Quasianalytic Wave Front Sets for Solutions of Linear Partial Differential Operators

Abstract

In the present paper, we introduce and study Beurling and Roumieu quasianalytic (and nonquasianalytic) wave front sets, W F * , of classical distributions. In particular, we have the following inclusionwhere Ω is an open subset of R n , P is a linear partial differential operator with coefficients in a suitable ultradifferentiable class, and Σ is the characteristic set of P . Some applications are also investigated.Mathematics Subject Classification (2010). Primary 46F05; Secondary 35A18, 35A21.

Global regularity in ultradifferentiable classes

“…We also recall the notion of wave front sets in the setting of ultradifferentiable classes (see [2]):…”

“…On the other hand, let u ∈ E (T N ) be such that P u ∈ E (ω) (T N ). Since, by condition (ii), P is elliptic in t, we have (t, x, τ, 0) ∈ W F (ω) (u) for any (t, x) ∈ T n and τ ∈ R m \ {0} (see [2,Theorem 4.1] or, also, [3, Theorem 3.15] for non quasi-analytic weight functions). So, we can apply Theorem 3.1 to conclude that u ∈ E (ω) (T N ).…”

“…As in [2], since ω satisfies (α 0 ) we can assume that it is equivalent to a sub-additive weight function and then…”

“…Since L and ξ are arbitrary, we can apply Lemma 3.1 of [2] to conclude that there exist ε > 0 and C > 0 such that…”

“…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 .…”