Abstract

We study some multilinear operators with rough kernels. For the multilinear fractional integral operators and the multilinear fractional maximal integral operators , we obtain their boundedness on weighted Morrey spaces with two weights when or . For the multilinear singular integral operators and the multilinear maximal singular integral operators , we show they are bounded on weighted Morrey spaces with two weights if and bounded on weighted Morrey spaces with one weight if for .

1. Introduction and Main Results

Let us consider the following multilinear fractional integral operator, and the multilinear fractional maximal operator: where is homogeneous of degree zero in , is a function defined on , and denotes the th order Taylor series remainder of at expanded about ; that is, , each , is a nonnegative integer, , , , and .

We notice that if , the above two operators , are the multilinear singular integral operator and the multilinear maximal singular integral operator whose definitions are given as follows, respectively: For , is obviously the commutator of and : , where is the fractional integral operator given by

There are numerous works on the study of multilinear operators with rough kernels. If , the boundedness of was obtained by means of a good- inequality by Cohen and Gosselin [1]. In 1994, Hofmann [2] proved that is a bounded operator on when and . Recently, Lu et al. [3] proved and are bounded from to when , while for multilinear fractional integral operators, Ding and Lu [4] showed the boundedness of and (their definitions will be given later) if . After that, Lu and Zhang [5] proved is a bounded operator from to when .

On the other hand, the classical Morrey spaces were first introduced by Morrey [6] to study the local behavior of solutions to second-order elliptic partial differential equations. From then on, a lot of works concerning Morrey spaces and some related spaces have been done; see [7–9] and the references therein for details. In 2009, Komori and Shirai [10] first studied the weighted Morrey spaces and investigated some classical singular integrals in harmonic analysis on them, such as the Hardy-Littlewood maximal operator, the Calderón-Zygmund operator, the fractional integral operator, and the fractional maximal operator. Recently, Wang [11] discussed the boundedness of the classical singular operators with rough kernels on the weighted Morrey spaces.

We note that many works concerning , , , and have been done on spaces or weighted spaces when belongs to some function spaces for . However, there is not any study about these operators on weighted Morrey spaces. Therefore, it is natural to ask whether they are bounded on weighted Morrey spaces. The aim of this paper is to investigate the boundedness of , , , and on weighted Morrey spaces if or . When , we show and are controlled pointwisely by the fractional singular integral operators and (their definition will be given later), respectively. Thus, the problem of studying the boundedness of and on weighted Morrey spaces with two weights could be reduced to that of and . When , the boundedness of on weighted Morrey spaces with two weights is proved by standard method. However, we could only obtain the boundedness of on weighted Morrey spaces with one weight for and , since we need the boundedness of in our proof, but to the best of our knowledge, there is not such bounds hold for when . For and , we show they are controlled pointwisely by and , respectively. Thus, it is easy to obtain the same results for and as those of and .

Before stating our main results, we introduce some definitions and notations at first.

A weight is a locally integrable function on which takes values in almost everywhere. For a weight and a measurable set , we define , the Lebesgue measure of by and the characteristic function of by . The weighted Lebesgue spaces with respect to the measure are denoted by with . We say a weight satisfies the doubling condition if there exists a constant such that for any ball , we have . When satisfies this condition, we denote for short.

Throughout this paper, denotes a ball centered at with radius . Let be a cube with sides parallel to the axes. For , denotes the cube with the same center as and side-length being times longer. When , we will denote , , by , , , respectively. And for any number , stands for the conjugate of . The letter denotes a positive constant that may vary at each occurrence but is independent of the essential variable.

Next, we give the definition of weighted Morrey space introduced in [10].

Definition 1. Let , let , and let be a weight. Then the weighted Morrey space is defined bywhere and the supremum is taken over all balls in .

When we investigate the boundedness of the multilinear fractional integral operator, we need to consider the weighted Morrey space with two weights. It is defined as follows.

Definition 2. Let , let , and let , be two weights. The two weights weighted Morrey space is defined by where and the supremum is taken over all balls in . If , then we denote for short.

As is pointed out in [10], we could also define the weighted Morrey spaces with cubes instead of balls. So we shall use these two definitions of weighted Morrey spaces appropriate to calculation.

