FALL 2021 Nonlinear Analysis Special Lecture Part 2 of 2

Mr. Julian Weigt, Aalto University

Title: Higher dimensional techniques for the regularity of maximal functions

Abstract:

It has been an open question if maximal operators M satisfy the endpoint regularity bound \(mathop{\mathrm{var}}(Mf) \leq C \mathop{\mathrm{var}}(f)\). So far the majority of the known results has been in one dimension. I give an overview of the progress on this question with a focus on the techniques. Next I present the techniques used in the recent proofs of \(mathop{\mathrm{var}}(Mf) \leq C \mathop{\mathrm{var}}(f)\) for several maximal operators in higher dimensions. They are mostly geometric measure theoretic in the spirit of the relative isoperimetric inequality and involve a stopping time and various covering arguments.

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