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.