Sharp Weighted Bounds for Multilinear Fractional Type Operators Associated with Bergman Projection

Abstract

We first introduce the multiple weights which are suitable for the study of Bergman type operators. Then, we give the sharp weighted estimates for multilinear fractional Bergman operators and fractional maximal function.

1. Introduction

Let be the area measure in the upper half plane and for . Then, the Bergman space () is defined to be the space of analytic functions in . The well-known Bergman projection and its maximal operator have the following integral representations: where is an appropriate real constant. It is well known that [1] both and map boundedly onto for . The boundedness of and in weighted spaces was studied by Békollé and Bonami in [2, 3]. They showed that the following assertions are equivalent: (i): ,(ii): ,(iii),

where is the family of positive locally integrable functions for which the following inequality holds: Here is denoted to be the Carleson cube associated with the interval (see (18)) and . Later on, for , Pott and Reguera [4] demonstrated the sharp Békollé estimates for the maximal Bergman projection as follows:where the constant depends only on , and the estimate is sharp in the sense that the power of is optimal. Since then, several nice works have been done in the direction of the sharp estimate of Bergman type operators, such as [5–7].

It is easy to see that the maximal Bergman operator coincides with the following operators in the case of : The operator and the following operator appear naturally in the problem of off-diagonal weighted inequalities for the Bergman operator [8].Obviously, is pointwise dominated by .

In order to state some known results, we first introduce the weights class . Let be a weight function and let ; we say if the following quantity is finite: The weights class is an extension and an analogue of the weight class introduced by Muckenhoupt and Wheeden in [9]. Recently, Sehba [8] showed that is bounded from into if and only if for , , or . Moreover, it holds thatand the power of in the above inequality is sharp. In the case , one may note that formula (7) coincides with formula (3). We summarize the results in [8, 10] for the estimates of and as follows.

Theorem A (see [8]). Let ,  ,  ,  , and . Then is bounded from into if and only if . Moreover, and the power of is sharp.

Theorem B (see [10]). Let ,  ,  , and . If then where the exponent of is sharp.

In order to state our results, we first introduce some definitions.

Definition 1 (multilinear weights ). Let and . Suppose that each   () is a positive locally integrable function defined on . Given ,  , and . A multiple weight associated with is said to satisfy the condition if the following inequality holds:

Definition 2 (multilinear fractional Bergman operator). For ,  , and , the multilinear positive fractional Bergman operator is defined by

Definition 3 (multilinear fractional maximal function). For ,  ,  , and , the multilinear fractional maximal function is defined by Note that if , then denote ; it is just the multilinear version of Bergman operators. This paper will be devoted to investigate the sharp weighted bounds for and . We summarize our results as follows.

Theorem 4. Let ,  ,  , and . Suppose that , , and Then, the following inequality holds: and the exponent of is sharp.

Theorem 5. Let ,  ,  , and . Suppose that ,   for some and . If  , then it holds that and the exponent of is sharp.

Although the estimate in Theorem 5 is sharp, it can be improved whenever mixed estimates are invoked.

Theorem 6. Let ,  ,  ,  , and . If is an tuple of weights satisfying for all and , then

As a consequence of Theorem 6, if and , then . The following Corollary 7 follows immediately from Theorem 6.

Corollary 7. Let ,  ,  ,  , and . Suppose that and . Then we have the following: and the exponents are sharp.

Throughout the paper, the symbol means that , where is used to denote a constant which is independent of the main parameters.

2. Notions and Preliminaries

We begin this section with some basic concepts and notions that will be used in our paper. The space we will focus on is the upper half of complex plane, . For any subset of , we denote . For an interval , the Carleson cube associated with will be denote by , and the upper half associated with will be denoted by ; namely,In order to establish the sparse domination for the Bergman type operators, let us first introduce the following system of dyadic grids [11], for . We need the following well-known lemma given by Pott and Reguera [4].

Lemma 8 (see [4]). Given an interval , there is a dyadic interval , for some , such that and .

