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

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

Fix $d\geq 2$, and $s\in (d-1,d)$. 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. As an application, the authors give a metric characterization of the removable sets for locally Lipschitz continuous solutions of the fractional Laplacian operator $(-\Delta )^\alpha /2$, $\alpha \in (1,2)$, in terms of a well-known capacity from non-linear potential theory. This result contrasts sharply with removability results for Lipschitz harmonic functions.