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

Fingerprint

Spherical geometry
Lobachevskian geometry
Riesz Transform
Convolution
Extremum Problem
Eigenvalue
Geodesic Distance
First Eigenvalue
Hyperbolic Space
Sectional Curvature
Integral Operator
Geodesic
Riemannian Manifold
Ball
Analogue
Operator

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

On first and second eigenvalues of Riesz transforms in spherical and hyperbolic geometries. / Ruzhansky, Michael; Suragan, Durvudkhan.

In: Bulletin of Mathematical Sciences, Vol. 6, No. 2, 01.07.2016, p. 325-334.

Research output: Contribution to journalArticle

@article{fcfe27ed60704c2d8c32d4138454224a,
title = "On first and second eigenvalues of Riesz transforms in spherical and hyperbolic geometries",
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–Szeg{\"o} type inequality. The main examples of the considered convolution type operators are the Riesz transforms with respect to the geodesic distance of the space.",
keywords = "Convolution operators, Hong–Krahn–Szeg{\"o} inequality, n-sphere, Rayleigh–Faber–Krahn inequality, Real hyperbolic space",
author = "Michael Ruzhansky and Durvudkhan Suragan",
year = "2016",
month = "7",
day = "1",
doi = "10.1007/s13373-016-0082-5",
language = "English",
volume = "6",
pages = "325--334",
journal = "Bulletin of Mathematical Sciences",
issn = "1664-3607",
publisher = "Springer Basel AG",
number = "2",

}

TY - JOUR

T1 - On first and second eigenvalues of Riesz transforms in spherical and hyperbolic geometries

AU - Ruzhansky, Michael

AU - Suragan, Durvudkhan

PY - 2016/7/1

Y1 - 2016/7/1

N2 - 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.

AB - 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.

KW - Convolution operators

KW - Hong–Krahn–Szegö inequality

KW - n-sphere

KW - Rayleigh–Faber–Krahn inequality

KW - Real hyperbolic space

UR - http://www.scopus.com/inward/record.url?scp=84976440502&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84976440502&partnerID=8YFLogxK

U2 - 10.1007/s13373-016-0082-5

DO - 10.1007/s13373-016-0082-5

M3 - Article

VL - 6

SP - 325

EP - 334

JO - Bulletin of Mathematical Sciences

JF - Bulletin of Mathematical Sciences

SN - 1664-3607

IS - 2

ER -