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

Jiale Chen, Bo He, Maofa Wang

Abstract

In this paper, we investigate the $r$-summing Carleson embeddings on weighted Fock spaces $F^p_{α,w}$. By using duality arguments, translating techniques and block diagonal operator skills, we completely characterize the $r$-summability of the natural embeddings $I_d:F^p_{α,w}\to L^p_α(μ)$ 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 $1<p<2$.

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

Abstract

In this paper, we investigate the -summing Carleson embeddings on weighted Fock spaces . By using duality arguments, translating techniques and block diagonal operator skills, we completely characterize the -summability of the natural embeddings for any and , where is a weight on the complex plane that satisfies an -type condition. As applications, we establish some results on the -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 , which were left open before for the case .

Paper Structure

This paper contains 11 sections, 26 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Absolutely summing operators
  4. Estimates involving weights
  5. Proof of Theorem \ref{['main']}
  6. The case $1<p<2$
  7. The case $p\geq2$
  8. Applications
  9. Differentiation and integration operators
  10. Volterra-type operators
  11. Composition operators

Key Result

Theorem 1.2

Let $\alpha>0$ and $\mu$ be a positive Borel measure on $\mathbb{C}$. Here, $\widehat{\mu}_{w}(z)=\frac{\mu(D(z,1))}{w(D(z,1))}$ for $z\in\mathbb{C}$.

Theorems & Definitions (43)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 2.1
  • proof
  • ...and 33 more