Trace theorems for pseudo-differential operators

An integral operator with smooth kernel can always be restricted to a hypersurfaceS. Acutally, it is again an integral operator and its kernel is the restriction (in both variables) of the original one toS. Here we study restrictions of pseudo-differential operators of arbitrary order. We find sufficient and (to some extent) necessary conditions on the symbol ensuring existence of the restriction. These conditions require the vanishing of some geometrical invariants defined on the conormal bundle of the hypersurface. In particular, for a pseudo-differential operator of orderm, the principal symbol should vanish of order [m]+2 and the subprincipal symbol of order [m]+1. These classical invariants are sufficient to treat the problem for the casem<1, but in the general case we need to introduce new higher order invariants related to the operator and the hypersurface.