Products of redial derivative and integral-type operators from Zygmund spaces to Bloch spaces

TL;DR

This paper studies the boundedness and compactness of products of radial derivative and integral-type operators acting from Zygmund spaces to Bloch spaces in the unit ball, extending understanding of operator behavior in complex analysis.

Contribution

It characterizes conditions for boundedness and compactness of these operator products between Zygmund and Bloch spaces, a novel analysis in several complex variables.

Findings

Established criteria for boundedness of the operators.

Derived conditions for compactness of the operators.

Extended operator theory in the context of several complex variables.

Abstract

Let $H(\mathbb{B})$ denote the space of all holomorphic functions on the unit ball $\mathbb{B}\in \mathbb{C}^n$. In this paper we investigate the boundedness and compactness of the products of radial derivative operator and the following integral-type operator $$ I_\phi^g f(z)=\int_0^1 \Re f(\phi(tz))g(tz)\frac{dt}{t},\ z\in\mathbb{B} $$ where $g\in H(\mathbb{B}), g(0)=0$, $\phi$ is a holomorphic self-map of $\mathbb{B}$,\ between Zygmund spaces and Bloch spaces.