The fractional Riesz transform and an exponential potential

Abstract:

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 $\mathcal {W}_{\Phi ,s}(\mu )(x) = \int _0^{\infty }\Phi \bigl (\frac {\mu (B(x,r))}{r^s}\bigr ) \frac {dr}{r},$ 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 nonlinear capacity associated to the gauge $\Phi$. It appears to be the first result of this type for $s>1$.