The fractional Riesz transform and an exponential potential

In this paper we study the $s$-dimensional Riesz transform of a finite measure $\mu$ in $\mathbf R^d$, with $s\in(d-1,d)$. We show that the boundedness of the Riesz transform of $\mu$ yields a weak type estimate for the Wolff potential $\mathcal W_{\Phi,s}(\mu)(x)=\int_0^\infty\Phi\bigl(\frac{\mu(B(x,r))}{r^s}\bigl)\frac{dr}r$, where $\Phi(t)=e^{-1/t^\beta}$ with $\beta>0$ depending on $s$ and $d$. In particular, this weak type estimate implies that $\mathcal W_{\Phi,s}(\mu)$ is finite $\mu$-almost everywhere. As an application, we obtain an upper bound for the Calderón–Zygmund capacity $\gamma_s$ in terms of the non-linear capacity associated to the gauge $\Phi$. It appears to be the first result of this type for $s>1$.