In this note we describe some recent advances in the area of maximal function inequalities. We also study the behaviour of the centered Hardy--Littlewood maximal operator associated to certain families of doubling, radial decreasing measures, and acting on radial functions. In fact, we precisely determine when the weak type $(1,1)$ bounds are uniform in the dimension.
"On high dimensional maximal operators." Banach J. Math. Anal. 7 (2) 225 - 243, 2013. https://doi.org/10.15352/bjma/1363784233