On some new embeddings in minimal bounded homogeneous domains in $C^n$

TL;DR

This paper explores new embeddings and sharp results for Bergman spaces in minimal bounded homogeneous domains, generalizing known results from symmetric domains and utilizing recent lattice properties.

Contribution

It introduces new embeddings and sharp estimates for Bergman spaces in minimal bounded homogeneous domains, extending previous results to more general settings.

Findings

New sharp results on Bergman type spaces in minimal bounded homogeneous domains.

Extension of embedding results from Yamaji's work to broader classes of domains.

Utilization of recent properties of r-lattices for these domains.

Abstract

In this short note we consider very general bounded minimal homogeneous domains. Under certain natural additional conditions new sharp results on Bergman type analytic spaces in minimal bounded homogeneous domains are obtained. Domains we consider here are direct generalizations of the well-studied so-called bounded symmetric domains in $C^n.$ In the unit disk and in the unit ball all our results were obtained by first author. Some results were obtained previously in tubular and bounded pseudoconvex domains. Our proofs are heavily based on properties of so called r-lattices for these general domains provided in recent papers of Yamaji. Our proofs are also based on arguments provided earlier in less general domains. We partially extend an embedding result from Yamaji's paper.