A dimension-free estimate on $L^2$ for the maximal Riesz transform in terms of the Riesz transform

Abstract

W prove a dimension-free estimate for the $L^2({\mathbb{R}}^d)$ norm of the maximal truncated Riesz transform in terms of the $L^2({\mathbb{R}}^d)$ norm of the Riesz transform. Consequently, the vector of maximal truncated Riesz transforms has a dimension-free estimate on $L^2({\mathbb{R}}^d)$. We also show that the maximal function of the vector of truncated Riesz transforms has a dimension-free estimate on all $L^p({\mathbb{R}}^d)$ spaces, $1<p<\infty .$

18 pages