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On duality theory and pseudodifferential techniques for Colombeau algebras: generalized delta functionals, kernels and wave front sets. (English) Zbl 1121.46037

The following topics are reviewed: 1.topological structures and duality theory, 2.pseudodifferential techniques and microlocal analysis. The authors are the founders of the theories which are presented so that their survey article essentially explains some of their fundamental results.
Part 1 recalls the definitions of locally convex topological \(\widetilde{\mathbb C}\)-modules and the strict inductive limit of such spaces. Then, Colombeau type spaces, already defined from the algebraic point of view, are described through corresponding topologies and dual pairing. The discussion includes generalized delta functions and generalized kernels, as well as Fourier integral operators with generalized amplitudes.
Part 2 begins with the generalized wave front set and its description through the characteristic sets of corresponding pseudodifferential operators, slow scale microellipticity and the fundamental estimate of the wave front set for a solution of a pseudodifferential equation. The propagation of \(\mathcal G^\infty\)-regularity along bicharacteristics of the equation \(U_t+Pu=0\), where \(P\) is a first order partial differential operator with real principal symbol, is described in a similar way as in the classical case for \(C^\infty\)-regularity.
In the last part, the authors consider a nonhomogeneous Cauchy problem of the form \(U_t+Pu=f\), where the coefficients and \(f\) are generalized functions so that the bicharacteristics are also generalized functions. With this, a more complex analysis of the propagation of singularities is provided with the final conclusion that the generalized wave front is not described completely through generalized bicharacteristics.

MSC:

46F30 Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.)
46A20 Duality theory for topological vector spaces
47G30 Pseudodifferential operators
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