Weighted boundedness of multilinear singular integral operator with general kernels for the extreme cases

Mingjun Zhang,Yanfeng Guo

doi:10.1186/1029-242x-2014-341

Abstract

We prove the weighted boundedness properties for the multilinear operator associated to the singular integral operator with general kernels for the extreme cases. MSC:42B20, 42B25.

Highlights

Introduction and preliminariesAs for the development of singular integral operators, their commutators and multilinear operators have been well studied
Throughout this paper, Q will denote a cube of Rn with sides parallel to the axes
For a locally integrable functions b and a weight function w, let w(Q) = Q w(x) dx, wQ = |Q|– Q w(x) dx, the weighted sharp function of b is defined by b#(x) = sup Q x w(Q)

Summary

The multilinear operator associated to T is defined by

Note that the classical Calderón-Zygmund singular integral operator satisfies Definition (see [ , ]). Theorem Let T be the singular integral operator as Definition , and we have the sequence {kmCk} ∈ l , < p < ∞, w ∈ A , and Dαbj ∈ BMO(Rn) for all α with |α| = mj and j = , . Lemma (see [ ]) Let T be the singular integral operator as Definition , and we have the sequence {Ck} ∈ l.

Dαj bj BMO

Fix a cube