On an algebra of pseudodifferential operators in spaces with weighted norms

Pseudodifferential operators on Euclidean space $\mathbf R^n$ are studied. These operators, whose definition must be modified in a natural way, act in Hilbert spaces with weighted norms. Using the Mellin transform, a pseudodifferential operator on the sphere $S^{n-1}$ (actually a meromorphic function of a complex parameter $\lambda$) is assigned to an operator on $\mathbf R^n$. This permits one to reduce the study of an algebra of operators on $\mathbf R^n$ to the investigation of an algebra of meromorphic operator-valued functions on $S^{n-1}$.
This paper consists of four sections. In § 1 preliminaries are presented. The second section is devoted to the study of an algebra of meromorphic operator-valued functions on the sphere $S^{n-1}$. In § 3 an algebra of pseudodifferential operators on $\mathbf R^n$ is considered. The last section contains rules for change of variables in operators on the sphere and on $\mathbf R^n$.
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