Ultraholomorphic extension theorems in the mixed setting

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.


Introduction
In the authors' recent works [14] and [11] 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]. In [11] we have transferred Thilliez's ideas to the weight function situation (by using ultradifferentiable Whitney extension results) and in [14] we have used complex methods treated by A. Lastra, S. Malek and the second author [18,19] in the single weight sequence approach.
In the ultradifferentiable setting also Whitney extension results involving two weight sequences M and N and weight functions σ and ω are known in the literature. In the weight sequence case we refer to [7] for the Whitney jet mapping and to [31] for the Borel mapping, in the weight function case see [5] for the Borel mapping and [17], [25] and [22] for the general Whitney jet mapping. In our recent paper [12], which has served as motivation for this article, by involving a ramification parameter r ∈ N >0 we have generalized the mixed setting results from [31] to r-ramification classes introduced in [30]. We have also generalized the Whitney extension results from [7] by using a parameter r > 0 (see [12,Theorem 5.10]). The possibility of an extension in these mixed settings has been characterized in terms of growth properties of weight sequences and functions. We refer also to Remarks 3.3 and 3.4 where more (historical) explanations will be given.
From this theoretical point of view it seems natural to ask whether in the ultraholomorphic framework we can also prove extension results in the mixed settings and this question will be treated in this present work. We will consider Roumieu type classes in both the weight sequence and weight function setting. By inspecting the proofs of the main results in [14], [11] and [32] it has turned out that, up to our ability, only the complex methods from [14] admit the possibility to generalize the result to a mixed situation, see Remark 5.8 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 [32], [13], [14] and [11] 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 [10]. Then a similar notion is required in the mixed setting to obtain satisfactory theorems. Therefore, motivated by the occurring mixed ramified conditions between M and N and their associated weight functions ω M and ω N appearing in [12] the definition of the mixed growth index for sequences γ(M, N ) and for weight functions γ(σ, ω) has been given, see Section 3.1. Under restrictions of the opening of the sectors in terms of these indices, we have stated the main extension result, Theorem 5.7, for a pair of two given weight functions, using the weight matrix tool described and used in [28] and [23]. Then the results are transferred to the weight sequence case thanks to the associated weight functions.
Compared with the previous known extension results for weight functions (in the ultraholomorphic setting) we will also treat "exotic" cases here, more precisely: The growth property ω(2t) = O(ω(t)) as t → +∞, denoted by (ω 1 ) in this article, will not be needed in general anymore in the mixed situation. This property is usually a very basic assumption when working with (Braun-Meise-Taylor) weight functions ω and it is equivalent to having γ(ω) > 0 as shown by the authors in [14]. Moreover (ω 1 ) has also been used to have that the class defined by ω admits a representation by using the so-called associated weight matrix Ω, see Section 2.4 for a summary. Our main extension result Theorem 5.7 is formulated between ultraholomorphic classes defined by weight matrices and we are able to treat such a general situation since in [14] we have worked with weight functions and their associated weight matrices also in a "nonstandard" setting, i.e. not assuming (ω 1 ) necessarily. More detailed explanations will be given in Remark 5.1 below. In Appendix A such nonstandard examples will be constructed explicitly and underlining the different situation in our work here.
In the preceding extension results for one sequence, the opening of the sector where the functions are defined is at most πγ(M ). As it will be seen in Section 3, for any sequences M and N satisfying standard assumptions the mixed index γ(M, N ) is always belonging to the interval [γ(N ), µ(N )], where µ(N ) is denoting the so-called order of quasianalyticity introduced by the second author, see [26] and [13]. We know that even for strongly regular sequences N one can have γ(N ) < µ(N ) and the gap can become as large as desired, see Remark 3.12. In these situations, we can provide an extension map for any opening πγ with γ(N ) ≤ γ < µ(N ) by limiting the size of the derivatives at the origin in terms of a smaller sequence M . Furthermore, this sequence M can be chosen optimal in some sense, thanks to a modified version of the technical construction in [24,Section 4.1]. Hence we can show that the Borel map will be not surjective necessarily anymore but admitting a controlled loss of regularity, so that µ(N ), usually related to the injectivity of the Borel mapping, does have also a meaning associated with the surjectivity. For weight functions the situation is analogous by introducing the order µ(ω) in Section 3.10.
The paper is organized as follows: First, in Section 2 all necessary notation and conditions on weight sequences and functions used in this article will be introduced. In Section 3 we will define and study the new mixed growth indices γ(M, N ) and γ(σ, ω) and investigate also the connection of these values to the orders µ(N ) and µ(ω). In Sections 4 and 5 we will transfer the results from [14] to the mixed settings and providing only the necessary changes in the proofs, the main results will be Theorem 5.7 for the general mixed weight function case, Corollary 5.10 for mixed Braun-Meise-Taylor weight functions having (ω 1 ) and Theorem 5.12 for the mixed weight sequence case. In Section 6 we will prove mixed extension results fixing only the weight that defines the function space for any sector with opening smaller than πγ(·), see Theorems 6.2 and 6.4. Finally, in the Appendix A, we are providing some (counter-)examples showing γ(M, N ), γ(σ, ω) > 0, but such that all nonmixed indices γ(·) are vanishing, see Theorem A.3.

