M 2-Type Sharp Estimates and Weighted Boundedness for Commutators Related to Singular Integral Operators Satisfying a Variant of Hörmander ’ s Condition

As the development of singular integral operators(see [14,15]), their commutators have been well studied(see [4]). In [13], the authors prove that the commutators generated by the singular integral operators and BMO functions are bounded on L(R) for 1 < p < ∞. Chanillo (see [1]) proves a similar result when singular integral operators are replaced by the fractional integral operators. In [8], some singular integral operators satisfying a variant of Hörmander’s condition are introduced, and the boundedness for the operators are obtained(see [8,16]). The purpose of this paper is to prove the sharp maximal function inequalities for the the commutators related to some singular integral operators satisfying a variant of Hörmander’s condition. As an application, we obtain the weighted boundedness of the commutator on Lebesgue and Morrey space.


Introduction
As the development of singular integral operators(see [14,15]), their commutators have been well studied(see [4]).In [13], the authors prove that the commutators generated by the singular integral operators and BM O functions are bounded on L p (R n ) for 1 < p < ∞.Chanillo (see [1]) proves a similar result when singular integral operators are replaced by the fractional integral operators.In [8], some singular integral operators satisfying a variant of Hörmander's condition are introduced, and the boundedness for the operators are obtained(see [8,16]).The purpose of this paper is to prove the sharp maximal function inequalities for the the commutators related to some singular integral operators satisfying a variant of Hörmander's condition.As an application, we obtain the weighted boundedness of the commutator on Lebesgue and Morrey space.

Preliminaries
First, let us introduce some notations.Throughout this paper, Q will denote a cube of R n with sides parallel to the axes.For any locally integrable function f , 2000 Mathematics Subject Classification: 42B20, 42B25 130 Daqing Lu and Lanzhe Liu the sharp maximal function of f is defined by where, and in what follows, Let M be the Hardy-Littlewood maximal operator defined by Let Φ be a Young function and Φ be the complementary associated to Φ, we denote that the Φ-average by, for a function f , and the maximal function associated to Φ by The Young functions to be using in this paper are Φ(t) = t(1 + logt) and Φ(t) = exp(t), the corresponding average and maximal functions denoted by || • || L(logL),Q , M L(logL) and || • || expL,Q , M expL .Following [13], we know the generalized Hölder's inequality and the following inequalities hold: The A p weight is defined by (see [7]) Given a weight function w.For 1 ≤ p < ∞, the weighted Lebesgue space L p (w) is the space of functions f such that Definition 2.1.Let Φ = {φ 1 , ..., φ m } be a finite family of bounded functions in R n .For any locally integrable function f , the Φ sharp maximal function of f is defined by where the infimum is taken over all m-tuples {c 1 , ..., c m } of complex numbers and Definition 2.3.Given a positive and locally integrable function f in R n , we say that f satisfies the reverse Hölder's condition (write this as f ∈ RH ∞ (R n )), if for any cube Q centered at the origin we have In this paper, we will study some singular integral operators as following(see [16]).

Daqing Lu and Lanzhe Liu
For f ∈ C ∞ 0 , we define the singular integral operator related to the kernel K by Let b be a locally integrable function on R n .The commutator related to T is defined by Remark 2.5.Note that the classical Calderón-Zygmund singular integral operator satisfies Definition 2.4(see [14,15]).
Definition 2.6.Let ϕ be a positive, increasing function on R + and there exists a constant D > 0 such that Let w be a weight function and f be a locally integrable function on R n .Set, for , which is the classical weighted Morrey spaces (see [11,12]).If ϕ(d) = 1, then L p,ϕ (R n , w) = L p (R n , w), which is the weighted Lebesgue spaces (see [7]).
As the Morrey space may be considered as an extension of the Lebesgue space, it is natural and important to study the boundedness of the operator on the Morrey spaces (see [2,5,6,9,10]).

Theorems and Lemmas
We shall prove the following theorems.Theorem 3.1.Let T be the singular integral operator as Definition 2.4, 0 < r < 1 and b ∈ BM O(R n ).Then there exists a constant C > 0 such that, for any To prove the theorems, we need the following lemmas.
Then T is bounded on L p (w) for 1 < p < ∞, w ∈ A 1 and weak (L 1 , L 1 ) bounded.
Lemma 3.7.(see [16]).Let Then, for any smooth function f for which the left-hand side is finite, Lemma 3.8.(see [2,5]) Let 1 < p < ∞, w ∈ A 1 and 0 < D < 2 n .Then, for any smooth function f for which the left-hand side is finite, Then, for any smooth function f for which the left-hand side is finite, Proof: w) .This finishes the proof.✷ Lemma 3.10.Let T be the singular integral operator as Definition 2.4, 1 < p < ∞, w ∈ A 1 and 0 < D < 2 n .Then The proof of the Lemma is similar to that of Lemma 3.9 by Lemma 3.6, we omit the details.

Proofs of Theorems
Proof of Theorem 3.1.It suffices to prove for f ∈ C ∞ 0 (R n ) and some constant C 0 , the following inequality holds: where Q is any a cube centered at x 0 , C 0 = m j=1 g j φ j (x 0 −x) and For I 1 , by Hölder's inequality and Lemma 3.6, we obtain For I 2 , by Lemma 3.4, 3.5 and 3.6, we obtain Daqing Lu and Lanzhe Liu For I 3 , we have ≤ C||b|| BMO ( T (f ) L p,ϕ (w) + f L p,ϕ (w) ) ≤ C||b|| BMO f L p,ϕ (w) .
This completes the proof of the theorem.

M 2 -
Type Sharp Estimates and Weighted Boundedness y)| p w(y)dy 1/p , where Q(x, d) = {y ∈ R n : |x − y| < d}.The generalized Morrey space is defined by

Theorem 3 . 2 .
Let T be the singular integral operator as Definition 2.4, 1 < p < ∞, w ∈ A 1 and b ∈ BM O(R n ).Then T b is bounded on L p (w).