On first and second eigenvalues of Riesz transforms in spherical and hyperbolic geometries

Michael Ruzhansky, Durvudkhan Suragan

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

In this note we prove an analogue of the Rayleigh–Faber–Krahn inequality, that is, that the geodesic ball is a maximiser of the first eigenvalue of some convolution type integral operators, on the sphere Sn and on the real hyperbolic space Hn. It completes the study of such question for complete, connected, simply connected Riemannian manifolds of constant sectional curvature. We also discuss an extremum problem for the second eigenvalue on Hn and prove the Hong–Krahn–Szegö type inequality. The main examples of the considered convolution type operators are the Riesz transforms with respect to the geodesic distance of the space.

Original languageEnglish
Pages (from-to)325-334
Number of pages10
JournalBulletin of Mathematical Sciences
Volume6
Issue number2
DOIs
Publication statusPublished - Jul 1 2016

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Keywords

  • Convolution operators
  • Hong–Krahn–Szegö inequality
  • n-sphere
  • Rayleigh–Faber–Krahn inequality
  • Real hyperbolic space

ASJC Scopus subject areas

  • Mathematics(all)

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