On first and second eigenvalues of Riesz transforms in spherical and hyperbolic geometries
File(s)
Published version
Author(s)
Ruzhansky, M
Suragan, D
Type
Journal Article
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 SnSn and on the real hyperbolic space HnHn . 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 HnHn 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.
Date Issued
2016-04-08
Date Acceptance
2016-03-30
Citation
Bulletin of Mathematical Sciences, 2016, 6 (2), pp.325-334
ISSN
1664-3615
Publisher
Springer Verlag
Start Page
325
End Page
334
Journal / Book Title
Bulletin of Mathematical Sciences
Volume
6
Issue
2
Copyright Statement
© The Author(s) 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
License URL
Sponsor
Engineering & Physical Science Research Council (EPSRC)
The Leverhulme Trust
Grant Number
EP/K039407/1
RPG-2014-002
Subjects
Science & Technology
Physical Sciences
Mathematics
Convolution operators
n-sphere
Real hyperbolic space
Rayleigh-Faber-Krahn inequality
Hong-Krahn-Szego inequality
ISOPERIMETRIC-INEQUALITIES
SOBOLEV INEQUALITIES
OPERATOR
math.SP
math.FA
35P99, 47G40, 35S15
Publication Status
Published