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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 651941, 15 pages
Boundedness for Parametrized Littlewood-Paley Operators with Rough Kernels on Weighted Weak Hardy Spaces
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.
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 , when satisfies the Lipschitz condition of , Sakamoto and Yabuta  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.  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  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 . 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 . The atomic decomposition theory of weak spaces on was given by Fefferman and Soria in . Later, Liu established the weak spaces on homogeneous groups in . In 2000, Quek and Yang introduced the weighted weak Hardy spaces in  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 . 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 . 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 ). 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 ). 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 7 (see ). 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 thatFirstly, let us estimate . As , , , it is easy to see thatSince , , 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 thatBy 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 5By using Lemma 7, we can getIt 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  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 havewhereSimilarly 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 (