Extended Ces$\acute{a}$RO Operators on Zygmund Spaces in the Unit Ball

TL;DR

This paper investigates the boundedness and compactness of extended Cesàro and related integral operators on Zygmund spaces within the unit ball, providing necessary and sufficient conditions for these properties.

Contribution

It introduces new criteria for the boundedness and compactness of extended Cesàro operators on Zygmund spaces in the unit ball.

Findings

Characterization of boundedness conditions for $T_g$ and $I_g$

Criteria for compactness of the operators

Extension of classical results to higher dimensions

Abstract

Let $g$ be a holomorphic function of the unit ball $B$ in the $n$-dimensional space, and denote by $T_g$ and $I_g$ the induced extended Ces$\acute{a}$ro operator and another integral operator. The boundedness and compactness of $T_g$ and $I_g$ acting on the Zygmund spaces in the unit ball are discussed and necessary and sufficient conditions are given in this paper.