Sobolev Regularity for Maximal Operators

Explore the recent developments in regularity theory for maximal operators acting on Sobolev or BV data in this 43-minute lecture by Emanuel Carneiro. Gain insights into this active research topic, starting with an introduction and illustrative examples. Delve into the Hardy-Littlewood maximal operator, Sobolev regularity, and progress made in recent years for both one-dimensional and higher-dimensional cases. Examine maximal functions of convolution type, fractional Hardy-Littlewood maximal operators, and discrete analogues. Learn about the strategy outline for tackling these problems and understand the continuity aspects. Clarify questions related to continuous cases and explore various kernels. This accessible talk is designed for a broad audience interested in mathematical analysis and operator theory.

Intro
An illustration
A basic idea...
A deceiving question
The Hardy-Littlewood maximal operator
Sobolev regularity
Progress over last years: dimension d = 1
Progress over last years: dimension d 1
Maximal functions of convolution type
Fractional Hardy-Littlewood maximal operator
Discrete analogues
Strategy outline
Progress in higher dimensions
Continuity
Clarifying the questions - continuous cases
Kernels