An endpoint estimate of the kunze-stein phenomenon on su(n,m)

The endpoint estimates for the Kunze-Stein phenomenon associated with real rank one semisimple Lie groups and the Jacobi hypergroup were obtained by A. Ionescu and J. Liu respectively. Recently, a modified endpoint estimate was also proved for complex semisimple Lie groups by J. Liu and the author. Hence the original estimate seems not to be valid for general semisimple Lie groups with higher rank. We treat the case of SU(n,m) and obtain a modified estimate similarly to the complex case. The process using the Abel transform makes it clear why the original endpoint estimate might not be valid for general semisimple Lie groups.