The fractional Riesz transform and an exponential potential

In this paper we study the $s$-dimensional Riesz transform of a finite measure $μ$ in $\mathbf{R}^d$, with $s\in (d-1,d)$. We show that the boundedness of the Riesz transform of $μ$ implies that a nonlinear potential of exponential type is finite $μ$-almost everywhere. It appears to be the first result of this type for $s>1$.

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