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

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.

Isoperimetric Inequalities for Some Integral Operators Arising in Potential Theory

“…We review an analogue of the Luttinger inequality for the Newton potential operator N Ω and provide related explicit examples. It is a particular case of our previous result with G. Rozenblum in [28] for the Newton potential (see also [30], [32] and [22] for a non-self adjoint operators). In Section 3 we present:…”

Isoperimetric inequalities for the Cauchy-Dirichlet heat operator

“…However, there are also many papers on this subject for other type of compact operators. For instance, in the recent work [8] the authors proved Rayleigh-Faber-Krahn type inequality and Hong-Krahn-Szegö type inequality for the Riesz potential (see also [9], [10] and [11]). All these works were for self-adjoint operators.…”

Isoperimetric inequalities for the heat potential operator

“…Nowadays, the Rayleigh-Faber-Krahn inequality has been generalised to many different operators; see e.g. [5][6][7][8] for further references. In the present paper we present an analogue of the Rayleigh-Faber-Krahn theorem for the heat potential operator H, i.e.…”

“…This paper is mainly inspired by recent works of Rozenblum, Ruzhansky and Suragan (see e.g. [5][6][7][8] ) in which analogues of Rayleigh-Faber-Krahn type inequalities were studied for self-adjoint convolution type integral operators. Thus, in this paper we proved:…”