Square functions of fractional homogeneity and Wolff potentials

In this paper it is shown that for anymeasure $μ$ in $\mathbb{R}^d$ and for a non-integer $0<s<d$, the Wolff energy $\displaystyle{\iint_0^\infty(\frac{μ(B(x,r))}{r^s})^2\,\frac{dr}{r}dμ(x)}$ is comparable to $\iint_0^\infty(\frac{μ(B(x,r))}{r^s} - \frac{μ(B(x,2r))}{(2r)^s})^2\,\frac{dr}rdμ(x),$ 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.

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