Embedding and compact embedding between Bergman and Hardy spaces

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