  The classical Fourier transform on the line sends the operator of
multiplication by $x$ to $i\frac{d}{d\xi}$ and the operator of differentiation
$\frac{d}{d x}$ to the 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.
