Boundedness of Toeplitz-type operator associated to general integral operator on L p spaces with variable exponent

Xin-Sheng Yuan,Zhi-Gang Wang

doi:10.1186/1029-242x-2013-344

Abstract

In this paper, the boundedness for some Toeplitz-type operator related to some general integral operator on L p spaces with variable exponent is obtained by using a sharp estimate of the operator. The operator includes Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.MSC:42B20, 42B25.

Highlights

As the development of singular integral operators, their commutators 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 < ∞
The theory of Lp spaces with variable exponent was developed because of its connections with some questions in fluid dynamics, calculus of variations, differential equations and elasticity

Summary

Introduction

As the development of singular integral operators (see [ , ]), their commutators 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 < ∞. Denote by M(Rn) the sets of all measurable functions p : Rn → [ , ∞) such that the HardyLittlewood maximal operator M is bounded on Lp(·)(Rn) and the following hold:. Let b be a locally integrable function on Rn. The Toeplitz-type operator related to T is defined by. Theorem Let T be the integral operator as defined in Definition, p(·) ∈ M(Rn) and b ∈. Lemma [ ] Let T be the integral operator as defined in Definition.

Rn f

Thus m

Set H to be the space

Assume that