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In this paper it is shown that, for any measure   $\mu $   in   ${\mathbb {R}}^d$   and for a non-integer   $0&lt;s&lt;d$  , the Wolff energy   $\iint _0^\infty (\frac {\mu (B(x,r))}{r^s})^2\,\frac {{\rm d} r}{r}\,{\rm d}\mu (x)$   is comparable to   \[ \iint_0^\infty\left(\frac{\mu(B(x,r))}{r^s}- \frac{\mu(B(x,2r))}{(2r)^s}\right)^2\,\frac{{\rm d} r}{r}\,{\rm d}\mu(x), \]   unlike in the case when   $s$   is an integer. We also study the relation with the   $L^2$  -norm of   $s$  -Riesz transforms,   $0&lt;s&lt;1$  , and we provide a counterexample in the integer case.

Square Functions of Fractional Homogeneity and Wolff Potentials