Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators

Dazhao Chen,Hui Huang

doi:10.1515/math-2021-0122

Abstract

Abstract In this paper, we establish some sharp maximal function estimates for certain Toeplitz-type operators associated with some fractional integral operators with general kernel. As an application, we obtain the boundedness of the Toeplitz-type operators on the Lebesgue, Morrey and Triebel-Lizorkin spaces. The operators include the Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

Highlights

Introduction and preliminariesThe classical Morrey space was introduced by Morrey in [1] to investigate the local behavior of solutions to second-order elliptic partial differential equations
As the Morrey space may be considered as an extension of the Lebesgue space, it is natural and important to study the boundedness of operator on the Morrey spaces
The boundedness of the maximal operator, the singular integral operator, the fractional integral operator and their commutators on Morrey spaces have been studied by many authors

Summary

Introduction and preliminaries

The classical Morrey space was introduced by Morrey in [1] to investigate the local behavior of solutions to second-order elliptic partial differential equations ( see [2]). As the Morrey space may be considered as an extension of the Lebesgue space, it is natural and important to study the boundedness of operator on the Morrey spaces. The boundedness of the maximal operator, the singular integral operator, the fractional integral operator and their commutators on Morrey spaces have been studied by many authors (see [3,4,5,6]). The weighted boundedness of these operators on Lebesgue spaces was obtained by Muckenhoupt, Wheeden, Coifman and Fefferman (see [7]). In [9,12], some Toeplitz-type operators related to the singular integral operators and strongly singular integral operators are introduced, and the boundedness for the operators generated by BMO and Lipschitz functions are obtained. We obtain the boundedness of the operators on the Lebesgue, Morrey and Triebel-Lizorkin space. The operators include Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator

Preliminaries

Theorems

Proofs of theorems

Applications