Embedding and compact embedding between Bergman and Hardy spaces
Guanlong Bao, Pan Ma, Fugang Yan, Kehe Zhu
TL;DR
This work provides a complete characterization of when Hardy spaces $H^p$ and weighted Bergman spaces $A^p_\alpha$ embed into each other on the unit ball in $\mathbb{C}^n$, and when these embeddings are compact. It splits the analysis into two regimes, proving sharp conditions: for $0<p<q<\infty$, $A^p_\alpha\subset H^q$ holds precisely when $\frac{n+1+\alpha}{p}\le\frac{n}{q}$ and $H^p\subset A^q_\alpha$ holds when $\frac{n}{p}\le\frac{n+1+\alpha}{q}$, with compactness equivalent to strict inequalities; for $0<q\le p<\infty$, the criteria involve thresholds at $\alpha=-1$ depending on $q$ and $p$. The paper also develops a Carleson-measure–type framework to handle the $q<p$ case and introduces tight fittings, conjecturing a complete description of contractive, non-compact embeddings and their extremal functions, extending known one-dimensional results to higher dimensions. These findings unify and extend classical results (e.g., Carleman, Beatrous–Burbea) and have implications for operator theory and several complex variables. The work thus provides a definitive map of when Hardy and Bergman spaces sit inside one another, including the subtle boundary cases and the nature of extremal embeddings.
Abstract
For Hardy spaces and weighted Bergman spaces on the open unit ball in ${\mathbb C}^n$, we determine exactly when $A^p_α\subset H^q$ or $H^p\subset A^q_α$, where $0<q<\infty$, $0<p<\infty$, and $-\infty<α<\infty$. 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.