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.
