Embedding and compact embedding between Bergman and Hardy spaces

Abstract

For Hardy spaces and weighted Bergman spaces on the open unit ball in ${\mathbb{C}}^n$, we determine exactly when $A_{\hat{I}}^p\pm \subset H^q$ or $H^p\subset A_{\hat{I}}^q\pm$, where $0<q<\infty$, $0<p<\infty$, and $-\infty <\hat{I}\pm <\infty$. 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.

21 pages, 1 figure