On first and second eigenvalues of Riesz transforms in spherical and hyperbolic geometries (Q300790)

The authors investigate the integral operator \({\mathcal K}_{\Omega}:L^2(\Omega)\to L^2(\Omega)\) defined by NEWLINE\[NEWLINE{\mathcal K}_{\Omega}f(x)=\int_{\Omega}K(d(x,y))f(y)\,dy,\quad f\in L^2(\Omega),NEWLINE\]NEWLINE supposed to be compact, where \(\Omega\subset \mathbb{S}^n\) or \(\Omega\subset \mathbb{H}^n\) is an open bounded set, \(\mathbb{S}^n\) is the sphere, \(\mathbb{H}^n\) is the real hyperbolic space, \(d(x,y)\) is the distance between the points \(x\) and \(y\), and the kernel \(K\in L^1(\mathbb{S}^n)\) or \(L^1_{\mathrm{loc}}(\mathbb{H}^n)\) is a real, positive and non-increasing function. They give a Rayleigh-Faber-Krahn type inequality, proving that the first eigenvalue of \({\mathcal K}_{\Omega}\) is maximised on the geodesic ball among all domains of a given measure in \(\mathbb{S}^n\) or \(\mathbb{H}^n\). An extremum problem for the second eigenvalue on \(\mathbb{H}^n\) and the Hong-Krahn-Szegő type inequality are also discussed.