Talk by Burat Hikmet Özcan
On July 17th, 2024, Burat Hikmet Özcan (Izmir Institute of Technology) gave a talk about "Hardy-Littlewood Maximal Function: Structural Results and Regularity Properties" as part of the research seminar Analysis of the FernUniversität in Hagen. This lecture is partially supported by the COST action Mathematical models for interacting dynamics on networks.
Abstract
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We define the Hardy-Littlewood maximal function for a locally integrable function f : R^n → R, where B_r(x) is the ball of radius r centered at x, and |B_r(x)| denotes its Lebesgue measure.
In this talk, we will focus on two main aspects: the structure of the maximal function and its regularity properties. First, we will discuss the fundamental theorem related to the Hardy-Littlewood Maximal Function and then explore its application through the Lebesgue Differentiation Theorem. Next, we will delve into the regularity theory of maximal operators in both continuous and discrete settings, summarizing key developments in this area over the past 20 years. Finally, I will present our new contributions to understanding the higher order regularity of the discrete non-centered maximal function.
