On the Symbol of Nonlocal Operators in Sobolev Spaces

We consider nonlocal operators generated by pseudodifferential operators and the operator of shift along the trajectories of an arbitrary diffeomorphism of a smooth closed manifold. We introduce the notion of symbol of such operators acting in Sobolev spaces. As examples, we consider specific diffeomorphisms, namely, isometries and dilations. © 2011 Pleiades Publishing, Ltd.