Convexity of the embedding parameter sets of some analytic function spaces

TL;DR

This paper investigates the geometric structure of parameter sets for embeddings between weighted Bergman-Orlicz spaces, demonstrating convexity properties under certain conditions and exploring interpolation results.

Contribution

It establishes convexity of admissible parameter sets and log-convexity of growth function pairs for embeddings, advancing understanding of the geometric structure of these spaces.

Findings

The set of admissible weight exponents is convex under specific conditions.

Growth function pairs that produce embeddings form a log-convex collection.

Interpolated embeddings between Bergman-Orlicz spaces are obtained.

Abstract

In this note, we study the geometric structure of the parameter sets governing continuous embeddings between weighted Bergman-Orlicz spaces. First, for a fixed pair of growth functions, we show that the set of admissible weight exponents $(\alpha, \beta)$ is convex, provided the growth functions satisfy specific log-convexity and log-concavity conditions of the inverses. Second, we consider the dual problem where the weight exponents are fixed. We prove that the collection of growth function pairs that yield such an embedding is log-convex under a natural interpolation of their inverses. We then obtain interpolated embeddings between Bergman-Orlicz spaces.