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

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  • University of Kent, Ohio
  • Universitat Autònoma de Barcelona


Fix d ≥ 2, and s ∈ (d-1, d). We characterize the non-negative locally finite non-atomic Borel measures μ in R dfor 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 languageEnglish
Pages (from-to)1-110
Number of pages110
JournalMemoirs of the American Mathematical Society
Issue number1293
Early online date21 Jul 2020
Publication statusE-pub ahead of print - 21 Jul 2020

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