Average radial integrability spaces, tent spaces and integration operators

TL;DR

This paper investigates Carleson measure conditions for tent spaces in the unit disk, characterizes the boundedness of integration operators between these spaces and classical Hardy spaces, and extends known results in complex analysis.

Contribution

It provides necessary and sufficient conditions for measures to induce bounded operators on average integrability and tent spaces, addressing a problem posed by D. Luecking and applying to the Pommerenke operator.

Findings

Characterization of Carleson measures for average tent spaces.

Boundedness criteria for the Pommerenke operator between these spaces.

Extension of results to mappings from average integrability spaces to Hardy spaces.

Abstract

We deal with a Carleson measure type problem for the tent spaces $AT_{p}^{q}(\alpha)$ in the unit disc of the complex plane. They consist of the analytic functions of the tent spaces $T_{p}^{q}(\alpha)$ introduced by Coifman, Meyer and Stein. Well known spaces like the Bergman spaces arise as a special case of this family. Let $s,t,p,q\in (0,\infty)$ and $\alpha >0\,.$ We find necessary and sufficient conditions on a positive Borel measure $\mu$ of the unit disc in order to exist a positive constant $C $ such that $$ \int_{\mathbb{T}} \left(\int_{\Gamma (\xi)} |f(z)|^{t}\ d\mu(z)\right)^{s/t}\ |d\xi|\leq C \|f\|^s_{T_{p}^{q}(\alpha)} \,,\quad f\in AT_{p}^{q}(\alpha)\,, $$ where $\Gamma (\xi) = \Gamma_M (\xi)=\{ z\in \mathbb{D} : |1-\bar{\xi} z |< M (1-|z|^2)\},$ $M> 1/2 $ and $\xi$ is a boundary point of the unit disk. This problem was originally posed by D. Luecking. We apply our results to the study of the action of the integration operator $T_g$, also known as Pommerenke operator, between the average integrability spaces $RM(p,q) ,$ for $p,q\in [1,\infty)$. These spaces have appeared recently in the work of the first author with M. D. Contreras and L. Rodr\'iguez-Piazza. We also consider the action from an $RM(p,q)$ to a Hardy space $H^s$, where $ p,q,s \in [1,\infty)$.