For any locally integrable function and , it follows that where is defined in the same way as but the supremum is taken only over dyadic intervals of the dyadic grid .

We say that is an -sparse family, if for any , there exists , such that and the sets are pointwise disjoint. We need one more definition.

Definition 9. Let be one of the dyadic grids described above, and a sparse family in , we define the dyadic fractional Bergman operators byThen we have the following sparse domination for , which will be used in proving Theorems 4 and 5.

Proposition 10. Let , ; then, it holds that

Proof. Let us fix , then there exists such that It is easy to see that there exists an interval such that ; by Lemma 8, we have and . Now it follows that Applying the same decomposition to the case yields that

3. Lemmas and the Proof of Theorems

We begin by preparing some lemmas.

Lemma 11 (see [10]). Let ,  ,  , and be a weight. Assume that is such that . Then there exists a constant such that

Lemma 12. Let ,  ,  , and . Suppose that ,  , and ; then we have

Proof. Let ; we have Given , we recall that it is upper-half is the set . It is clear that if is a dyadic grid in , then the family forms a tiling of . As , and , i.e., ,  .
We remark that Then, it follows thatSince , we deduce that Hence, we have Combining the former estimate, we get the desired result,

Lemma 13. Let ,  ,  , and . Suppose that ,   for some ,   and . Furthermore, assume that is a sparse family. Then we have

Proof. Let ; one may obtain that By (31) and the fact , we have Let , by the Hölder inequality, we obtain

We are now ready to prove Theorem 4.

Proof of Theorem 4. We know that when , we can use the duality. However, when , we cannot use the method of the duality. Thus, we need to consider these two cases separately, and here we will borrow some ideas from [12, 13].
(i) The Case  . By (22), it is sufficient to show that If , then using Lemma 12, we may get the desired result. If , we assume that . By duality, we have (ii) The Case  . In this case, we first notice that By (23), it is enough to show that By Lemma 13, it yields that

Proof of Theorem 5. Let and . Then we have , where and is a family of dyadic cube which is maximal with respect to the inclusion and enjoys the properties that Define ; then it is easy to see that the sets are disjoint. By the Hölder inequality, it follows that Hence, we obtain Thus, , and then is a sparse family.
Therefore, it holds that By Lemma 13, we complete the proof of Theorem 5.

To prove Theorem 6, the following Carlson embedding Theorem (see, for examples, [10, 14, 15]) will be needed. And we give the definition of the Carlson sequence first.

Definition 14 (see [10]). Let and be a positive weight. For any , a sequence of positive numbers is called a -Carleson sequence, if there is a constant such that for any , The smallest constant in the above definition is called the Carleson constant of the sequence.

Lemma 15 (Carlson embedding theorem [10]). Let , and . Let be a weight on and . Assume is a sequence of positive numbers. If there exists some constant such that for any interval , it holds that Then, for any , we have

Proof of Theorem 6. Let with and denote by . Using the Hölder inequality, one deduces that We claim that the sequence is a -Carlson sequence. Indeed, when , we have So by Carlson embedding theorem, we have By Lemma 11, we have Then we have

4. Examples

In this part, we give some examples to show that our estimates are sharp. First, we show that Theorem 5 is sharp, we consider the simple case, , recall that , let , we suppose that , then consider and , and one can easily check that and We also consider the function and . We obtained that and . Let ; then Then we have That is, Let us also check that Theorem 4 is sharp. We notice that . If , taking the same and as the above, then we may obtain the sharpness immediately. If , we merely prove the case is sharp. Indeed, ; that is, . But in this case, we have already showed that the result is sharp. So the sharpness just follows from the standard duality argument in the proof of Theorem 4.

Finally, we give the example to show Corollary 7 is sharp. We proceed to use the similar calculation as before. For and , we take and ,  ; then . One can easily check that ,  , and . Since the process is similar as the above, we omit the proof. Let ; then Therefore, we have

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The second author was supported partly by NSFC (no. 11371295). The third author was supported by NSFC (nos. 11471041 and 11671039) and NSFC-DFG (no. 11761131002).