Compact embedding derivatives of Hardy spaces into Lebesgue spaces

TL;DR

This paper characterizes measures for which the differentiation operator of any order is compact from Hardy spaces to Lebesgue spaces, advancing understanding of operator behavior between these function spaces.

Contribution

It provides a complete characterization of measures ensuring the compactness of differentiation operators from Hardy to Lebesgue spaces.

Findings

Identifies conditions on measures for compactness of differentiation operators

Extends previous results to higher order derivatives

Clarifies the relationship between Hardy and Lebesgue spaces

Abstract

We characterize the positive Borel measures such that the differentiation operator of order $n\in\mathbb{N}\cup\{0\}$ is compact from the Hardy space $H^p$ into $L^q(\mu)$, $0<p,q<\infty$.