Sharp maximal function inequalities and boundedness for Toeplitz type operator associated to singular integral operator with non-smooth kernel

Xiaosha Zhou

doi:10.1186/1029-242x-2014-141

Abstract

In this paper, we establish the sharp maximal function inequalities for the Toeplitz type operator associated to some singular integral operator with non-smooth kernel. As an application, we obtain the boundedness of the operator on Morrey and Triebel-Lizorkin spaces. MSC:42B20, 42B25.

Highlights

Introduction and preliminariesAs the development of the singular integral operators, their commutators and multilinear operators have been well studied
In [, ], the authors prove 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

The generalized fractional Morrey spaces are defined by

Theorem Let T be the singular integral operator with non-smooth kernel as Definition ,. Theorem Let T be the singular integral operator with non-smooth kernel as Definition , < s < ∞ and b ∈ BMO(Rn). Corollary Let [b, T](f ) = bT(f ) – T(bf ) be the commutator generated by the singular integral operator T with non-smooth kernel and b. Lemma ([ , ]) Let T be the singular integral operator with non-smooth kernel as Definition. Lemma (See [ ]) Let {At, t > } be an ‘approximation to the identity’ and Kα,t(x, y) be the kernel of difference operator Iα – AtIα. Proof of Theorem It suffices to prove for f ∈ C ∞(Rn), the following inequality holds: Mβ,s IαT k, (f ) (x) + Mβ+α,s T k, (f ) (x) , k=. For I , by Hölder’s inequality and Lemma , we obtain

Rn m

Lq m