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

Maria Carmen Reguera, Xavier Tolsa, Benjamin Jaye, Fedor Nazarov

ISBN 10: 1470442132 / ISBN 13: 9781470442132

Verlag: American Mathematical Society, US, 2020

Beschreibung

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.

Bestandsnummer des Verkäufers LU-9781470442132

Bibliografische Details

• Titel: The Riesz Transform of Codimension Smaller Than One and the Wolff Energy
• Verlag: American Mathematical Society, US
• Erscheinungsjahr: 2020
• Sprache: Englisch
• ISBN-10: 1470442132
• ISBN-13: 9781470442132
• Einband: Paperback
• Zustand: New
• Gewicht: 210 Gramm

Über diesen Titel

Inhaltsangabe

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.

Über die Autorin bzw. den Autor

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