Abstract:
The paper is devoted to the study of the uniqueness problem for convolution
equations on groups of motions of homogeneous spaces. The main results relate to the case of the
motion group $G=\mathrm{PSL}(2,\mathbb{R})$ of the hyperbolic plane $\mathbb{H}^2$ and are as follows:
1) John type uniqueness theorems for solutions of convolution equations on the group $G$ are proved;
2) exact conditions for the uniqueness of the solution of the system of convolution equations
on regions in $G$ are found.
To prove these results, a technique based on the study of generalized convolution equations on
$\mathbb{H}^2$ is developed. These equations, in turn, are investigated using
transmutation operators of a special kind constructed in the work. The proposed method also allows
us to establish a number of other results related to generalized convolution equations on
$\mathbb{H}^2$ and the group $G$.
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