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Carleson-type embeddings with closed range

Konstantin M. Dyakonov

Abstract

We characterize the Carleson measures $μ$ on the unit disk for which the image of the Hardy space $H^p$ under the corresponding embedding operator is closed in $L^p(μ)$. In fact, a more general result involving $(p,q)$-Carleson measures is obtained. A similar problem is solved in the setting of Bergman spaces.

Carleson-type embeddings with closed range

Abstract

We characterize the Carleson measures on the unit disk for which the image of the Hardy space under the corresponding embedding operator is closed in . In fact, a more general result involving -Carleson measures is obtained. A similar problem is solved in the setting of Bergman spaces.

Paper Structure

This paper contains 4 sections, 7 theorems, 45 equations.

Table of Contents

  1. Introduction and statement of results
  2. Preliminaries
  3. Proof of Theorem \ref{['thm:hardy']}
  4. Proofs of Theorem \ref{['thm:bergman']} and Proposition \ref{['prop:discon']}

Key Result

Theorem 1.1

Let $0<p,q<\infty$ and let $\mu$ be a $(p,q)$-Carleson measure supported on an infinite subset of ${\mathbb D}$. The following are equivalent: (i) The range of the embedding operator eqn:embpqhar is closed in $L^q(\mu)$; (ii) $p=q$ and $\mu$ has the form $\sum_na_n\delta_{z_n}$, where $\{z_n\}$ is a

Theorems & Definitions (9)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3