Uniqueness of solutions of generalized convolution equations on the

V. V. Volchkov ; Vit. V. Volchkov

Izvestiya. Mathematics , Tome 88 (2024) no. 6, pp. 1050-1086

Résumé

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 $G$.

Détail

Keywords: mean periodicity, hyperbolic plane, John uniqueness theorem, spherical transform, transmutation operators.