$L^2$-norm and estimates from below for Riesz transforms on Cantor sets

TL;DR

This paper provides lower bounds for the $L^2$-norms of Riesz transforms on Cantor sets, demonstrating the distribution of their values and establishing sharpness of previous estimates for these fractal sets.

Contribution

It introduces a new regularization process for Cantor sets and analyzes the distribution and norms of Riesz transforms on these sets, extending prior work.

Findings

Values of $|R_{ u}^s|^2$ are uniformly distributed on large portions of the Cantor set.

Established lower bounds for $L^2$-norms of Riesz transforms on these sets.

Provided examples showing the sharpness of earlier estimates.

Abstract

The aim of this paper is to estimate the $L^2$-norms of vector-valued Riesz transforms $R_{\nu}^s$ and the norms of Riesz operators on Cantor sets in $\R^d$, as well as to study the distribution of values of $R_{\nu}^s$. Namely, we show that this distribution is "uniform" in the following sense. The values of $|R_{\nu}^s|^2$ which are comparable with its average value are attended on a "big" portion of a Cantor set. We apply these results to give examples demonstrating the sharpness of our previous estimates for the set of points where Riesz transform is large, and for the corresponding Riesz capacities. The Cantor sets under consideration are different from the usual corner Cantor sets. They are constructed by means a certain process of regularization introduced in the paper.