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.
- Convolution operators
- Hong–Krahn–Szegö inequality
- Rayleigh–Faber–Krahn inequality
- Real hyperbolic space
ASJC Scopus subject areas