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Pseudodifferential operators with generalized symbols and regularity theory. (English) Zbl 1087.35102
The authors study an equation of the form \[ a_\varepsilon(x,D)u_\varepsilon= f_\varepsilon, \] where \(a_\varepsilon(x,D)\) is a pseudodifferential operator with amplitude \(a_\varepsilon(x,\xi)\) depending on a singular parameter \(\varepsilon\to 0+,\) where the point of view is that of asymptotic analysis. In this setting, they develop a full symbol-calculus (with formal-series expansions of symbols) for pseudodifferential operators acting on algebras of Colombeau generalized functions. The calculus allows them to construct parametrices and obtain hypoellipticity results for generalized pseudodifferential equations.

MSC:
35S50 Paradifferential operators as generalizations of partial differential operators in context of PDEs
35S30 Fourier integral operators applied to PDEs
46F10 Operations with distributions and generalized functions
46F30 Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.)
35D10 Regularity of generalized solutions of PDE (MSC2000)
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