Ultraholomorphic extension theorems in the mixed setting

Abstract

The aim of this work is to generalize the ultraholomorphic extension theorems from V. Thilliez in the weight sequence setting and from the authors in the weight function setting (of Roumieu type) to a mixed framework. Such mixed results have already been known for ultradifferentiable classes and it seems natural that they have ultraholomorphic counterparts. In order to have control on the opening of the sectors in the Riemann surface of the logarithm for which the extension theorems are valid we are introducing new mixed growth indices which are generalizing the known ones for weight sequences and functions. As it turns out, for the validity of mixed extension results the so-called order of quasianalyticity (introduced by the second author for weight sequences) is becoming important. KeywordsSpaces of ultraholomorphic functions • Weight sequences and weight functions • Growth indices • (non)quasianalyticity of function classes • Restriction and extension mappings • Surjectivity of the Borel map Mathematics Subject Classification 26E10 • 30D60 • 46A13 • 46E10 Communicated by Mieczyslaw Mastylo.

On the maximal extension in the mixed ultradifferentiable weight sequence setting

“…Note that in [21, Lemma 5.7] it is shown that for M, N ∈ LC with µ ≤ ν, (M, N ) γ 1 implies (ω M , ω N ) snq (see also [12,Lemma 4,Rem. 6] for r = 1).…”

“…The parameter r is then used to introduce the (mixed) growth index γ(•) which measures the opening of the sector under consideration. For more details we refer to [10]- [12] and the references therein.…”

“…It has been introduced in [5] and also used in [19]. Finally, in [11] and [12] a ramified generalization of this condition appeared (see again Section 3.4). Note that in (M, N ) γ 1 the sequence of quotients of M , denoted by (µ j ) j , appears whereas (M, N ) SV is connected with the sequence of roots (M j ) 1/j , and these sequences are comparable (up to a constant) if and only if M satisfies the technical assumption of moderate growth (by [19,Lemma 2.2]).…”

Equality of Ultradifferentiable Classes by Means of Indices of Mixed O-regular Variation

“…( 1.1 ) and ( 1.2 ) are the mixed versions of the standard assumption as on weight functions (denoted by in this work). These types of mixed conditions, for either weight sequences or weight functions, have frequently appeared in the literature in the study of mixed extension results in the ultradifferentiable or ultraholomorphic settings, see [ 3 , 6 , 7 , 10 , 16 ].…”

Equality of ultradifferentiable classes by means of indices of mixed O-regular variation

“…The main aim of this article is to give a complete solution to ( * * ) and to ( * * ) ′ expressed in terms of growth properties for M. In order to do so, instead of (M {L} ) we consider the weaker requirement 2) are the mixed versions of the standard assumption ω(2t) = O(ω(t)) as t → +∞ on weight functions (denoted by (ω 1 ) in this work). These types of mixed conditions, for either weight sequences or weight functions, have frequently appeared in the literature in the study of mixed extension results in the ultradifferentiable or ultraholomorphic settings, see [3], [16], [6], [7], [10].…”