Journal of Function Spaces
Volume 2016 (2016), Article ID 4876146, 11 pages
http://dx.doi.org/10.1155/2016/4876146

Research Article

Multilinear Square Functions with Kernels of Dini’s Type

Zengyan Si¹ and Qingying Xue²

¹School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China
²School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

Received 9 December 2015; Accepted 31 January 2016

Academic Editor: Rodolfo H. Torres

Copyright © 2016 Zengyan Si and Qingying Xue. 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.

Let be a multilinear square function with a kernel satisfying Dini(1) condition and let be the corresponding multilinear maximal square function. In this paper, first, we showed that is bounded from to Secondly, we obtained that if each , then and are bounded from to and if there is , then and are bounded from to , where Furthermore, we established the weighted strong and weak type boundedness for and on weighted Morrey type spaces, respectively.

1. Introduction and Main Results

Let and . In 1978, Coifman and Meyer [1] introduced a class of multilinear operators as a multilinearization of Littlewood-Paley -function as follows:

where and have compact support with . They studied the estimate of by using the notion of Carleson measures. In 1982, letting and , Yabuta [2] obtained the boundedness and BMO type estimates of by weakening the assumptions in [1]. In 2002, Sato and Yabuta [3] studied the boundedness of the following multilinear Littlewood-Paley -function:

The kernels in (1) and (2) are restricted to separable variable kernels. Thus, efforts have been made to study the above operators with kernels of nonseparated type. In 2015, Xue et al. [4] introduced and studied the weighted estimates for the following multilinear Littlewood-Paley -function with convolution type kernel:

where

with .

The importance of multilinear Littlewood-Paley -function and related multilinear Littlewood-Paley type estimates were shown in PDE and other fields (see [5–10]). For other recent works about multilinear Littlewood-Paley type operators, see [11, 12] and the references therein.

In this paper, our aim is to study the boundedness of multilinear square functions with more rough kernels. To begin with, we give some notations and introduce some definitions. The following multiple weights classes were introduced and studied by Lerner et al. [13].

Definition 1 ( weights class [13]). Let , Given , set . One says that satisfies the condition if

when and is understood as

Definition 2 ( condition). Suppose that is a nondecreasing function with For , one says that , if

Definition 3 (kernels of type ). For any , a locally integrable function defined away from the diagonal in is called a kernel of type , if there is a positive constant , such that

whenever ; and

whenever .
We say that is a multilinear square function with a kernel of type , if

whenever and each
The corresponding multilinear maximal square function is defined by

where

We always assume that and can be extended to be a bounded operator from to for some with

The aim of this paper is to study the bounded properties of multilinear square function and multilinear maximal square function with nonconvolution type kernels. It should be pointed out that the methods used in [4, 11, 12] do not work for Littlewood-Paley operators with more general nonconvolution type kernels, for the reason that the estimates there rely heavily on the convolution type kernels and the well-known Marcinkiewicz function studied in [14].

We formulate the main results of this paper as follows.

Theorem 4. Let be a multilinear square function of type and . Then, can be extended to be a bounded operator from to

Theorem 5. Let be a multilinear square function of type and . Let and . Then, one has the following weighted estimates:

(i)If , then

(ii)If , then

Theorem 6. Let be a multilinear maximal square function of type and . Let and Then, one has the following:

(i)If , then

(ii)If , then

Remark 7. When for some , Theorems 4 and 5 were proved in [15].

Remark 8. The theory on multilinear Calderón-Zygmund operators has attracted much attention recently. In 2009, Maldonado and Naibo [16] established the weighted norm inequalities, with the Muckenhoupt weights, for bilinear Calderón-Zygmund operators of type . In 2014, Lu and Zhang [17] obtained some multiple-weighted norm inequalities for multilinear Calderón-Zygmund operators of type and related operators.

The paper is organized as follows. Section 2 contains some preliminary information and proofs of Theorems 4–6. In Section 3, we establish the weighted strong and weak type boundedness of and on Morrey type spaces.

2. Proofs of the Main Theorems

First, we give the proof of Theorem 4.

Proof. The basic idea of the following arguments is essentially taken from [15, 17].
Set Without loss of generality, we can assume that , where For any fixed number , we need to show that there is constant such that

We perform Calderón-Zygmund decomposition to each function at level , where is a positive number to be determined later. Then, we obtain a sequence of pairwise disjoint cubes and decomposition Moreover, we have

(P1),(P2),(P3),(P4),(P5),(P6) for

Let be the center of cube and let be its side length. For , set , where and . And let

It follows from property (P4) that

By the boundedness of and property (P6), we get

Thus, we have

To complete the proof, we need to estimate for Suppose that for some there are bad functions and good functions appearing in , where For simplicity, we assume that the bad functions appear at the entries , and denote the corresponding term by to distinguish it from the other terms. That is, we will consider

and the other terms can be estimated similarly.
We will show that Obviously, can be controlled by

By properties (P4) and (P6) and Minkowski’s inequality, we can further control the above term by

Let for and . It was proved that ; see [17, p. 105]. Assume that has the smallest length in ; then, by (9) one has

This together with Chebyshev’s inequality gives

where
Now, by repeating the same arguments as in [17, p. 106–108], we can easily obtain Thus, setting , we get

This completes the proof of Theorem 4.

