MICROLOCAL ANALYSIS OF GENERALIZED FUNCTIONS: PSEUDODIFFERENTIAL TECHNIQUES AND PROPAGATION OF SINGULARITIES

[摘要] We characterize microlocal regularity, in the $mathcal{G}^{infty}$-sense, 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 of determining generalized wavefront sets that is analogous to the original definition of the wavefront 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 non-characteristic regularity result for distributional solutions and consider propagation of $mathcal{G}^{infty}$-singularities for homogeneous first-order hyperbolic equations.

[学科分类] 数学（综合）

[关键词] Primary 46F30;Secondary 35S99;35D10;algebras of generalized functions;wavefront sets;pseudodifferential operators