On Lie groups and hyperbolic symmetry—From Kunze–Stein phenomena to Riesz potentials

Sharp forms of Kunze–Stein phenomena on SL(2,R)SL(2,R) are obtained by using symmetrization and Stein–Weiss potentials. A new structural proof with remarkable simplicity can be given on SL(2,R)SL(2,R) which effectively transfers the analysis from the group to the symmetric space corresponding to a manifold with negative curvature. Our methods extend to include the Lorentz groups and nn-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 here.