The Riesz Transform of Codimension Smaller Than One and the Wolff Energy

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The authors characterize the non-negative locally finite non-atomic Borel measures $\\\mu $ in $\\\mathbb R^d$ for which the associated $s$-Riesz transform is bounded in $L^2(\\\mu )$ in terms of the Wolff energy. This extends the range of $s$ in which the Mateu-Prat-Verdera characterization of measures with bounded $s$-Riesz transform is known.