Absolutely summing Hankel operators on Bergman spaces

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 $p=q$.