A sharp inequality for multilinear singular integral operators with non-smooth kernels

Guangze Gu,Mingjie Cai

doi:10.1186/1029-242x-2013-439

Abstract

In this paper, we establish a sharp inequality for some multilinear singular integral operators with non-smooth kernels. As an application, we obtain the weighted L p -norm inequality and LlogL-type inequality for the multilinear operators.MSC:42B20, 42B25.

Highlights

In this paper, we establish a sharp inequality for some multilinear singular integral operators with non-smooth kernels
We study some multilinear operator associated to the singular integral operators with non-smooth kernels as follows
Definition A linear operator T is called a singular integral operator with non-smooth kernel if T is bounded on L (Rn) and associated with a kernel K(x, y) such that

Summary

The multilinear operator associated to T is defined by

The main purpose of this paper is to prove a sharp function inequality for the multilinear singular integral operator with non-smooth kernel when Dαbj ∈ BMO(Rn) for all α with |α| = mj. In [ ], the boundedness of the commutator associated to the singular integral operator with non-smooth kernel is obtained. The Young functions used in this paper are (t) = t( + log t)r and (t) = exp(t /r), the corresponding average and maximal functions are denoted by · L(log L)r,Q, ML(log L)r and. Theorem If T is a singular integral operator with non-smooth kernel as given in Definition , let Dαbj ∈ BMO(Rn) for all α with |α| = mj and j = , . Tb is bounded on Lp(w) for any < p < ∞ and w ∈ Ap, that is, l

Dαj bj BMO

Qis the cube centered at x and having side length

Dα b

Dαbj BMO