Mon, 27 Feb 2023
LuboŇ° Pick
Charles University

In mathematical modelling, data and solutions are often represented as measurable functions, and their quality is being captured by their membership to a certain function space. One of the core questions arising in applications of this approach is the comparison of the quality of the data and that of the solution. A particular attention is being paid to optimality of the results obtained. A delicate choice of scales of suitable function spaces is required in order to balance the expressivity (the ability to capture fine mathematical properties of the model) and the accessibility (the level of its technical difficulty) for a practical use. We will give an overview of the research area which grew out of these questions and survey recent results obtained in this direction as well as challenging open questions. We will describe a development of a powerful method based on the so-called reduction principles and demonstrate its use on specific problems including the continuity of Sobolev embeddings or boundedness of pivotal integral operators such as the Hardy - Littlewood maximal operator and the Laplace transform.

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