Abstract

Let be a singular integral operator with its kernel satisfying , , where and are appropriate functions and and are positive constants. For with , the multilinear commutator generated by and is formally defined by . In this paper, the weighted -boundedness and the weighted weak type estimate for the multilinear commutator are established.

1. Introduction and Results

In the classical Calderón-Zygmund theory, the Hörmander's condition introduced by Hörmander [1], plays a fundamental role in the theory of Calderón-Zygmund operators. On the other hand, singular integral operators whose kernels do not satisfy the Hörmander's condition have been extensively studied.

In 1997, in order to study the -boundedness of certain singular integral operators, Grubb and Moore [2] introduced the following variant of the classical Hörmander's condition, where and 's are appropriate functions (see Theorem 3 below). As an example we note that the kernel verifies (2), but it is not a Calderón-Zygmund kernel since its derivative does not decay quickly enough at infinity (see [2] or [3]).

Obviously, if we take , and , then condition (2) is exactly the classical Hörmander's condition (1).

Definition 1. We say that a nonnegative locally integrable function defined on satisfies the reverse Hölder condition, in short, , if there is a constant such that for every cube centered at the origin we have The smallest constant is said to be the constant of .

Remark 2. It is easy to see that if , then also (see [3] Remark 2.4).

In [2], Grubb and Moore established the -boundedness and the weak type estimates for the singular integral operators with kernels satisfying (2).

It is well known that the classical Hörmander's condition (1) is too weak to get weighted inequalities for the classical Calderón-Zygmund operators by any known method. The usual hypothesis on the kernel to obtain them is the Lipschitz condition Conditions, the so-called -Hörmander's condition, weaker than (4), but stronger than (1), have been also considered in [4, 5] (also see [6, 7]).

In 2003, Trujillo-González [3] establishes the weighted norm inequalities for when satisfies a variant of the Lipschitz condition (see (6) below).

As usual, we denote by the Muckenhoupt weights classes (see [8], or [9] and [10]). For a weight , and a measurable set , we write

Theorem 3 (see [3]). Let . Suppose that there is a constant , such that; ; there exist functions and such that , where and ; for a fixed and for any ,  For , we defined the convolution operator associated to the kernel by (1)Let and . Then there exists a constant such that (2)Let . Then there exists a constant such that for all

It is easy to see that any kernel satisfies condition (6) and also verifies (2). Obviously, if we take , , and , then condition (6) is exactly the classical Lipschitz condition (4). We remark that the function satisfies conditions , but does not satisfy the Hörmander's condition (1) (see [11] page 5).

Under the assumption of Theorem 3, several authors have studied two-weight inequalities for the convolution operator , for example [11–13]. Recently, the authors [14] introduce a variant of the classical -Hörmander's condition in the scope of (2) and establish the weighted norm inequalities for singular integral operator with its kernel satisfying such a variant of the classical -Hörmander's condition.

On the other hand, the commutators of singular integral operators have been widely studied by many authors; see, for example, [15–22] and the references therein. Given a locally integrable function and a linear operator with kernel , the linear commutator is formally defined by

For with . The generalized commutator, the so-called the multilinear commutator, is formally defined by

In 2002, Pérez and Trujillo-González [22] studied the sharp weighted estimates for the multilinear commutators of the classical Calderón-Zygmund operators. In 2006, Zhang [23] studied the weighted estimates for maximal multilinear commutators.

In 1993, Alvarez et al. [15] established a generalized boundedness criterion for the commutators of linear operators. Now, we restate Theorem 2.13 in [15] in the following strong form.

Theorem 4 (see [15]). Let be a linear operator and . Suppose that for all , the linear operator satisfies the following weighted estimate where the constant depends only on , , and the constant of . Then for and any weight function , the commutator is bounded from to with bound depending on , , and the constant of .

The goal of this paper is to study the weighted norm inequalities for multilinear commutator of the convolution operator defined by (7) with its kernel satisfying .

By Theorem 3 and applying Theorem 4-times, we can easily get the following weighted inequalities for the multilinear commutator .

Theorem 5. Let be the singular integral operator defined by (7) with its kernel satisfying . If , , and , then there exists a positive constant such that

It is well-known that, in general, the linear commutator of Calderón-Zygmund operator fails to be of weak type and does not map into when ; see [20] for more details. Instead, an endpoint theory was provided for this operator, such as the weak type estimate and the weak type estimate (see [20, 24]).

The main result of this paper is the following weak type estimate for multilinear commutator of the singular integral operator defined in Theorem 3.

Theorem 6. Let be the singular integral operator defined by (7) with its kernel satisfying . If and , then, for all , where is a positive constant independent of and .

Throughout this paper, denotes the positive number appeared in (6). As usual, the letter stands for a positive constant which is independent of the main parameters and not necessary the same at each occurrence. A cube in always means a cube whose sides parallel to the coordinate axes. For a cube and a number , we denote by the cube with the same center and -times the side length as . The symbol means there exist positive constants and such that .

This paper is arranged as follows. In Section 2, we formulate some preliminaries and lemmas we need. In Section 3 we will prove Theorem 6 for the case , and in the last section we prove Theorem 6 for the general case .

2. Preliminaries and Lemmas

In this section, we give some notations and results needed for the proof of the main result.

2.1. Muckenhoupt Weight Classes

A nonnegative locally integrable function defined on is called a weight. We say a weight , if there exists a constant such that for all cubes

We say a weight , if there exists a constant such that for all cubes

The weights class is defined by . There is also another characterization of the class, that is, we say a weight , if there exist positive constants and such that, for any cube and any measurable set , there exist

2.2. Projection of Function

Now, let us recall the definition of the projection of a function (see [2] or [3]). By the projection of an -function onto a finite-dimensional subspace we refer to such an element, if it exists of verifying

Lemma 7 (see [2]). Suppose is a finite family of bounded functions on such that . Then, for any cube centered at the origin and any , there exists the projection of onto and satisfies where the constant depends only on , , and the constant of .

2.3. Notations Related to Orlicz Spaces

A function is said to be a Young function, if is continuous, convex, and increasing with and . We use to denote the complementary Young function associated to ; that is,

The -average of a locally integrable function over a cube is defined by which satisfies the following inequalities (see [25], p. 69, or formula in [21]):

The Young function that we are going to use is with its complementary Young function . Denote

When , we simply write and , and and .

The following generalized Hölder’s inequality holds (see in [22]):

We also need the following notations (see [26] pages 1712-1713). For and a cube , denote

Similarly to (22), we have

There also holds the following generalized Hölder's inequality:

2.4. Lemmas

The following generalized Young's inequality is from [22] Lemma 8. We note that when , it is proved by O'Neil in [27].

Lemma 8 (the generalized Young's inequality). are real-valued, nonnegative, nondecreasing, left continuous functions defined on . For , define . If for all Then, for all , there exist

For and , we have and (see [21] page 35). Then for any integer with , we have

Noting that since , then it follows from Lemma 8 that, for all , , we have

For a locally integrable function and a cube , denote

Lemma 9 (see [26]). Let and . Then, for any cube , where and are positive constants independent of and , and is the -norm of .

Lemma 10 (see [28]). Let , , , and be a cube. Then for any positive integer and ,

3. Proof of Theorem 6: The Case

When , we write and for simplicity. We need to prove that, for and , there exists constant such that, for all ,

For any fixed , we consider the Calderón-Zygmund decomposition of at height and get a sequence of nonoverlapping cubes , where is a cube centered at with radius , such that

Denote by the restriction of to . Let be the projection of onto . We decompose into two parts, , where and with for .