Optimality problems in Orlicz spaces

TL;DR

This paper develops a unified approach to determine the existence of optimal Orlicz spaces for various mathematical problems, balancing expressivity and accessibility in function space analysis.

Contribution

It introduces a general principle providing necessary and sufficient conditions for optimality in Orlicz spaces across multiple applications.

Findings

Established a verifiable criterion for optimality in Orlicz spaces.

Applied the principle to Sobolev embeddings and integral operators.

Demonstrated the approach's versatility in different analysis problems.

Abstract

In mathematical modelling, the data and solutions are represented as measurable functions and their quality is oftentimes captured by the membership to a certain function space. One of the core questions for an analysis of a model is the mutual relationship between the data and solution quality. The optimality of the obtained results deserves a special focus. It requires a careful choice of families of function spaces balancing between their expressivity, i.e. the ability to capture fine properties of the model, and their accessibility, i.e. its technical difficulty for practical use. This paper presents a unified and general approach to optimality problems in Orlicz spaces. Orlicz spaces are parametrized by a single convex function and neatly balance the expressivity and accessibility. We prove a general principle that yields an easily verifiable necessary and sufficient condition for the existence or the non-existence of an optimal Orlicz space in various tasks. We demonstrate its use in specific problems, including the continuity of Sobolev embeddings and boundedness of integral operators such as the Hardy--Littlewood maximal operator and the Laplace transform.