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 ${\mathrm{M}}_{\phi }$, associated to non-negative and radial decreasing kernels $\phi$ of integral $1$. In dimension $d=1$ we prove that $\mathrm{var}({\mathrm{M}}_{\phi }f)\le \mathrm{var}(f)$ when $f$ is a characteristic function. In higher dimensions we prove that $\mathrm{var}({\mathrm{M}}_{\phi }f)\le C_d\mathrm{var}(f)$ when $f$ is a radial characteristic function, with a constant $C_d$ depending only on the dimension $d$ but not on the kernel $\phi$. 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.

arXiv comment36 pages, 7 figures