Weighted sharp maximal function inequalities and boundedness of multilinear singular integral operator satisfying a variant of Hörmander’s condition

Changhong Wu,Meng Zhang

doi:10.1186/1029-242x-2014-57

Changhong Wu, Meng Zhang

Open Access

https://doi.org/10.1186/1029-242x-2014-57

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Abstract

In this paper, we establish the weighted sharp maximal function inequalities for the multilinear operator associated with the singular integral operator satisfying a variant of Hörmander’s condition. As an application, we obtain the boundedness of the operator on weighted Lebesgue spaces. MSC:42B20, 42B25.

Highlights

As the development of singular integral operators, their commutators and multilinear operators have been well studied
In [ – ], the authors proved that the commutators generated by the singular integral operators and BMO functions are bounded on Lp(Rn) for < p < ∞
In [, ], the boundedness for the commutators generated by the singular integral operators and Lipschitz functions on Triebel-Lizorkin and Lp(Rn) ( < p < ∞) spaces are obtained

Summary

Introduction

As the development of singular integral operators (see [ , ]), their commutators and multilinear operators have been well studied. In [ , ], the boundedness for the commutators generated by the singular integral operators and Lipschitz functions on Triebel-Lizorkin and Lp(Rn) ( < p < ∞) spaces are obtained. In [ , ], the boundedness for the commutators generated by the singular integral operators and the weighted BMO and Lipschitz functions on Lp(Rn) ( < p < ∞) spaces are obtained ( see [ ]). In [ ], some singular integral operators satisfying a variant of Hörmander’s condition are introduced, and the boundedness for the operators and their commutators is obtained (see [ , ]) Motivated by these results, in this paper, we will study the multilinear operator generated by the singular integral operator satisfying a variant of Hörmander’s condition and the weighted Lipschitz and BMO functions, that is, Dαb ∈ BMO(w) or Dαb ∈ Lipβ (w) for all α with |α| = m. For any locally integrable function f , the sharp maximal function of f is defined by

Qx inf

Qis the cube centered at x and having side length

Dα bf WL