Dimension free bounds for the Hardy--Littlewood maximal operator associated to convex sets

This survey is based on a series of lectures given by the authors at the working seminar "Convexit\'e et Probabilit\'es" at UPMC Jussieu, Paris, during the spring 2013. It is devoted to maximal inequalities associated to symmetric convex sets in high dimensional linear spaces, a topic mainly developed between 1982 and 1990 but recently renewed by further advances. The series focused on proving for these maximal functions inequalities in $L^p(\mathbb{R}^n)$ with bounds independent of the dimension $n$, for all $p \in (1, +\infty]$ in the best cases. This program was initiated in 1982 by Elias Stein, who obtained the first theorem of this kind for the family of Euclidean balls in arbitrary dimension. We present several results along this line, proved by Bourgain, Carbery and M\"uller during the period 1986--1990, and a new one due to Bourgain (2014) for the family of cubes in arbitrary dimension. We complete the cube case with negative results for the weak type $(1, 1)$ constant, due to Aldaz, Aubrun and Iakovlev--Str\"omberg between 2009 and 2013.