Then, we give the definitions of Lipschitz space and space.

Definition 3. The Lipschitz space of order , , is defined by and the smallest constant is the Lipschitz norm .

Definition 4. A locally integrable function is said to be in if where and the supremum is taken over all balls in .

At last, we give the definition of two weight classes.

Definition 5. A weight function is in the Muckenhoupt class with if there exists such that for any ball , We define .
When , we define if there exists such that for almost every ,

Definition 6. A weight function belongs to for if there exists such that such that for any ball , When , then we define with if there exists such that

Remark 7 (see [10]). If with , then (a).(b) with .

Now we state the main results of this paper.

Theorem 8. If , is homogeneous of degree zero, , , , , , then

Theorem 9. If , is homogeneous of degree zero, , , , , , then

Theorem 10. If , is homogeneous of degree zero, , , , , , then

When and , we denote , by , , and , , respectively, in order to distinguish from and that are defined for any . To be more precise, Then for the above operators, we have the following results on weighted Morrey spaces with one weight.

Theorem 11. If is homogeneous of degree zero and satisfies the vanishing condition , , , , , then

Theorem 12. If is homogeneous of degree zero and satisfies the moment condition , , , , , then

Remark 13. Here we point out that for and , when , the analogues of Theorems 11 and 12 are open for .

Remark 14. Define where , . When , they are a class of multilinear fractional integral operators and multilinear fractional maximal operators. When , they are a class of multilinear singular integral operators and multilinear maximal singular integral operators. Repeating the proofs of the theorems above, we will find that for and , the conclusions of Theorems 8 and 9 above with the bounds and Theorem 10 with the bounds also hold, respectively.

The organization of this paper is as follows. In Section 2, we give some requisite lemmas and well-known results that are important in proving the theorems. The proof of the theorems will be shown in Section 3.

2. Lemmas and Well-Known Results

Lemma 15 (see [1]). Let be a function on with th order derivatives in for some . Then where is the cube centered at with sides parallel to the axes, whose diameter is .

Lemma 16 (see [12]). For , , we have For , the formula should be interpreted appropriately.

Lemma 17 (see [13]). Let , . Then

Theorem 18 (see [14]). Suppose that , , , and is homogeneous of degree zero. Then is a bounded operator from to , if the index set satisfies one of the following conditions:(a) and ;(b) and ;(c) and there is , such that .

Lemma 19 (see [10]). If , then there exists a constant , such that

We call the reverse doubling constant.

Theorem 20 (see [4]). Suppose that , , , is homogeneous of degree zero. Moreover, for , , , and , if the index set satisfies one of the following conditions:(a) and ;(b) and ;(c) and there is , , such that .Then there is a , independent of and , such that

Lemma 21 (see [15]). (a) (John-Nirenberg Lemma) Let . Then if and only if (b) Assume ; then for cubes , (c) If , then

Theorem 22 (see [16]). Suppose that is homogeneous of degree zero and satisfies the vanishing condition . If , then is bounded on if the index set satisfies one of the following conditions:(a), and ;(b), and ;(c) and .

Theorem 23 (see [2]). If is homogeneous of degree zero and satisfies the moment condition , , , , then we have

Lemma 24 (see [15]). The following are true:(1)If for some , then . More precisely, for all we have (2)If for some , then there exist and such that for any cube and a measurable set ,

Lemma 25 (see [17]). Let . Then the norm of is equivalent to the norm of , where

3. Proofs of the Main Results

Before proving Theorem 8, we give a pointwise estimate of at first. Set where is homogeneous of degree zero in . Then we have the following estimate.

Theorem 26. If , , , then there exists a constant independent of such that

Proof. For fixed , , let be a cube with center at and diameter . Denote and set where is the average of on . Then we have, when , and it is proved in [1] thatHence, By Lemma 15 we get Note that, if , then . By Lemmas 16 and 17 we have, when , It is obvious that when , Thus, Therefore, It follows that Thus, we finish the proof of Theorem 26.

The following theorem is a key theorem in proving (18) of Theorem 8.

