Dahlberg's theorem in higher co-dimension

Guy David, Joseph Feneuil, Svitlana Mayboroda

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Abstract

In 1977 the celebrated theorem of B. Dahlberg established that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a Lipschitz graph of dimension n−1 in R n , and later this result has been extended to more general non-tangentially accessible domains and beyond. In the present paper we prove the first analogue of Dahlberg's theorem in higher co-dimension, on a Lipschitz graph Γ of dimension d in R n , d<n−1, with a small Lipschitz constant. We construct a linear degenerate elliptic operator L such that the corresponding harmonic measure ω L is absolutely continuous with respect to the Hausdorff measure on Γ. More generally, we provide sufficient conditions on the matrix of coefficients of L which guarantee the mutual absolute continuity of ω L and the Hausdorff measure.

Original languageEnglish (US)
Pages (from-to)2731-2820
Number of pages90
JournalJournal of Functional Analysis
Volume276
Issue number9
DOIs
StatePublished - May 1 2019

Bibliographical note

Funding Information:
Mayboroda is supported in part by the Alfred P. Sloan Fellowship, the NSF INSPIRE Award DMS 1344235, NSF CAREER Award DMS 1220089. Svitlana Mayboroda was also supported by: - the NSF RAISE-TAG grant DMS 1839077 (USA), - the Simons Fellowship (no grant number), - the Simons Foundation grant 563916, SM (USA). David is supported in part by the ANR, programme blanc GEOMETRYA ANR-12-BS01-0014. Guy David was also supported by: - the FP7 grant MANET 607643 (European Union), - the H2020 grant GHAIA 777822 (European Union), the Simons Foundation grant 601941, GD (USA). This work was supported by a public grant as part of the Investissement d'avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH. Part of this work was completed during Mayboroda's visit to Universit? Paris-Sud, Laboratoire de Math?matiques, Orsay, and Ecole Polytechnique, PMC. We thank the corresponding Departments, together with the Fondation Jacques Hadamard, the ?cole des Mines, and the Mathematical Sciences Research Institute (NSF grant DMS 1440140) for support and hospitality.

Funding Information:
Mayboroda is supported in part by the Alfred P. Sloan Fellowship, the NSF INSPIRE Award DMS 1344235 , NSF CAREER Award DMS 1220089 . Svitlana Mayboroda was also supported by: - the NSF RAISE-TAG grant DMS 1839077 (USA), - the Simons Fellowship (no grant number), - the Simons Foundation grant 563916 , SM (USA). David is supported in part by the ANR , programme blanc GEOMETRYA ANR-12-BS01-0014 . Guy David was also supported by: - the FP7 grant MANET 607643 (European Union), - the H2020 grant GHAIA 777822 (European Union), the Simons Foundation grant 601941 , GD (USA). This work was supported by a public grant as part of the Investissement d'avenir project, reference ANR-11-LABX-0056-LMH , LabEx LMH . Part of this work was completed during Mayboroda's visit to Université Paris-Sud , Laboratoire de Mathématiques, Orsay, and Ecole Polytechnique, PMC . We thank the corresponding Departments, together with the Fondation Jacques Hadamard , the École des Mines , and the Mathematical Sciences Research Institute ( NSF grant DMS 1440140 ) for support and hospitality.

Publisher Copyright:
© 2019 Elsevier Inc.

Keywords

  • Boundary with co-dimension higher than 1
  • Dahlberg's theorem
  • Degenerate elliptic operators
  • Harmonic measure in higher codimension

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