ON SHARP RESULTS ON COEFFICIENT MULTIPLIERS OF HARMONIC FUNCTION SPACES OF SEVERAL VARIABLES AND RELATED PROBLEMS

Abstract

The first sharp results concerning coefficient multipliers in various spaces of harmonic functions in the unit disk were obtained by A. Shields and D. Williams several decades ago. Since then, extending these theorems to higher-dimensional harmonic function spaces remained an open problem for a long time. In 1998, R. F. Shamoyan provided the first such sharp results in higher dimensions using spherical harmonics. These results were subsequently extended in various directions to harmonic function spaces on the unit ball by R. F. Shamoyan and M. Arsenović, as well as by E. Malinnikova and A. Eikrem.

The main goal of this note is to collect and survey these sharp results within a single unified framework. In the second part of this note, we also briefly review recent advances on this topic for analytic function spaces of several variables.