The precise representative for the gradient of the Riesz potential of a finite measure

Julià Cufí; Augusto C. Ponce; Joan Verdera

doi:10.1112/jlms.12607

The precise representative for the gradient of the Riesz potential of a finite measure

Julià Cufí, Augusto C. Ponce^*, Joan Verdera

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

1 Citation (Scopus)

Abstract

Given a finite nonnegative Borel measure (Formula presented.) in (Formula presented.), we identify the Lebesgue set (Formula presented.) of the vector-valued function (Formula presented.) for any order (Formula presented.). We prove that (Formula presented.) if and only if the integral above has a principal value at (Formula presented.) and (Formula presented.) In that case, the precise representative of (Formula presented.) at (Formula presented.) coincides with the principal value of the integral. We also study the existence of Lebesgue points for the Cauchy integral of the intrinsic probability measure associated with planar Cantor sets, which leads to challenging new questions.

Original language	English
Pages (from-to)	1603-1627
Number of pages	25
Journal	Journal of the London Mathematical Society
Volume	106
Issue number	2
DOIs	https://doi.org/10.1112/jlms.12607
Publication status	Published - Sept 2022

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title = "The precise representative for the gradient of the Riesz potential of a finite measure",

abstract = "Given a finite nonnegative Borel measure (Formula presented.) in (Formula presented.), we identify the Lebesgue set (Formula presented.) of the vector-valued function (Formula presented.) for any order (Formula presented.). We prove that (Formula presented.) if and only if the integral above has a principal value at (Formula presented.) and (Formula presented.) In that case, the precise representative of (Formula presented.) at (Formula presented.) coincides with the principal value of the integral. We also study the existence of Lebesgue points for the Cauchy integral of the intrinsic probability measure associated with planar Cantor sets, which leads to challenging new questions.",

author = "Juli{\`a} Cuf{\'i} and Ponce, {Augusto C.} and Joan Verdera",

note = "Funding Information: The authors would like to thank L. Grafakos and P. Mattila for an interesting correspondence. A. C. Ponce is grateful for the invitation, hospitality, and support of the Departament de Matem{\`a}tiques of the Universitat Aut{\`o}noma de Barcelona. Publisher Copyright: {\textcopyright} 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.",

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doi = "10.1112/jlms.12607",

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AU - Cufí, Julià

AU - Ponce, Augusto C.

AU - Verdera, Joan

N1 - Funding Information: The authors would like to thank L. Grafakos and P. Mattila for an interesting correspondence. A. C. Ponce is grateful for the invitation, hospitality, and support of the Departament de Matemàtiques of the Universitat Autònoma de Barcelona. Publisher Copyright: © 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.

PY - 2022/9

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N2 - Given a finite nonnegative Borel measure (Formula presented.) in (Formula presented.), we identify the Lebesgue set (Formula presented.) of the vector-valued function (Formula presented.) for any order (Formula presented.). We prove that (Formula presented.) if and only if the integral above has a principal value at (Formula presented.) and (Formula presented.) In that case, the precise representative of (Formula presented.) at (Formula presented.) coincides with the principal value of the integral. We also study the existence of Lebesgue points for the Cauchy integral of the intrinsic probability measure associated with planar Cantor sets, which leads to challenging new questions.

AB - Given a finite nonnegative Borel measure (Formula presented.) in (Formula presented.), we identify the Lebesgue set (Formula presented.) of the vector-valued function (Formula presented.) for any order (Formula presented.). We prove that (Formula presented.) if and only if the integral above has a principal value at (Formula presented.) and (Formula presented.) In that case, the precise representative of (Formula presented.) at (Formula presented.) coincides with the principal value of the integral. We also study the existence of Lebesgue points for the Cauchy integral of the intrinsic probability measure associated with planar Cantor sets, which leads to challenging new questions.

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JF - Journal of the London Mathematical Society

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The precise representative for the gradient of the Riesz potential of a finite measure

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