# FALL 2021 Nonlinear Analysis Special Lecture Part 2 of 2

### Date

Wednesday, December 15, 2021 - 15:00 to 17:30

Online via Zoom

### Description

Mr. Julian Weigt, Aalto University

### 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.