On Lie groups and hyperbolic symmetry -- from Kunze-Stein phenomena to Riesz potentials (Q494209)

The author obtains sharp forms of Kunze-Stein phenomena on \(\mathrm{SL}(2, \mathbb{R})\) by using symmetrization and Stein-Weiss potentials. A new structural proof with remarkable simplicity can be given on \(\mathrm{SL}(2, \mathbb{R})\) which effectively transfers the analysis from the group to the symmetric space corresponding to a manifold with negative curvature. The methods are extended to include the Lorentz groups and \(n\)-dimensional hyperbolic space through application of the Riesz-Sobolev rearrangement inequality. A new framework is developed for Riesz potentials on semisimple symmetric spaces and the semi-direct product of groups analogous to the Iwasawa decomposition for semisimple Lie groups. Extensions to higher-rank Lie groups and analysis on multidimensional connected hyperboloids including anti de Sitter space are suggested by the analysis outlined.