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

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 $\mathbb{S}^{n}$ and on the real hyperbolic space $\mathbb{H}^{n}$. 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 $\mathbb{H}^{n}$ 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.