Sectorial Extensions, via Laplace Transforms, in Ultraholomorphic Classes Defined by Weight Functions

Abstract

We prove an extension theorem for ultraholomorphic classes defined by so-called Braun-Meise-Taylor weight functions ω and transfer the proofs from the single weight sequence case from V. Thilliez [28] to the weight function setting. We are following a different approach than the results obtained in [11], more precisely we are working with real methods by applying the ultradifferentiable Whitney-extension theorem. We are treating both the Roumieu and the Beurling case, the latter one is obtained by a reduction from the Roumieu case.

On the Construction of Large Algebras Not Contained in the Image of the Borel Map

“…In particular we have ω∼ω W (λ) for all λ > 0. We are summarizing some facts for this conjugate, see also [17,Section 3.1]. The function h ⋆ is clearly nondecreasing, continuous and concave, and lim t→+∞ h ⋆ (t) = +∞, see [3, (8), p. 156].…”

“…Let ω be given satisfying (ω 0 ), (ω 3 ) and (ω 4 ), then as shown in [21,Section 5], respectively [25, Theorem 4.0.3, Lemma 5.1.3] and reproved in [17,Lemma 2.5] in a more precise way, we have…”

“…(ii) Let ω ∈ W be given. Then ω∼(ω ι ) ⋆ implies γ(ω) = +∞, with γ denoting the growth index studied in detail in [15] and used in the extension results in [17], [16] (the fact that ω has (ω 1 ) is equivalent to having γ(ω) > 0, see […”

“…any ω satisfying (ω 0 ), (ω 3 ) and (ω 4 ) condition (5.1) does always imply (ω 1 ), see[16, Appendix A]. In [16, Lemma A.1] it has been shown that for any ω ∈ W with (5.1) the associated matrix Ω does have both (4.11) and (4.12) (by having a precise relation between the indices λ and κ).…”

Almost analytic extensions of ultradifferentiable functions with applications to microlocal analysis

“…The arguments in the proof of the implication (4) ⇒ (5) in [8,Lemma 12] show that also (1.3) holds (in fact, ω Q (t) = o(t) as t → ∞). By [23,Lemma 3.4], ω Q is equivalent to a concave weight function. Hence ω Q is a weight function that is equivalent to a concave weight function.…”