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

Abstract

In this paper, we investigate the $r$-summing Carleson embeddings on weighted Fock spaces $F_{\hat{I}\pm ,w}^p$. By using duality arguments, translating techniques and block diagonal operator skills, we completely characterize the $r$-summability of the natural embeddings $I_d:F_{\hat{I}\pm ,w}^p\to L_{\hat{I}}^p\pm (\hat{I}¼)$ for any $r\ge 1$ 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 $1<p<2$.

To appear in Journal of Operator Theory