On the Hardy--Littlewood maximal functions in high dimensions: Continuous and discrete perspective

Abstract

This is a survey article about recent developments in dimension-free estimates for maximal functions corresponding to the Hardy--Littlewood averaging operators associated with convex symmetric bodies in ${\mathbb{R}}^d$ and ${\mathbb{Z}}^d$.

24 pages, no figures. Dedicated to Fulvio Ricci on the occasion of his 70th birthday. Revised following the referee report