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Abstract
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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.
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Keywords
ultradifferentiable wavefront set, elliptic regularity,
uniqueness theorems, Denjoy–Carleman classes, Bony's
theorem, quasianalytic uniqueness theorems
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Mathematical Subject Classification 2010
Primary: 26E10, 35A18
Secondary: 35A02, 35A30
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Milestones
Received: 5 June 2019
Revised: 5 April 2020
Accepted: 6 April 2020
Published: 4 September 2020
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