Boundedness of Eisenstein integrals on split rank-one semisimple symmetric spaces

Sanjoy Pusti and Iswarya Sitiraju

Vol. 344 (2026), No. 2, 395–430

DOI: 10.2140/pjm.2026.344.395

Abstract

We characterize the bounded left $K$-invariant normalized Eisenstein integrals on split rank-one semisimple symmetric spaces. As a consequence we prove the Hausdorff–Young inequality on these spaces. We prove a similar result for $K$-finite Eisenstein integrals on pseudo-Riemannian real hyperbolic spaces.

Keywords

semisimple symmetric spaces, pseudo-Riemannian real hyperbolic spaces, Eisenstein integral, Helgason–Johnson theorem

Mathematical Subject Classification

Primary: 43A85, 43A90

Secondary: 22E30, 33C67

Milestones

Received: 9 December 2025

Revised: 31 May 2026

Accepted: 9 June 2026

Published: 8 September 2026