Embedding and compact embedding between Bergman and Hardy spaces

TL;DR

This paper precisely characterizes when Hardy and weighted Bergman spaces on the unit ball are embedded into each other, including conditions for compactness, and introduces a new concept called tight fitting to unify known results.

Contribution

It provides a complete classification of embedding and compact embedding conditions between Hardy and Bergman spaces, covering all cases and introducing the novel notion of tight fitting.

Findings

Exact conditions for space embeddings and compactness.

Complete coverage of all embedding cases.

Introduction of the tight fitting concept and related conjecture.

Abstract

For Hardy spaces and weighted Bergman spaces on the open unit ball in ${\mathbb C}^n$, we determine exactly when $A^p_\alpha\subset H^q$ or $H^p\subset A^q_\alpha$, where $0<q<\infty$, $0<p<\infty$, and $-\infty<\alpha<\infty$. For each such inclusion we also determine exactly when it is a compact embedding. Although some special cases were known before, we are able to completely cover all possible cases here. We also introduce a new notion called {\it tight fitting} and formulate a conjecture in terms of it, which places several prominent known results about contractive embeddings in the same framework.