Sectorial extensions for ultraholomorphic classes defined by weight functions

Abstract

We prove an extension theorem for ultraholomorphic classes defined by so-called Braun-Meise-Taylor weight functions and transfer the proofs from the single weight sequence case from V. Thilliez to the weight function setting. We are following a different approach than the results obtained in a recent paper by the authors, 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. K E Y W O R D S extension operators, functions and matrices, growth indices, Legendre conjugates, spaces of ultradifferentiable/ultraholomorphic functions, weight sequences, Whitney extension theorem in the ultradifferentiable setting M S C (2 0 1 0)

Ultraholomorphic extension theorems in the mixed setting

“…Such results have already been known before for the weight sequence approach, see [32]. In [14] we have transferred Thilliez's ideas to the weight function situation (by using ultradifferentiable Whitney extension results) and in [10] we have used complex methods treated by Lastra, Malek and the second author [18,19] in the single weight sequence approach.…”

“…see [19,26]). At this point we want to make the reader aware that here we are using the same notation as it has already been used by the authors in [10,14], whereas in [32] and also in [11] the sequence M is precisely m in the notation in this work.…”

“…In the authors' recent works [10,14] we have shown extension theorems in the ultraholomorphic weight function framework, in the first article for spaces of Roumieu type and in the second one also for the Beurling type classes. Such results have already been known before for the weight sequence approach, see [32].…”

“…We will consider Roumieu type classes in both the weight sequence and weight function setting. By inspecting the proofs of the main results in [10,14,32] it has turned out that, up to our ability, only the complex methods from [10] admit the possibility to generalize the result to a mixed situation, see Remark 13 below for further details.…”

“…The existence of ultraholomorphic extension results is tightly connected to the opening of the sectors where the functions are defined. In the previous results [10,12,14,32] growth indices γ (M) and γ (ω) have been introduced to measure the maximum size of these sectors, for a detailed study and comparison of these values we refer to [11]. Then a similar notion is required in the mixed setting to obtain satisfactory theorems.…”