In order to prove Theorem 5, the following lemmas are needed.

Lemma 9 (see [15]). Suppose that ; then, there is a constant such that, for all , the following inequality holds:

Lemma 10. Let be a multilinear square function of type and . For any , there is a constant such that for any bounded and compact supported , , the following inequality holds:

Proof. The proof of Lemma 10 involves a routine application of the method used in Lemma  4.1 in [15]. For the sake of similarity, we sketch the proof. Given , for a fixed point and a cube , it is sufficient to show that there exists a constant such that

For each , let satisfying or . We next introduce two notations:

Therefore,

where

By using the boundedness of , we get immediately that
Using the smooth condition (8), we obtain

Then, by Minkowski’s inequality, it yields that

Thus, we have finished the proof of Lemma 10.

Proof of Theorem 5. Theorem 5 follows from using Lemmas 9 and 10 and repeating the same steps as in [15], here, we omit the proof.

To prove Theorem 6, we need some preliminary lemmas.

Lemma 11 (see [18]). If and , then maps from to .

Lemma 12 (see [13]). Let with and .

(1)If for all , then is bounded from to if and only if .(2)If for all , then is bounded from to if .

Lemma 13 (see [19]). If and , then maps from to .

Lemma 14. Let be a multilinear square function of type and . For any , there is a constant depending on such that for all in any product of spaces, with , the following inequality holds for all :

Proof. The basic idea is due to [18, 20]. Set . For a fixed point and a cube centered at with radius , it is clear that

By using the size condition (7) and Minkowski’s inequality, we get

We are ready to estimate the second term. Set For any , we introduce an operator

Let be the sets in , where , such that for we have if and only if Using the smooth condition (8), we obtain that

where , for
Thus, we obtain

Raising the above inequality to the power , integrating over , and dividing by , we conclude that

Next we estimate the last term in (42). By Theorem 4, we know that is bounded from to ; then we can deduce that

Letting , we obtain that

where we have used the fact (it suffices to prove the lemma for arbitrarily small). Finally, if we insert estimate (44) into (42) and raise to the power , we obtain the desired estimate. This finishes the proof of Lemma 14.

Proof of Theorem 6. Theorem 6 follows by using Lemmas 13 and 14. Using the pointwise estimate for in Lemma 14, we obtain that

Notice that for all (see [13, Theorem 3.6]). By Lemma 13, we have

Thus, we obtain the desired estimates by applying Theorem 5 and Lemma 12:

If we use Lemma 11, by using the same arguments, we can get the weak type estimates; we omit its proof here.

3. Weighted Boundedness on Morrey Type Spaces

The classical Morrey space was first introduced by Morrey in [21] to study the local behavior of solutions to second order elliptic partial differential equations. Later, Komori and Shirai [22] introduced the weighted Morrey space for and investigated the boundedness of classical operators, including Hardy-Littlewood maximal operator, Calderón-Zygmund operator, and fractional integral operator. In order to deal with the multilinear case , Wang and Yi [23] extended the range to

Motivated by the works on multilinear Calderón-Zygmund operators and multilinear square functions, as demonstrated in [4, 12, 15, 24–26], we are going to study the boundedness of multilinear square function of type on weighted Morrey type spaces.

Let and and let be a weighted function on . Then, the weighted Morrey space is defined by

where

Furthermore, the weighted weak Morrey space is defined by

where

The main results in this section are the following Theorem.

Theorem 15. Let be a multilinear square function of type and . Suppose that and with . For and , the following two weighted inequalities hold:

(i)If , then

(ii)If and , then

Remark 16. Theorem 15 also holds with replaced by .

In order to prove Theorem 15, we will use the following lemmas.

Lemma 17 (see [23]). Let , , and with . Assume that and set Then, for any ball , there exists a constant such that

Lemma 18 (see [27]). Let with Then, for any ball , there exists an absolute constant such that

Lemma 19 (see [27]). Let Then, for any ball and all measurable subsets of , there exists such that

Now we are in the position to prove Theorem 15.

Proof. First, let us prove (i). For a fixed point and a ball , we split each as , where for . Write

where , or for
Then, we have

It was shown in [13, Theorem 3.6] that . This fact together with Theorem 6(i) and Lemmas 18 and 17 yields that

We now estimate with . For any , by Minkowski’s inequality, the size condition (7), and Lemma 17, we have

Then, Lemma 19 implies that

It remains for us to consider with , or for We may assume that and Minkowski’s inequality and the size condition (7) imply that

Together with Lemma 19, we obtain

Combining the above estimates and then taking the supremum over all balls , we complete the proof of Theorem 15(i).
We are now in a position to demonstrate (ii). For any , we have

Using Theorem 6(ii) and Lemmas 18 and 17, we obtain

We assume that and . If , recall that in (60) and (62) we have proved the following fact:

Hence, we may obtain

To prove Theorem 15(ii), we may assume that

Otherwise, there is nothing needing to be proved.
Then, by the above estimates, we obtain that

Therefore, we get

This completes the proof of Theorem 15.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

The first author was supported by the National Natural Science Foundation of China (no. 11401175 and no. 11501169). The second author was supported partly by NSFC (no. 11471041), the Fundamental Research Funds for the Central Universities (no. 2014kJJCA10), and NCET-13-0065.

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