Pseudo-differential operators on radial sections of line bundles over the Poincar\'e upper half plane

TL;DR

This paper studies the boundedness of pseudo-differential operators on radial sections over the Poincaré upper half plane, extending results to symbols with limited regularity and linking operator norms to Euclidean analogues.

Contribution

It establishes boundedness results for pseudo-differential operators with both smooth and rough symbols on radial sections over the Poincaré upper half plane, including new estimates for matrix coefficients.

Findings

Boundedness for smooth symbols established

Introduction of rough symbol class with boundedness results

Asymptotic estimates for matrix coefficients provided

Abstract

In this article, we explore the boundedness properties of pseudo-differential operators on radial sections of line bundles over the Poincar\'e upper half plane, even when dealing with symbols of limited regularity. We first prove the boundedness of these operators when the symbol is smooth. To achieve this, we establish a connection between the operator norm of the local part of our pseudo-differential operators and the corresponding Euclidean pseudo-differential operators. Additionally, we introduce a class of rough symbols that lack any regularity conditions in the space variable and investigate the boundedness properties of the associated pseudo-differential operators. As a crucial part of our proof, we provide asymptotic estimates and functional identities for certain matrix coefficients of the principal and discrete series representations of the group $\mathrm{SL(2,\mathbb{R})}$.