Given a non-quasianalytic subadditive weight function ω we consider the weighted Schwartz space S ω and the short-time Fourier transform on S ω , S ′ ω and on the related modulation spaces with exponential weights. In this setting we define the ω-wave front set WF ′ ω (u) and the Gabor ω-wave front set WF G ω (u) of u ∈ S ′ ω , and we prove that they coincide. Finally we look at applications of this wave front set for operators of differential and pseudo-differential type.2010 Mathematics Subject Classification. Primary 35A18; Secondary 46F05, 42C15, 35S05.

Abstract. We study the behaviour of linear partial differential operators with polynomial coefficients via a Wigner type transform. In particular, we obtain some results of regularity in the Schwartz space S and in the space S ω as introduced by Björck for weight functions ω. Several examples are discussed in this new setting.

We consider p-evolution equations in (t,x) with real characteristics. We give sufficient conditions for the well-posedness of the Cauchy problem in Sobolev spaces, in terms of decay estimates of the coefficients as the space variable x goes to infinity

We use techniques from time-frequency analysis to show that the space S ω of rapidly decreasing ω-ultradifferentiable functions is nuclear for every weight function ω(t) = o(t) as t tends to infinity. Moreover, we prove that, for a sequence (M p ) p satisfying the classical condition (M 1) of Komatsu, the space of Beurling type S (Mp) when defined with L 2 norms is nuclear exactly when condition (M 2) ′ of Komatsu holds.

We consider the Cauchy problem for overdetermined systems of linear partial differential operators with constant coefficients. In order to avoid the problem of the formal coherence of the initial data, which leads to the notion of formally non-characteristic introduced by Andreotti-Nacinovich in the 80's, we allow formal solutions (in the sense of Whitney) of the given system on any closed subset as initial data. The problem is then to find classical smooth solutions of the system, whose restrictions in the sense of Whitney are the given initial data. This gives a unifying point of view encompassing several different problems, ranging from questions of smoothness of the solutions, to the classical Cauchy problem, to comparison of formal to actual solutions, to Hartog's type phenomena