## Abstract

Fix d ≥ 2, and s ∈ (d-1, d). We characterize the non-negative locally finite non-atomic Borel measures μ in R
^{d}for which the associated s-Riesz transform is bounded in L
^{2}(μ) 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 (-Δ)
^{α/2}, α ∈ (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.

Original language | English |
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Pages (from-to) | 1-110 |

Number of pages | 110 |

Journal | Memoirs of the American Mathematical Society |

Volume | 266 |

Issue number | 1293 |

Early online date | 21 Jul 2020 |

DOIs | |

Publication status | E-pub ahead of print - 21 Jul 2020 |

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics