Canonical Operator on Punctured Lagrangian Manifolds and Commutation with Pseudodifferential Operators: Local Theory

Maslov's canonical operator on punctured Lagrangian manifolds provides a solution to the Cauchy problem with initial data concentrated near a point or a submanifold of positive codimension for equations and wave-type systems for which the roots of the characteristic equation have singularities such as nonsmoothness and/or change of multiplicities at zero values of momenta. The theory of the canonical operator on punctured Lagrangian manifolds was constructed in the article [1] by S. Yu. Dobrokhotov, A. I. Shafarevich, and the author, in which, however, the formula for commutation of the canonical operator with pseudodifferential operators was not given. This formula is proved in the present article; moreover, the construction of the canonical operator on punctured Lagrangian manifolds is presented in an equivalent, more convenient form. We restrict ourselves to the local theory (the precanonical operator, or the operator in a separate chart of the Lagrangian manifold corresponding to some nondegenerate phase function), since the transition to the global construction does not contain anything new compared to the standard case.