , , there exists a constant such that

2. Notations and Preliminaries

In this section, we will introduce some notations and preliminary lemmas used in the proofs of our main theorems in the next section.

The classical weighted theory was first introduced by Muckenhoupt in the study of weighted boundedness of Hardy-Littlewood maximal functions in [20]. A weight is a locally integrable function on which takes values in at almost everywhere. Given a ball and , denotes the ball with the same center as whose radius is times that of . We also denote the weighted measure of by ; that is, . We say that with if there exists a constant , such that for every ball ,

We say that if

A weight function if it satisfies the condition for some . It is well known that if , , then for all , and for some . We thus write to denote the critical index of .

Lemma 4 (see [21]). Let , . Then, for any ball , there exists an absolute constant , such that

In general, for any , we have

where does not depend on nor on .

Lemma 5 (see [19]). Let , . For every belongs to , there exists a sequence of bounded measurable functions such that

(i), in ,(ii)each can be further decomposed into , where satisfies the following conditions.

(a), where denotes the ball with center and radius . Moreover,

where denotes the characteristic function of the set and .(b), where is independent of , .(c) for every multi-index with .

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

In the end of this section, we need the following lemmas used in the next section.

Lemma 6 (see [22]). Suppose that satisfies (1) and the following condition

. Then, for , , and , there is a constant independent of , such that

Lemma 7 (see [23]). Suppose that , is homogeneous of degree zero and satisfies the -Dini condition. If there exists a constant such that , then we have

where the constant is independent of , .

3. Proof of Main Results

Proof of Theorem 1. In order to prove Theorem 1, it suffices to show that there exists a constant , for any and , such that

Take such that ; then by Lemma 5 we can write

where , , and satisfies (a)–(c) in Lemma 5. Then, we have

First, we claim that the following inequality holds:

In fact, since , , then it follows from Minkowski's integral inequality that

Let us estimate . By Chebyshev's inequality, Lemma 6 and (27), we have

Now we turn our attention to the estimate of . If we set

where and is a fixed positive number such that , therefore,

Since , then by Lemma 4 we can get

An application of Chebyshev’s inequality and Minkowski integral inequality gives us that

Firstly, let us estimate . As , , , it is easy to see that

Since , , then . By Lemma 4, we obtain that

Noticing that , , we have

Now we consider . Write

Take . First we deal with . As , we have

Using the Minkowski inequality, we get that

By Lemma 5, we have

Now let us consider . It is easy to check that as , . Thus, we can obtain by the condition (c) of in Lemma 5

By using Lemma 7, we can get

It is easy to see that

Using the same method as what used to deal with the inequality (39), we can obtain that

For , we have

Hence, by the inequalities (44) and (45), we have

Now we give the estimate for . Since , , then

Thus,

Repeating this process which is similar to the one of estimating (from (42) to (46)), we may have

Thus by (36) and (40), we get

This completes the proof of Theorem 1.

Proof of Theorem 2. Combining the idea of proving Theorem 1 with the similar steps as in [12] and the following inequalities

it is not difficult to get the proof of Theorem 2. We omit the details here.

Proof of Theorem 3. We follow the strategy of the proof of Theorem 1. It suffices to show that there exists a constant , such that, for any , ,

Take such that , we have

where the notations , are the same as in the proof of Theorem 1. Using the same method of the proof of Theorem 1, we can get

Below, we will give the estimate of . If we set

where , is a fixed positive number such that ; thus,

Noting that , then by Lemmas 4 and 5, we have

where

Similarly as in the proof of Theorem 1, for , if the integration domains of is , , then we denote it by . If , , we denote by . Moreover, if in the integration domains of , we denote it by , otherwise denote it by . Further, we divide again by the integration domains; namely, if , we denote it by , otherwise denote it by . Now, we are in a position to give the estimates of , , , , , respectively. First, we take in the whole proof of Theorem 2. Obviously,

By the proof of Theorem 1, we have

Notice that if , , , , it is easy to check that

 (), . If , , , , then () , , for ; () ; () , for ; () , for .

Now, let us estimate . By the fact () and the Minkowski inequality, we have

As for , notice that , , , , using the Minkowski inequality and the previous facts () and (), we have

Now we consider . By the fact () and , we have

For , since , then we have . Thus, by the cancellation of , we have . By using the fact (), we get

Noting that , we have . Hence

Thus,

From (61) to (66), we can obtain

We conclude the proof of Theorem 3.

Acknowledgments

The authors would like to express their deep thanks to the referee for his/her very careful reading and many valuable comments and suggestions. Shuangping Tao is supported by National Natural Foundation of China (Grants nos. 11161042 and 11071250).

References

E. M. Stein, “The development of square functions in the work of A. Zygmund,” Bulletin of the American Mathematical Society, vol. 7, no. 2, pp. 359–376, 1982.

C. E. Kenig, Harmonic Ayalysis Techniques for Second Order Elliptic Boundary Value Problems, vol. 83 of CBMS, American Mathematical Society, 1991.

S.-Y. A. Chang, J. M. Wilson, and T. H. Wolff, “Some weighted norm inequalities concerning the Schrödinger operators,” Commentarii Mathematici Helvetici, vol. 60, no. 2, pp. 217–246, 1985.

C. Fefferman and E. M. Stein, “ spaces of several variables,” Acta Mathematica, vol. 129, no. 3-4, pp. 137–193, 1972.

C. Fefferman and E. M. Stein, “Some maximal inequalities,” American Journal of Mathematics, vol. 93, pp. 107–115, 1971.

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

E. M. Stein, “On some funcions of Littlewood-Paley and Zygmund,” Bulletin of the American Mathematical Society, vol. 67, pp. 99–101, 1961.

E. M. Stein, Singular integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, USA, 1970.

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

M. Sakamoto and K. Yabuta, “Boundedness of Marcinkiewicz functions,” Studia Mathematica, vol. 135, no. 2, pp. 103–142, 1999.

Y. Ding, S. Z. Lu, and K. Yabuta, “A problem on rough parametric Marcinkiewicz functions,” Journal of the Australian Mathematical Society, vol. 72, no. 1, pp. 13–21, 2002.

Y. Ding, S. Z. Lu, and Q. Y. Xue, “Parametrized area integrals on Hardy spaces and weak Hardy spaces,” Acta Mathematica Sinica (English Series), vol. 23, no. 9, pp. 1537–1552, 2007.

Y. Ding, S. Z. Lu, and Q. Y. Xue, “Parametrized Littlewood-Paley operators on Hardy and weak Hardy spaces,” Mathematische Nachrichten, vol. 280, no. 4, pp. 351–363, 2007.

H. B. Wang and Z. G. Liu, “Weighted estimates for parametrized Littlewood-Paley operators,” Frontiers of Mathematics in China, vol. 6, no. 3, pp. 517–534, 2011.

H. Wang, “Boundedness of intrinsic square functions on the weighted weak Hardy spaces,” Integral Equations and Operator Theory, vol. 75, no. 1, pp. 135–149, 2013.

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 H ¹,” Studia Mathematica, vol. 85, no. 1, pp. 1–16, 1987.

H. P. Liu, “The weak spaces on homogeneous groups,” in Harmonic analysis, Lecture Notes in Math., pp. 113–118, Springer, Berlin, Germany, 1991.

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

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. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, The Netherlands, 1985.

Y. Ding and Q. Y. Xue, “Weighted boundedness for parametrized Littlewood-Paley operators,” Taiwanese Journal of Mathematics, vol. 11, no. 4, pp. 1143–1165, 2007.

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