Abstract
We introduce a global wave front set using Weyl quantizations of pseudodifferential operators of infinite order in the ultradifferentiable setting. We see that in many cases it coincides with the Gabor wave front set already studied by the last three authors of the present work. In this sense, we also extend, to the ultradifferentiable setting, previous work by Rodino and Wahlberg. Finally, we give applications to the study of propagation of singularities of pseudodifferential operators.
Highlights
In the theory of partial differential equations, the wave front set locates the singularities of a distribution and, at the same time, describes the directions of the high frequencies responsible for those singularities
[28] Hormander introduces two different types of global wave front sets addressed to the study of quadratic hyperbolic operators: the C∞ wave front set, in the Beurling setting, for temperate distributions u ∈ S (Rd) using Weyl quantizations, and the analytic wave front set, in the Roumieu setting, for ultradistributions SA(Rd) of Gelfand-Shilov type, defined in terms of a very general known version of the FBI transform as introduced originally by Sjostrand [40]
Rodino and Wahlberg [37] recover the concept of C∞ wave front set of [28] and show that it can be reformulated in terms of the short-time Fourier transform, very related to the FBI transform
Summary
In the theory of partial differential equations, the wave front set locates the singularities of a distribution and, at the same time, describes the directions of the high frequencies (in terms of the Fourier transform) responsible for those singularities. In the classical context of Schwartz distributions theory, it was originally defined by Hormander [27]. There is a huge literature on wave front sets for the study of the regularity of linear partial differential operators in spaces of distributions or ultradistributions in a local sense; see, for instance, [1,2,9,10,23,36,37] and the references therein. In global classes of functions and distributions (like the Schwartz class S(Rd) and its dual) the concept of singular support does not make sense, since we require the information on the whole Rd. we can still define a
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