  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 $\mathrm{SL(2,\mathbb{R})}$.
