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

TL;DR

This paper establishes a dimension-free bound for the maximal truncated Riesz transform in L^2, and extends this to all L^p spaces, improving understanding of the transform's behavior across dimensions.

Contribution

It provides the first dimension-free L^2 estimate for the maximal truncated Riesz transform and extends the result to all L^p spaces for 1<p<∞.

Findings

Dimension-free L^2 estimate for maximal truncated Riesz transform

Dimension-free estimate for vector of maximal truncated Riesz transforms in L^2

Dimension-free estimate for maximal function of truncated Riesz transforms in all L^p spaces

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, $1<p<\infty.$