Wave front sets for ultradistribution solutions of linear partial differential operators with coefficients in non‐quasianalytic classes

Abstract

This is an author version of the contribution published on:Questa è la versione dell'autore dell'opera: [A.A. Albanese, D. Jornet e A. Oliaro, ``Wave front sets for ultradistribution solutions of linear partial differential operators with coefficients in non-quasianalytic classes'',

Ultra‐differentiable classes and intersection theorems

“…Also compare with Proposition 7.6 in [8], Corollary 3.8 and Remark 3.9 in [16] and Proposition 1.4.11 in [22]. More recently [2], [1], [15], all contain intersection theorems which are somehow close to the spirit of the result of T. Bang. The results in this section are related to those of [1], [15].…”

“…We conclude that we have 2] (|α|) if we also use (2.1). The constant c 5 /2 is now already essentially the constant "λ" of (7.1), so we mention for the Beurling case that in order to have a large c 5 /2, it will suffice to start from sufficiently small c 3 .…”

“…Wave front sets for Gevrey classes have been considered practically from the beginning of micro‐local theories. For general ultra‐differentiable classes see e.g., , , , , , and for ‐classes, . In the present section only the definition of the wave front set in Beurling type inhomogeneous Gevrey classes had not been given explicitly before.…”

Global regularity in ultradifferentiable classes

“…Motivated by the recent work developed in [15,16,32,33,43] and in [2,3], we investigate the global hypoellipticity of linear partial differential operators defined on the torus T N in a bigger scale of spaces, namely, in the setting of ultradifferentiable classes as introduced in [10]. Actually, we prove the ω-regularity of solutions of operators of type P = P (t, D t , Dx) defined on the torus T m+n with real valued coefficients in E * (T m ) and which are globally hypoelliptic in T m+n .…”

“…On the other hand, if u ∈ E (T N ) is such that P u ∈ E (ω) (T N ), then by [2, Theorem 4.1] (see also [3,Theorem 3.15] if ω is a non quasi-analytic weight) we have (t, x, τ, 0) ∈ W F (ω) (u) for any (t, x) ∈ T n and τ ∈ R m \ {0} as P is elliptic at every point t ∈ T m . We can apply Theorem 3.1 to conclude that u ∈ E (ω) (T N ).…”