Absolutely summing Hankel operators on Bergman spaces
Zhijie Fan, Bo He, Xiaofeng Wang, Zhicheng Zeng
TL;DR
The paper initiates the study of absolute summability (r-summing) for big and little Hankel operators between weighted Bergman and Lebesgue spaces on the unit ball. It reduces the problem to sharp Carleson-embedding criteria by developing a general necessity principle based on testing with a separated lattice, and then provides complete characterizations of $r$-summing Hankel operators in broad ranges of $(p,q,r)$ via lattice and invariant-mean quantities like $G_{q,\delta}(f)$, $MO_{q,\delta}(f)$, and $V_{b,\beta}(f)$. The results yield precise norm equivalences and extend known diagonal cases, contributing a systematic framework for absolutely summing operators on analytic function spaces. Overall, this work bridges Hankel operator theory with Carleson-embedding summability, offering new tools and characterizations that deepen our understanding of summability in Bergman-space settings.
Abstract
In this paper we initiate the study of absolute summability for big and little Hankel operators $ H_f^β,h_f^β:A_α^p(\mathbb{B}_n)\to L^q(\mathbb{B}_n,dv_β), $ 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 $H_f^β$ is $r$-summing, and those for which the little Hankel operator $h_f^β$ is $r$-summing. Our approach relies on a deep revisit of the absolute summability of the associated Carleson embedding operators from $A_α^p(\mathbb{B}_n)$ to $L^q(\mathbb{B}_n,dv_β)$, from which we obtain characterizations of absolutely summing big and little Hankel operators that appear to be new even in the diagonal case $p=q$.