  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.
