Square functions of fractional homogeneity and Wolff potentials

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 ${\iint }_0^{\infty }(\frac{\mu (B(x,r))}{r^s}{)}^2\frac{dr}{r}d\mu (x)$ is comparable to $${\iint }_0^{\infty }(\frac{\mu (B(x,r))}{r^s}-\frac{\mu (B(x,2r))}{(2r{)}^s}{)}^2\frac{dr}{r}d\mu (x),$$ 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.

20 pages, 4 figures