Optimality problems in Orlicz spaces

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Authors

MUSIL Vít PICK Luboš TAKÁČ Jakub

Year of publication 2023
Type Article in Periodical
Magazine / Source Advances in Mathematics
MU Faculty or unit

Faculty of Informatics

Citation
Web https://www.sciencedirect.com/science/article/pii/S0001870823004164
Doi http://dx.doi.org/10.1016/j.aim.2023.109273
Keywords Orlicz space; Rearrangement-invariant space; Optimality; Sobolev embedding; Maximal operator; Laplace transform
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Description 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. (c) 2023 Elsevier Inc. All rights reserved.
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