Overview

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.

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9781470442132

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Benjamin Jaye, Kent State University, OH Fedor Nazarov, Kent State University, OH Maria Carmen Reguera, University of Birmingham, UK Xavier Tolsa, Institucio Catalana de Recerca i Estudis Avancats, Barcelona, Catalonia, Spain, and Universitat Autonoma de Barcelona, Catalonia, Spain

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