부산대학교 도서관

Let $M$ be a manifold of bounded geometry which satisfies the following condition: the bottom of the essential spectrum of the Laplace operator on 1-forms on the cotangent sphere bundle $S^*M$ is strictly positive. The main purpose of the paper is to define an ILH (inverse limit of Hilbert) Lie group structure on the group ${\scr U}F^{0,k}(M)$ of invertible formal uniform Fourier integral operators on $M$ whose homogeneous canonical relation belongs to the identity component of the group of contact diffeomorphisms of $S^*M$. This result extends a similar result of M. R. Adams, T. S. Ratiu\ and R. Schmid\ [Math. Ann. {\bf 273} (1986), no.~4, 529--551; MR0826458 (88c:58067); Math. Ann. {\bf 276} (1986), no.~1, 19--41; MR0863703 (88c:58068)] stated in the case when $M$ is compact. \par The proof follows the lines of the above papers. First, the authors introduce structures of ILH group on the identity component ${\scr D}^\infty_{\theta,0}(T^*M\sbs\{0\})$ of the group of contact diffeomorphisms of $S^*M$ and on the group ${\scr U}\Psi^{0,k}(M)$ of invertible formal uniform pseudodifferential operators on $M$. Then they use an exact sequence of groups $1\rightarrow {\scr U}\Psi^{0,k}(M)\rightarrow {\scr U}F^{0,k}(M)\rightarrow {\scr D}^\infty_{\theta,0}(T^*M\sbs \{0\})\rightarrow 1$, a local splitting $\pi\colon {\scr D}^\infty_{\theta,0}(T^*M\sbs\{0\}) \rightarrow {\scr U}F^{0,k}(M)$ of this sequence satisfying compatibility conditions, constructed by means of a global phase function for uniform Fourier integral operators, and some group theoretical results.