Absolutely summing Hankel operators on Bergman spaces

TL;DR

This paper characterizes when big and little Hankel operators on weighted Bergman spaces are absolutely summing, providing new insights into their properties across different integrability exponents and extending previous results.

Contribution

It introduces a comprehensive characterization of absolutely summing Hankel operators on Bergman spaces, including the diagonal case, using a novel analysis of associated Carleson embedding operators.

Findings

Characterization of $r$-summing big Hankel operators.

Characterization of $r$-summing little Hankel operators.

New results even in the case $p=q$.

Abstract

In this paper we initiate the study of absolute summability for big and little Hankel operators $ H_f^\beta,h_f^\beta:A_\alpha^p(\mathbb{B}_n)\to L^q(\mathbb{B}_n,dv_\beta), $ acting between weighted Bergman and weighted Lebesgue spaces on the unit ball, for possibly different integrability exponents $p$ and $q$. We characterize those symbols $f$ for which the big Hankel operator $H_f^\beta$ is $r$-summing, and those for which the little Hankel operator $h_f^\beta$ is $r$-summing. Our approach relies on a deep revisit of the absolute summability of the associated Carleson embedding operators from $A_\alpha^p(\mathbb{B}_n)$ to $L^q(\mathbb{B}_n,dv_\beta)$, from which we obtain characterizations of absolutely summing big and little Hankel operators that appear to be new even in the diagonal case $p=q$.