Weighted Composition Operators and Integral-Type Operators between Weighted Hardy Spaces on the Unit Ball

Stevo Stević,Sei-Ichiro Ueki

doi:10.1155/2009/952831

Abstract

We study the boundedness and compactness of the weighted composition operators as well as integral-type operators between weighted Hardy spaces on the unit ball.

Highlights

Let B denote the open unit ball of the n-dimensional complex vector space Cn, ∂B its boundary, and let H B denote the space of all holomorphic functions on B
Assume g ∈ H B, g 0 0, and φ is a holomorphic self-map of B, we define an operator on the unit ball as follows: Pφg f z f φ tz g tz dt t
Suppose that u ∈ H B and φ is a holomorphic selfmap of B which induce the bounded operator uCφ : Hαp B → Hβq B

Summary

Introduction

Let B denote the open unit ball of the n-dimensional complex vector space Cn, ∂B its boundary, and let H B denote the space of all holomorphic functions on B. Φ and u induce a weighted composition operator uCφ on H B which is defined by uCφf u f ◦ φ This type of operators has been studied on various spaces of holomorphic functions in Cn, by many authors; see, for example, 4 , recent papers 5–17 , and the references therein. Assume g ∈ H B , g 0 0, and φ is a holomorphic self-map of B, we define an operator on the unit ball as follows: Pφg f z f φ tz g tz dt t. In this paper we study the boundedness and compactness of the weighted composition operators as well as the integral-type operator Pφg, between different weighted Hardy spaces on the unit ball. If both a b and b a hold, one says that a b

Weighted Composition Operators

Integral-Type Operators