Vector-valued inequalities for the commutators of rough singular kernels

Yanping Chen,Yong Ding

doi:10.1186/1029-242x-2014-139

Abstract

Vector-valued inequalities are considered for the commutator of the singular integral with rough kernel. The results obtained in this paper are substantial improvement and extension of some known results. MSC: 42B20, 42B25.

Highlights

The homogeneous singular integral operator T is defined by (x – y) T f (x) = p.v
Calderón and Zygmund [ ] proved that T is bounded in Lp(Rn) for < p < ∞ if is odd or ∈ L log+ L(Sn– )
In [ ], Grafakos and Stefanov gave a nice survey, which contains a thorough discussion of the history of the operator T

Summary

Introduction

1 Introduction The homogeneous singular integral operator T is defined by (x – y) T f (x) = p.v. Rn |x – y|n f (y) dy, when ∈ L (Sn– ) satisfies the following conditions: (a) is a homogeneous function of degree zero on Rn \ { }, i.e., (tx) = (x) for any t > and x ∈ Rn \ { }. The authors in [ ] proved that if ∈ Lip(Sn– ), the commutator [b, T ] for T and a BMO function b is bounded on Lp for < p < ∞ Combining Theorem A with the well-known results by Duoandikoetxea [ ] on the weighted Lp boundedness of the rough singular integral T , we know that if ∈ Lq(Sn– ) for some q > , [b, T ] is bounded on Lp for < p < ∞.

Results

Conclusion