Global Wave Front Sets in Ultradifferentiable Classes

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.

The Wigner global wave front set in spaces of tempered ultradistributions

“…Remark 32 Notice that if ω is a subadditive weight function satisfying the hypotheses in [2, Cor. 5.9] and 0 < τ < 1, then the chain of equalities in Theorem 30 is extended to the global wave front set defined in [2,Defin. 4.3] for all u ∈ S ω (R d ).…”

“…4.3] for all u ∈ S ω (R d ). In particular, this whole chain holds for Gevrey weight functions ω(t) = t a with 0 < a < 1 small enough (see [2,Example 5.10]).…”

“…See [10,11] for wave front sets of Gel'fand-Shilov type. Recently, the author, Boiti, Jornet, and Oliaro [2], using the theory developed in [1], generalized the concept of C ∞ wave front set to spaces of ω-tempered ultradistributions following the ideas given in [20,26], and provided conditions on the weight function under which this wave front set and the analytic wave front set in [7] are equal.…”