On embedding theorems in analytic and harmonic function spaces of several variables in some domains in $C^n$

TL;DR

This survey reviews recent sharp embedding theorems in analytic and harmonic function spaces of several variables, extending classical results to higher dimensions and unbounded domains with new sharpness and generalizations.

Contribution

It provides new sharp embedding results for analytic and harmonic function spaces in higher dimensions and unbounded domains, extending classical one-dimensional theorems.

Findings

New sharp embedding theorems in higher-dimensional domains

Extensions of classical one-dimensional embedding results

Analysis of properties of $r$-lattices in $C^n$ domains

Abstract

In this survey we collect some recent advances concerning embedding theorems in analytic and harmonic function spaces of several variables in various domains in $C^n.$ Some sharp embedding results presented in this survey paper extend sharp embeddings in analytic function spaces obtained in higher dimension previously by J. Ortega, J. Fabrega, J. Cima, D. Luecking, M. Abate and many others in recent decades in the unit ball and bounded strongly pseudoconvex domains with the smooth boundary for onefunctional case to similar type analytic multyfunctional Bergman type function spaces. Same type multifunctional sharp extensions of some recent embedding results of one functional analytic function spaces of B. Sehba and D. Bekolle will be provided by us in this survey for unbounded domains namely for tubular domains over symmetric cones.In this paper we collect many new sharp embeddings in higher dimension obtained recently by first author though some one dimensional new sharp embeddings will be also mentioned shortly at the end of paper. Our results can be seen as direct sharp extensions of various sharp embedding theorems in analytic spaces in the unit disk obtained many years ago by L. Carleson, P. Duren, I. Verbitsky, D. Luecking and many others. Nice properties of so called $r$-lattices in various domains in $C^n$ are very important in all proofs of theorems of this paper.