Microlocal Analysis of Generalized Functions: Pseudodifferential Techniques and Propagation of Singularities

Abstract

We characterize microlocal regularity, in the G ∞ -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 G ∞ -singularities for homogeneous first-order hyperbolic equations.

Geometrical embeddings of distributions into algebras of generalized functions

“…The characteristic set of P is char(P ) = σ(P ) −1 (0) ⊆ T * M \ 0. We follow the results in [11] to define the generalized wavefront set of u ∈ G(M) to be…”

Linear and Non-Linear Theory of Generalized Functions and its Applications

“…The recent investigation of the G-and G ∞ -regularity properties of generalized differential and pseudodifferential operators in the Colombeau context [6,12,13,17,18] has provided several sufficient conditions of G-and G ∞ -hypoellipticity. The search for necessary conditions for G-and G ∞ -hypoellipticity has been a long-standing open problem.…”

“…They have proved to be a valuable tool for treating partial differential equations with singular data or coefficients [3,12,17,20]. Recently, intense research has been done in the context of partial differential operators with generalized coefficients, leading to the development of a theory of generalized Fourier and pseudodifferential operators acting on Colombeau algebras [6,12,14,17] as well as to microlocal analysis in the Colombeau context [9,13,18]. Increasing importance is also attached to an understanding of topological structures [4,5,7,8] in spaces of generalized functions and to the development of functional analytic methods [10].…”

G- and G∞-hypoellipticity of partial differential operators with constant Colombeau coefficients

“…They have proved to be a valuable tool for treating partial differential equations with singular data or coefficients [3,12,17,20]. Recently, intense research has been done in the context of partial differential operators with generalized coefficients, leading to the development of a theory of generalized Fourier and pseudodifferential operators acting on Colombeau algebras [6,12,14,17] as well as to microlocal analysis in the Colombeau context [9,13,18]. Increasing importance is also attached to an understanding of topological structures [4,5,7,8] in spaces of generalized functions and to the development of functional analytic methods [11].…”

“…Indeed, G ∞ (Ω)∩D (Ω) = C ∞ (Ω) [20]. By means of pseudodifferential techniques elaborated 112 C. GARETTO in [6,12,13], sufficient conditions of G ∞ -hypoellipticity has been provided involving the symbol of the operator P (x, D) = |α|≤m c α (x)D α , c α ∈ G(Ω), or more in general the symbol of a generalized pseudodifferential operator [12,13,17,18]. These are conditions which allow to deduce from P (x, D)u ∈ G ∞ (Ω) that the generalized function u belongs to G ∞ (Ω) as well.…”