### 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 S^{n} and on the real hyperbolic space H^{n}. 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 H^{n} 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 language | English |
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Pages (from-to) | 325-334 |

Number of pages | 10 |

Journal | Bulletin of Mathematical Sciences |

Volume | 6 |

Issue number | 2 |

DOIs | |

Publication status | Published - 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)