Spherical maximal functions and fractal dimensions of dilation sets
Spherical maximal functions and fractal dimensions of dilation sets
abstract: For the spherical mean operators $\scr{A}_t$ in $\Bbb{R}^d$, $d\ge 2$, we consider the maximal functions $M_Ef=\sup_{t\in E}|\scr{A}_t f|$, with dilation sets $E\subset [1,2]$. In this paper we give a surprising characterization of the closed convex sets which can occur as closure of the sharp $L^p$ improving region of $M_E$ …