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The Riesz Transform of Codimension Smaller Than One and the Wolff Energy

About this Title

Benjamin Jaye, Department of Mathematical Sciences, Kent State University, Kent, Ohio 44240, Fedor Nazarov, Department of Mathematical Sciences, Kent State University, Kent, Ohio 44240, Maria Carmen Reguera, School of Mathematics, University of Birmingham, Birmingham, United Kingdom and Xavier Tolsa, Institució Catalana de Recerca i Estudis Avançats (ICREA), Passeig de Lluís Companys, 23, 08010 Barcelona, Catalonia—and—Departament de Matemàtiques and BGSMath, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia

Publication: Memoirs of the American Mathematical Society
Publication Year: 2020; Volume 266, Number 1293
ISBNs: 978-1-4704-4213-2 (print); 978-1-4704-6249-9 (online)
Published electronically: July 21, 2020
MSC: Primary 42B37, 31B15

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Table of Contents


  • 1. Introduction
  • 2. Preliminaries
  • 3. The general scheme: Finding a large Lipschitz oscillation coefficient
  • 4. Upward and Downward Domination

Part I: The blow-up procedures

  • 5. Preliminary results regarding reflectionless measures
  • 6. The basic energy estimates
  • 7. Blow up I: The density drop
  • 8. The choice of the shell
  • 9. Blow up II: Doing away with $\varepsilon$
  • 10. Localization around the shell

Part II: The non-existence of an impossible object

  • 11. The scheme
  • 12. Suppressed kernels
  • 13. Step I: Calderón-Zygmund theory (From a distribution to an $L^p$-function)
  • 14. Step II: The smoothing operation
  • 15. Step III: The variational argument
  • 16. Contradiction


  • A. The maximum principle
  • B. The small boundary mesh
  • C. Lipschitz continuous solutions of the fractional Laplacian equation
  • D. Index of Selected Symbols and Terms


Fix $d\geq 2$, 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.

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