Commutators of Singular Integral Operators Satisfying a Variant of a Lipschitz Condition

Let T be a singular integral operator with its kernel satisfying |K(x − y) − ∑k=1 ℓB k(x)ϕ k(y)| ≤ C | y|γ/|x − y|n+γ, |x | > 2 | y | > 0, where B k and ϕ k  (k = 1,…, ℓ) are appropriate functions and γ and C are positive constants. For b→=(b1,…,bm) with b j ∈ BMO(ℝn), the multilinear commutator Tb→ generated by T and b→ is formally defined by Tb→f(x)=∫ℝn∏j=1m(bj(x)-bj(y))K(x,y)f(y)dy. In this paper, the weighted L p-boundedness and the weighted weak type Llog⁡L estimate for the multilinear commutator Tb→ are established.


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 ( ) = sin / 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 ℓ = 1, 1 ( ) = ( ) and 1 ( ) ≡ 1, then condition (2) is exactly the classical Hörmander's condition (1). Definition 1. We say that a nonnegative locally integrable function defined on R satisfies the reverse Hölder ∞ condition, in short, ∈ ∞ (R ), if there is a constant > 0 such that for every cube ⊂ R centered at the origin we have The smallest constant is said to be the ∞ constant of .
In [2], Grubb and Moore established the -boundedness and the weak type (1, 1) 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. 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).
Under the assumption of Theorem 3, several authors have studied two-weight inequalities for the convolution operator , for example [11][12][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.
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 K be a linear operator and 1 < < ∞. Suppose that for all ∈ (R ), the linear operator K satisfies the following weighted estimate where the constant depends only on , , and the constant of . Then for ∈ (R ) and any weight function ] ∈ , the commutator [ , K] 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 ( 1 )-( 4 ).
By Theorem 3 and applying Theorem 4 -times, we can easily get the following weighted inequalities for the multilinear commutator ⃗ .
The main result of this paper is the following weak type log 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 ( 1 )-( 4 ). If ∈ 1 and ∈ (R ) ( = 1, . . . , ), then, for all > 0, 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 R always means a cube whose sides parallel to the coordinate axes. For a cube and a number > 0, we denote by the cube with the same center and -times the side length as . The symbol ≈ means there exist positive constants 1 and 2 such that 1 ≤ ≤ 2 .
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 = 1, and in the last section we prove Theorem 6 for the general case > 1.

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

Muckenhoupt Weight Classes.
A nonnegative locally integrable function defined on R is called a weight. We say a weight ∈ (1 < < ∞), if there exists a constant > 0 such that for all cubes ⊂ R We say a weight ∈ 1 , if there exists a constant > 0 such that for all cubes ⊂ R The ∞ weights class is defined by ∞ = ⋃ 1< <∞ . 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 1 -function onto a finite-dimensional subspace we refer to such an element, if it exists ( ) of verifying Then, for any cube centered at the origin and any ∈ 1 ( ), there exists the projection of onto span{ 1 , . . . , ℓ } ⊂ 1 ( ) and satisfies where the constant depends only on , ℓ, and the

Proof of Theorem 6: The General Case > 1
In this section, we will use an induction argument to prove Theorem 6 for the general case. To this end, we first introduce some notation. As in [22], where is a cube in R and ⃗ = (( 1 ) , . . . , ( ) ). We also need the following notation: Proof of Theorem 6 (the general case > 1). We have proved that Theorem 6 is true for = 1 in Section 3. Now, we assume that Theorem 6 holds for all positive integer < ; namely, for all 1 ≤ < and any ∈ C , we have For any fixed > 0, we consider the Calderón-Zygmund decomposition of at height as in Section 3 and use the notations { }, * , , ℎ, ℎ , and Ω as there.