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.

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... The main examples of the considered convolution type operators are the Riesz transforms with respect to the geodesic distance of the space.