Integration operators in average radial integrability spaces of analytic functions

TL;DR

This paper characterizes the boundedness, compactness, and weak compactness of integration operators on average radial integrability spaces of analytic functions, providing new tools like bidual descriptions and Littlewood-Paley inequalities.

Contribution

It introduces new characterizations and tools for analyzing integration operators on $RM(p,q)$ spaces, including bidual descriptions and norm estimates via derivatives.

Findings

Characterization of boundedness, compactness, and weak compactness of $T_g$

Development of Littlewood-Paley type inequalities for $RM(p,q)$ spaces

Description of the bidual of $RM(p,0)$

Abstract

In this paper we characterize the boundedness, compactness, and weak compactness of the integration operators \begin{align*} T_g (f)(z)=\int_{0}^{z} f(w)g'(w)\ dw \end{align*} acting on the average radial integrability spaces $RM(p,q)$. For these purposes, we develop different tools such as a description of the bidual of $RM(p,0)$ and estimates of the norm of these spaces using the derivative of the functions, a family of results that we call Littlewood-Paley type inequalities.