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 $\mu$ implies that a nonlinear potential of exponential type is finite $\mu$-almost everywhere. It appears to be the first result of this type for $s>1$.

48 pages, 2 figures