The fractional Riesz transform and an exponential potential

TL;DR

This paper establishes that the boundedness of the s-dimensional Riesz transform of a measure implies the finiteness of a nonlinear exponential potential almost everywhere, extending previous results to the case where s>1.

Contribution

It presents the first result linking Riesz transform boundedness to exponential potentials for s in (d-1,d), broadening understanding in harmonic analysis.

Findings

Bounded Riesz transform implies finite exponential potential almost everywhere.

First such result for s>1 in this context.

Advances the theory connecting singular integrals and nonlinear potentials.

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 $s>1$.