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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 someconvolution type integral operators, on the sphere $\mathbb{S}^{n}$ and on thereal hyperbolic space $\mathbb{H}^{n}$. It completes the study of such questionfor complete, connected, simply connected Riemannian manifolds of constantsectional curvature. We also discuss an extremum problem for the secondeigenvalue on $\mathbb{H}^{n}$ and prove the Hong-Krahn-Szeg\"{o} typeinequality. The main examples of the considered convolution type operators arethe Riesz transforms with respect to the geodesic distance of the space.

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