On the variation and asymptotics of maximal functions

Abstract

In this paper we present partial progress towards two conjectures on the variation of maximal operators of convolution type $\mathrm{M}_φ$, associated to non-negative and radial decreasing kernels $φ$ of integral $1$. In dimension $d=1$ we prove that $\mathrm{var}(\mathrm{M}_φ f) \leq \mathrm{var}(f)$ when $f$ is a characteristic function. In higher dimensions we prove that $\mathrm{var}(\mathrm{M}_φ f) \leq C_d \mathrm{var}(f)$ when $f$ is a radial characteristic function, with a constant $C_d$ depending only on the dimension $d$ but not on the kernel $φ$. We further show that $C_d \sim \sqrt{d}\log d $ is the sharp dependence on $d$ for such an inequality. This last claim relies on a novel explicit asymptotic estimate for the Hardy-Littlewood maximal function of the unit ball in $\mathbb R^d$, a result of independent interest.