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
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.
