Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities

Abstract

We characterize microlocal regularity of Colombeau generalized functions by an appropriate extension of the classical notion of micro-ellipticity to pseudodifferential operators with slow scale generalized symbols. Thus we obtain an alternative, yet equivalent, way to determine generalized wave front sets, which is analogous to the original definition of the wave front set of distributions via intersections over characteristic sets. The new methods are then applied to regularity theory of generalized solutions of (pseudo-)differential equations, where we extend the general noncharacteristic regularity result for distributional solutions and consider propagation of generalized singularities for homogeneous first-order hyperbolic equations.