For a given function $a(x,\xi)$ on $\mathbb{R}^n \times \mathbb{R}^n$, consider the pseudo-differential operator $a(x,D)$ defined by

\begin{equation} a(x,D) (f)(x) =\int_{\mathbb{R}^n} a(x,\xi) \widehat f(\xi) e^{2\pi i x\cdot \xi} d\xi, \end{equation}

where $\widehat{f}$ denotes the Fourier transform of a function $f$. Let $S^0$ be the set of all smooth functions $a: \mathbb{R}^n\times \mathbb{R}^n \rightarrow \mathbb{C}$ satisfying

\begin{equation} \left| \frac{\partial^\beta_x}{\partial_{\xi} ^\alpha} a (x,\xi)\right| \leq {C_{\alpha,\beta} }\, {( 1+ |\xi| )^{-|\alpha|}} \end{equation}

for all $x,\xi \in \mathbb{R}^n $ and for all multi indices $\alpha$ and $\beta$. It is well known that for $a\in S^0$, the associated pseudo-differential operator $a(x,D)$ extends to a bounded operator on $L^p(\mathbb{R}^n)$ to itself, for $1<p<\infty$.

In this talk, we will discuss an analogue of this result on radial sections of line bundles over the Poincaré upper half plane. More precisely, we will focus on the group $G=\mathrm{SL}(2,\mathbb{R})$, where we will explore the boundedness properties of the pseudo-differential operator defined on functions of fixed $K=\mathrm{SO}(2)$-type in $G$. Additionally, we will explore the case where the symbol exhibits restricted regularity in the spatial variable.

This talk is based on a joint work with Michael Ruzhansky.

The video of this talk is available on the IISc Math Department channel.