Square functions of fractional homogeneity and Wolff potentials

TL;DR

This paper establishes a comparison between Wolff energy and a related square function for measures in  when the fractional parameter is non-integer, and explores connections with Riesz transforms, highlighting differences from the integer case.

Contribution

It demonstrates that Wolff energy is comparable to a specific square function for non-integer s, revealing new insights into fractional homogeneity and potential theory.

Findings

Wolff energy is comparable to a related square function for non-integer s

The relation between Wolff energy and Riesz transforms is analyzed for 0<s<1

A counterexample shows the difference in behavior when s is an integer

Abstract

In this paper it is shown that for anymeasure $\mu$ in $\mathbb{R}^d$ and for a non-integer $0<s<d$, the Wolff energy $\displaystyle{\iint_0^\infty(\frac{\mu(B(x,r))}{r^s})^2\,\frac{dr}{r}d\mu(x)}$ is comparable to $$\iint_0^\infty(\frac{\mu(B(x,r))}{r^s} - \frac{\mu(B(x,2r))}{(2r)^s})^2\,\frac{dr}rd\mu(x),$$ unlike in the case when $s$ is an integer. We also study the relation with the $L^2-$norm of $s$-Riesz transforms, $0<s<1$, and we provide a counterexample in the integer case.