On the variation of maximal operators of convolution type II

Abstract

In this paper we establish that several maximal operators of convolution type, associated to elliptic and parabolic equations, are variation-diminishing. Our study considers maximal operators on the Euclidean space ${\mathbb{R}}^d$, on the torus ${\mathbb{T}}^d$ and on the sphere ${\mathbb{S}}^d$. The crucial regularity property that these maximal functions share is that they are subharmonic in the corresponding detachment sets.