Weighting of derivatives (Q1316884)

It is convenient to study anisotropic differential operators, such as the nonstationary Schrödinger operator, the operator of heat conductivity, etc. by assigning different weights to derivatives along various directions. A relatively minor modification of the technique of pseudodifferential and integral operators developed for the isotropic case allows us to investigate (local) solvability of equations and singularities of solutions. The definition of weighting used in a number of papers on linear differential equations is a coordinate nature. In this paper, a definition of weighting in an invariant form is given, its connection with the coordinate one is studied, and the notation of a weighted wavefront set of a linear functional over finite smooth cuts of an arbitrary vector bundle over a differential manifold is introduced. The notion of a linear differential operator defined on the cuts of a vector bundle serves as the starting point of the construction.