Embedding derivatives and derivative Area operators of Hardy spaces into Lebesgue spaces

TL;DR

This paper characterizes when derivatives of Hardy space functions can be embedded into Lebesgue spaces and analyzes the boundedness and compactness of derivative area operators, extending results to higher dimensions using tent space theory.

Contribution

It provides a complete characterization of the boundedness and compactness of derivative embeddings and area operators from Hardy spaces to Lebesgue spaces, especially in higher dimensions.

Findings

Characterization of compactness of derivative embeddings from Hardy to Lebesgue spaces.

Complete characterization of boundedness and compactness of derivative area operators.

Extension of one-dimensional results to higher dimensions using tent space theory.

Abstract

We characterize the compactness of embedding derivatives from Hardy space $H^p$ into Lebesgue space $L^q(\mu)$. We also completely characterize the boundedness and compactness of derivative area operators from $H^p$ into $L^q(\mathbb{S}_n)$, $0<p, q<\infty$. Some of the tools used in the proof of the one-dimensional case are not available in higher dimensions, such as the strong factorization of Hardy spaces. Therefore, we need the theory of tent spaces which was established by Coifman, Mayer and Stein in 1985.