The Riesz transform of codimension smaller than one and the Wolff energy

Abstract

Fix $d\ge 2$, and $s\in (d-1,d)$. We 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, we 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.

arXiv comment85 pages. In this version the removability result for the fractional Laplacian is local

Cited by in corpus (2)

• The measures with $L^2$-bounded Riesz transform and the Painlevé problem for Lipschitz harmonic functions
• The measures with $L^2$-bounded Riesz transform satisfying a subcritical Wolff-type energy condition