### 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)

### Cite this

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

Research output: Contribution to journal › Article

*Bulletin of Mathematical Sciences*, vol. 6, no. 2, pp. 325-334. https://doi.org/10.1007/s13373-016-0082-5

}

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

AN - SCOPUS:84976440502

VL - 6

SP - 325

EP - 334

JO - Bulletin of Mathematical Sciences

JF - Bulletin of Mathematical Sciences

SN - 1664-3607

IS - 2

ER -