The overdetermined Cauchy problem for $$\omega $$ ω -ultradifferentiable functions

Abstract

Abstract. In this paper we study the Cauchy problem for overdetermined systems of linear partial differential operators with constant coefficients in some spaces of ω-ultradifferentiable functions in the sense of Braun, Meise and Taylor [BMT], for non-quasianalytic weight functions ω. We show that existence of solutions of the Cauchy problem is equivalent to the validity of a Phragmén-Lindelöf principle for entire and plurisubharmonic functions on some irreducible affine algebraic varieties.

Regularity of partial differential operators in ultradifferentiable spaces and Wigner type transforms

“…The spaces of Roumieu type are not used here and a definition can be found in [8] with a stronger condition instead of our (γ). The use of (γ) is clarified for the Beurling case in [3] (see also [14]).…”

“…8.1] (cf. also [3], since we assume condition (γ) of Definition 4.1 instead of log(t) = o(ω(t)) as t → ∞).…”