Abschätzungen für das Spektrum von \(\Delta_ p\) auf Räumen konstanter Krümmung (Q799944)

The author presents estimations for the first eigenvalue of \(\Delta_ p\) on spaces of constant curvature in terms of vol(M), the diameter \(d_ M\) and the radius of the greatest geodesic ball contained in M. Main tool of the investigation is the explicit study of the eigenvalue problem \(\Delta_{p,F}\omega =\lambda\omega \) for the Friedrichs' extension of \(\Delta_ p\), considered over a geodesic ball in M. In geodesic polar coordinates \(ds^ 2\) has the form \(ds^ 2=dr^ 2+f(r)^ 2d\sigma^ 2\). Computation of \(\Delta_ p\) leads to Bessel, respectively differential equations for spherical functions. Thus estimations for the zeros of spherical functions enter into the final results. Finally estimations for the m-th eigenvalue of \(\Delta_ p\) are given.