# Public defence in Mathematics, M.Sc. Julian Weigt

How strongly can maximal functions oscillate?

Opponent: Doctor Emanuel Carneiro, ICTP, Italy
Custos: Professor Juha Kinnunen, Aalto University School of Science, Department of Mathematics and Systems Analysis

The public defence will be organised on campus (Otakaari 1, lecture hall U147 U5).

The thesis is publicly displayed 10 days before the defence in the publication archive Aaltodoc of Aalto University
Electronic thesis can be found here: https://aaltodoc.aalto.fi/handle/123456789/117016

Public defence announcement:

Title of the doctoral thesis: Endpoint regularity of maximal functions in higher dimensions

The variation of a function is a quantity that measures how strongly that function oscillates. It is one of many ways to describe the regularity of a function, or how smooth and well-behaved the function looks. Studying the regularity of functions is one of the main goals in the area of partial differential equations. Almost all systems in physics can be modeled by differential equations, and also quantities in other fields such as chemistry, biology or economics satisfy differential equations. This is why we are interested in the regularity of functions, it can be important to know if the path of a particle, or certain substance concentrations or market indicators can behave erratically or always follow a smooth curve.

In this thesis we investigate one particular type of regularity of so-called maximal functions: We prove that some maximal functions have bounded variation. Maximal functions are classically used as tools in the study of partial differential equations and in related mathematical areas. A maximal function is defined using maxima over averages of a function. The average is of course a very important mathematical notion, and thus features crucially not only in partial differential equations and but also in many other fields of mathematics. This is why the maximal function is often used to estimate functions in mathematical analysis.

The regularity of maximal functions has now been studied for about twenty-five years. It has been shown that maximal functions in one dimensional space have bounded variation, and it has been shown that they satisfy several notions of regularity in all dimensions. This thesis proves the natural conjecture that maximal functions in higher dimensions also have bounded variation, at least for some maximal functions, and opens the field to investigate this conjecture for further classical maximal functions.

Contact information of doctoral candidate: [email protected], +358504754400

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