Fourier Transform on the Lobachevsky Plane and Operational Calculus

The classical Fourier transform on the line sends the operator of multiplication by $x$ to $i\frac{d}{d\xi}$ and the operator $\frac{d}{d x}$ of differentiation to multiplication by $-i\xi$. For the Fourier transform on the Lobachevsky plane, we establish a similar correspondence for a certain family of differential operators. It appears that differential operators on the Lobachevsky plane correspond to differential-difference operators in the Fourier image, where shift operators act in the imaginary direction, i.e., a direction transversal to the integration contour in the Plancherel formula.