Theorem 27. Under the same assumptions of Theorem 8, is bounded from to .

Proof. Fix a ball , we decompose with . Then we have We estimate at first. By Remark 7(a) we know that . Then by Theorem 18(a) and the fact that we get,Now we consider the term . By Hölder’s inequality, we have We will estimate , , respectively. Let ; then for , , we have . Noticing that is homogeneous of degree zero and , then we have where . For , , we have . Thus, By Hölder’s inequality and , we get Thus, So we get We know from Remark 7(a) and Lemma 19 that satisfies inequality (33), so the above series converges since the reverse doubling constant is larger than one. Hence, Therefore, the proof of Theorem 27 is completed.

Remark 28. It is worth noting that Theorem 27 is essentially verifying the multilinear fractional operator is bounded on weighted Morrey spaces.

Now we are in the position of proving Theorem 8.

We will obtain (18) immediately in combination of Theorems 26 and 27.

Then let us turn to prove (19).

Set where is homogeneous of degree zero in . It is easy to see inequality (18) also holds for . On the other hand, for any , we have Taking the supremum for on the inequality above, we get Thus, we can immediately obtain (19) from (65) and (18).

Similarly as before, we give the following theorem at first before proving Theorem 9, since it plays an important role in the proof of Theorem 9. Set where is homogeneous of degree zero in .

Theorem 29. Under the assumptions of Theorem 9, is bounded from to .

The proof of Theorem 29 can be treated as that of Theorem 27 with only slight modifications; we omit its proof here.

Now, let us prove Theorem 9. It is not difficult to see that (20) can be easily obtained from Theorems 26 and 29. Then we can immediately arrive at (21) from (65) and (20).

From now on, we are in the place of showing Theorem 10. We prove (22) at first. Fixing any cube with center at and diameter , denote and set Noticing that equality (67) is the special case of equality (44) when . Thus, equalities (45) and (46) also hold for . We decompose as . Then we have By Theorem 20(a) and Remark 7(a) that , we have Next, we consider the term contained in . By Lemma 15 and equality (45), (46), we have We estimate and , respectively. By Lemma 21(a) and (b), Hölder’s inequality, and , we get For , , we have , so we obtain By Hölder’s inequality, we get We estimate the part containing the function as follows: For the term , since , we then have by Remark 7(b). Thus, by Lemma 25 that the norm of is equivalent to the norm of and , we have For the term , by Lemma 21(a), there exist such that for any cube and , since . Then by Lemma 24(2), we have for some . Hence it implies As a result, Thus, For the term , by Lemma 21(c), Hölder’s inequality, and , we get Hence, Therefore, where is the reverse doubling constant. Consequently, Taking supremum over all cubes in on both sides of the above inequality, we complete the proof of (22) of Theorem 10.

It is not difficult to see that inequality (23) is easy to get from (22) and (65).

Proof of Theorem 11. We consider (25) firstly. Let be the same as in the proof of (22) and denote ; we decompose as . Then we have By Theorem 22(a) and Lemma 24(1) that , we get For , by Hölder’s inequality, we obtain Next we estimate and , respectively. By Hölder’s inequality and , we have We estimate the part containing as follows: For the term , notice that ; we thus get by Lemma 25 that Next we estimate . By Lemmas 21(c) and 25, we have Hence,As a result, For , by Hölder’s inequality and , we getTherefore, So far, we have completed the proof of (25).

Inequality (26) can be immediately obtained from (65) and (25).

Proof of Theorem 12. As before, we prove (27) at first. Assume to be the same as in the proof of (22), denote , and set We also decompose according to . Then we get For , Theorem 23 and Lemma 24(1) imply We will omit the proof for since it is similar to and even easier than the part of in the proof of (22), except that we use the conditions , , , and . For inequality (28), it can be easily proved by (27) and (65). Thus, we complete the proof of Theorem 12.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

The research was partially supported by the National Nature Science Foundation of China under Grant no. 11171306 and no. 11571306 and sponsored by the Scientific Project of Zhejiang Provincial Science Technology Department under Grant no. 2011C33012 and the Scientific Research Fund of Zhejiang Provincial Education Department under Grant no. Z201017584.