Absolutely summing Carleson embeddings on weighted Fock spaces with $A_{\infty}$-type weights

TL;DR

This paper characterizes the r-summability of Carleson embeddings on weighted Fock spaces with $A_p$-type weights, and applies these results to operators like differentiation, integration, Volterra, and composition operators.

Contribution

It provides a complete characterization of r-summability of embeddings and operators on weighted Fock spaces with $A_p$-type weights, including cases previously unresolved.

Findings

Complete characterization of r-summability of embeddings for all r ≥ 1 and p > 1.

Results on the boundedness of Volterra-type and composition operators on vector-valued Fock spaces.

Extension of known results to the case 1 < p < 2 for certain operators.

Abstract

In this paper, we investigate the $r$-summing Carleson embeddings on weighted Fock spaces $F^p_{\alpha,w}$. By using duality arguments, translating techniques and block diagonal operator skills, we completely characterize the $r$-summability of the natural embeddings $I_d:F^p_{\alpha,w}\to L^p_{\alpha}(\mu)$ for any $r\geq1$ and $p>1$, where $w$ is a weight on the complex plane $\mathbb{C}$ that satisfies an $A_p$-type condition. As applications, we establish some results on the $r$-summability of differentiation and integration operators, Volterra-type operators and composition operators. Especially, we completely characterize the boundedness of Volterra-type operators and composition operators on vector-valued Fock spaces for all $1<p<\infty$, which were left open before for the case $1<p<2$.