Sharp mapping properties of Poisson transforms and the Baum-Connes conjecture

TL;DR

This paper establishes sharp mapping properties of Poisson transforms for real rank one semisimple Lie groups, proving a key conjecture in the Baum-Connes program by analyzing Sobolev space mappings and compactness of commutators.

Contribution

It proves a sharp, quantitative analogue of Helgason's conjecture for Poisson transforms, confirming the remaining open conjecture in Julg's Baum-Connes program for rank one groups.

Findings

Poisson transforms map Sobolev spaces boundedly with closed range to L^2 spaces.

Commutators of Poisson transforms with smooth functions are compact.

Results generalize to a broader class of Poisson transforms studied by Knapp-Wallach.

Abstract

We prove a sharp, quantitative analogue of Helgason's conjecture at the level of distributions: For a semisimple Lie group $G$ of real rank one, Poisson transforms map a Sobolev space on $P\backslash G$ boundedly with closed range to an $L^2$-space on $K\backslash G$. The result is obtained for the Poisson transform studied by Knapp-Wallach under the name Szeg\"o map, and the appropriate Sobolev spaces are defined using van Erp-Yuncken's Heisenberg calculus. The proof generalizes to show that commutators of this Poisson transform with smooth functions on the Furstenberg compactification are compact. This proves the remaining open conjecture in Julg's seminal program to establish the Baum-Connes conjecture for closed subgroups of semisimple Lie groups of real rank one.