Vol. 307, No. 2, 2020

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Geometric microlocal analysis in Denjoy–Carleman classes

Stefan Fürdös

Vol. 307 (2020), No. 2, 303–351

A systematic geometric theory for the ultradifferentiable (nonquasianalytic and quasianalytic) wavefront set similar to the well-known theory in the classic smooth and analytic setting is developed. In particular an analogue of Bony’s theorem and the invariance of the ultradifferentiable wavefront set under diffeomorphisms of the same regularity is proven using a theorem of Dynkin about the almost-analytic extension of ultradifferentiable functions. Furthermore, we prove a microlocal elliptic regularity theorem for operators defined on ultradifferentiable vector bundles. As an application, we show that Holmgren’s theorem and several generalizations hold for operators with quasianalytic coefficients.

ultradifferentiable wavefront set, elliptic regularity, uniqueness theorems, Denjoy–Carleman classes, Bony's theorem, quasianalytic uniqueness theorems
Mathematical Subject Classification 2010
Primary: 26E10, 35A18
Secondary: 35A02, 35A30
Received: 5 June 2019
Revised: 5 April 2020
Accepted: 6 April 2020
Published: 4 September 2020
Stefan Fürdös
Faculty of Mathematics, University of Vienna
University of Vienna