Operator covariant transform and local principle (Q2896231)

\textit{I. B. Simonenko} [Izv. Akad. Nauk SSSR, Ser. Mat. 29, 567--586, 757--782 (1965; Zbl 0146.13101)] introduced the localization technique to the theory of operators to study singular integral operators that are convolutions on the Euclidean group. In this paper, the author presents a different point of view on the original works of Simonenko in the context of the covariant transform. The new interpretation highlights the role of groups in the constructions, and links a certain geometrical meaning of locality with a homogeneous structure of nilpotent Lie groups. Thus, it allows one to make a straightforward modification of the localization technique for non-commutative nilpotent Lie groups. The key role is played by the one-parameter group of automorphisms realized as dilations.