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On the Theory of Multilinear Singular Operators with Rough Kernels on the Weighted Morrey Spaces
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 . In 1994, Hofmann  proved that is a bounded operator on when and . Recently, Lu et al.  proved and are bounded from to when , while for multilinear fractional integral operators, Ding and Lu  showed the boundedness of and (their definitions will be given later) if . After that, Lu and Zhang  proved is a bounded operator from to when .
On the other hand, the classical Morrey spaces were first introduced by Morrey  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  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  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 .
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 , 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 ). 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 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 ). 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 ). For , , we have For , the formula should be interpreted appropriately.
Lemma 17 (see ). Let , . Then
Theorem 18 (see ). 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 ). If , then there exists a constant , such that
We call the reverse doubling constant.
Theorem 20 (see ). 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 ). (a) (John-Nirenberg Lemma) Let . Then if and only if (b) Assume ; then for cubes , (c) If , then
Theorem 22 (see ). 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 ). If is homogeneous of degree zero and satisfies the moment condition , , , , then we have
Lemma 24 (see ). 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 ). 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  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