Extended Cesaro Operator from to Bloch Space

Mingzhu Yang

doi:10.5539/mas.v3n4p142

Abstract

Let g be a holomorphic function of the unit ball B in several complex variables, and denote by the induced extended Cesaro operator. This paper discussed the boundedness and compactness of acting from to Bloch space in the unit ball.

Highlights

Let B be the unit ball of Cn, and H (B) denotes the class of analytic functions in B
It is well known that the operator C is bounded on the usual Hardy spaces H p and Bergman space, as well as the Dirichlet space
The boundedness and compactness of this operator on weighted Bergman, mixed norm, Bloch, and Dirichlet spaces in the unit ball have been studied by Xiao and Hu

Summary

Introduction

Let B be the unit ball of Cn , and H (B) denotes the class of analytic functions in B. Let H p be the standard Hardy space on the unit disc D. Boundedness and compactness of extended Cesaro operator between several spaces of holomorphic functions have been studied by many mathematicians. It is well known that the operator C is bounded on the usual Hardy spaces H p and Bergman space, as well as the Dirichlet space. There is no easy way to determine when an extended Cesaro operator is bounded or compact. The boundedness and compactness of this operator on weighted Bergman, mixed norm , Bloch, and Dirichlet spaces in the unit ball have been studied by Xiao and Hu. In this paper, we continue this line of research.

Some Lemmas

Main Theorem