Singular integral operators on a manifold with a distinguished submanifold

Abstract

Let $X$ be a compact manifold without boundary and $X^0$ its smooth submanifold of codimension one. In this work we introduce classes of integral operators on $X$ with kernels $K_A(x,y)$, being smooth functions for $x\notin X^0$ and $y\notin X^0$, and admitting an asymptotic expansion of certain type, if $x$ or $y$ approaches $X^0$. For operators of these classes we prove theorems about action in spaces of conormal functions and composition. We show that the trace functional can be extended to a regularized trace functional $\operatorname{r-Tr}$ defined on some algebra $\mathcal K(X,X^0)$ of singular integral operators described above. We prove a formula for the regularized trace of the commutator of operators from this class in terms of associated operators on $X^0$. The proofs are based on theorems about pull-back and push-forward of conormal functions under maps of manifolds with distinguished codimension one submanifolds.