Optimal Flat Functions in Carleman–Roumieu Ultraholomorphic Classes in Sectors

Abstract

We construct optimal flat functions in Carleman–Roumieu ultraholomorphic classes associated to general strongly nonquasianalytic weight sequences, and defined on sectors of suitably restricted opening. A general procedure is presented in order to obtain linear continuous extension operators, right inverses of the Borel map, for the case of regular weight sequences in the sense of Dyn’kin. Finally, we discuss some examples (including the well-known q-Gevrey case) where such optimal flat functions can be obtained in a more explicit way.

On Orlicz classes defined in terms of associated weight functions

“…to simplify arguments. More precisely, in [ 6 , Lemma 3.18] it has been shown that even which clearly implies . We write if ( 2.6 ) holds for the corresponding sequences of quotients , .…”

“… Note that for all j does not hold automatically for all sequences belonging to the set . However, when given , then we can always find such that M and are equivalent and such that the corresponding sequence of quotients is strictly increasing, see [ 6 , Lemma 3.18]. This formal switch allows to avoid technical complications resp.…”

“…In Sect. 6 we introduce and study the complementary N-function F c M (see Theorem 6.4 and Corollary 6.5) and establish the connection between F c M and the dual sequence D, see the main statement Theorem 6.9. Finally, in Sect.…”