Ultradifferentiable classes defined by weight sequences and functions
Similarly we will use this notation for sequences N, S, L as well. M is called normalized if 1 = M 0 ≤ M 1 holds true and which can always be assumed without loss of generality. For any given weight sequence M and r > 0 we will write M 1/r := ( If M is log-convex and normalized, then M and the mapping j → (M j ) 1/j are nondecreasing, e.g. see [27,Lemma 2.0.4]. In this case we get M k ≥ 1 for all k ≥ 0 and We can replace in this condition M by m and by M 1/r (r > 0 arbitrary) by changing the constant C.
More generally, for arbitrary r > 0 we call M to be r-nonquasianalytic, denoted by (nq r ), if and so M has (nq r ) if and only if M 1/r has (nq).
Due to technical reasons it is often convenient to assume several properties for M at the same time and hence we define the class M ∈ SR, if M is normalized and has (slc), (mg) and (γ 1 ).
Using this notation we see that M ∈ SR if and only if m is a strongly regular sequence in the sense of [32, 1.1] (and this terminology has also been used by several authors so far, e.g. see [26], [19]). 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 [14] and [11], whereas in [32] and also in [10] the sequence M is precisely m in the notation in this work.
(5) For two weight sequences M = (M p ) p and N = (N p ) p we write M ≤ N if and only if M p ≤ N p ⇔ m p ≤ n p holds for all p ∈ N (and similarly for the sequence of quotients µ and ν) and write M N if In the relations above one can replace M and N simultaneously by m and n because M N ⇔ m n.
Some properties for weight sequences are very basic and so we introduce for convenience the following set: It is well-known (e.g. see [24,Lemma 2.2]) that for any M ∈ LC condition (mg) is equivalent to sup p∈N µ2p µp < ∞ and to sup p∈N>0 µp+1 (Mp) 1/p < ∞. A prominent example are the Gevrey sequences G r := (p! r ) p∈N , r > 0, which belong to the class SR for any r > 1.
Moreover we consider the following conditions, this list of properties has already been used in [28].
An interesting example is σ s (t) := max{0, log(t) s }, s > 1, which satisfies all listed properties except (ω 6 ). It is well-known that the ultradifferentiable class defined by using the weight t → t 1/s coincides with the ultradifferentiable class given by the weight sequence G s = (p! s ) p∈N of index s > 1.
Let σ, τ be weight functions, we write σ τ if τ (t) = O(σ(t)) as t → +∞ and call them equivalent, denoted by σ ∼ τ , if σ τ and τ σ. Motivated by the notion of a strong weight function given in [3] ω will be called a strong weight, if ω ∈ W 0 and in addition (ω snq ) is satisfied.
Concerning condition (ω nq ) we point out that hence it makes sense to consider the following generalization (ω nq r ) (analogously to (nq r )): Then ω r has (ω nq ) if and only if ω has (ω nq r ).

2.3.
Weight matrices. For the following definitions see also [23,Section 4]. Let I = R >0 denote the index set (equipped with the natural order), a weight matrix M associated with I is a (one parameter) family of weight sequences M : For convenience we will write (M) for this basic assumption on M. We call a weight matrix M standard log-convex, denoted by (M sc ), if M has (M) and Moreover, we put m x p := 2.4. Weight matrices obtained by weight functions. We summarize some facts which are shown in [23, Section 5] and will be needed in this work. All properties listed below will be valid for ω ∈ W 0 , except (2.3) for which (ω 1 ) is necessary.
(i) The idea was that to each ω ∈ W 0 we can associate a (M sc ) weight matrix Ω := {W l = (W l j ) j∈N : l > 0} by W l j := exp 1 l ϕ * ω (lj) . In general it is not clear that W x is strongly log-convex, i.e. w x is log-convex, too.
(ii) Ω satisfies In case ω has moreover (ω 1 ), Ω has also Consequently (ω 6 ) is characterizing the situation when Ω is constant. For an abstract introduction of the associated function we refer to [20, Chapitre I], see also [15, Definition 3.1]. If lim inf p→∞ (M p ) 1/p > 0, then ω M (t) = 0 for sufficiently small t, since log t p Mp < 0 ⇔ t < (M p ) 1/p holds for all p ∈ N >0 . Moreover under this assumption t → ω M (t) is a continuous increasing function, which is convex in the variable log(t) and tends faster to infinity than any log(t p ), p ≥ 1, as t → +∞. lim p→∞ (M p ) 1/p = +∞ implies that ω M (t) < +∞ for each finite t and which shall be considered as a basic assumption for defining ω M . For all t, r > 0 we get The functions h M and ω M are related by g. see also [7, p. 11]). If M ∈ LC, then M has (mg) if and only if Lemma 2.8. Let ω ∈ W 0 be given and Ω = {W l : l > 0} the matrix associated with ω. Then we have 2.9. Classes of ultraholomorphic functions. We introduce now the classes under consideration in this paper, see also [14,Section 2.5] and [11,Section 2.5]. For the following definitions, notation and more details we refer to [26,Section 2]. Let R be the Riemann surface of the logarithm. We wish to work in general unbounded sectors in R with vertex at 0, but all our results will be unchanged under rotation, so we will only consider sectors bisected by direction 0: For γ > 0 we set i.e. the unbounded sector of opening γπ, bisected by direction 0. Let M be a weight sequence, S ⊆ R an (unbounded) sector and h > 0. We define Similarly as for the ultradifferentiable case, we now define ultraholomorphic classes associated with a normalized weight function ω satisfying (ω 3 ). Given an unbounded sector S, and for every l > 0, we first define < +∞}.
(A ω,l (S), · ω,l ) is a Banach space and we put A ω,l (S). In any of the considered ultraholomorphic classes, an element f is said to be flat if f (p) (0) = 0 for every p ∈ N, that is, B(f ) is the null sequence.

Mixed growth indices for extension results
3.1. The indices γ(M, N ) and γ(σ, ω). First, for r > 0 we introduce the following condition which will be denoted by (γ r ), see [30] for r ∈ N >0 and [32, Lemma 2.2.1] for r > 0: It is immediate that M has (γ r ) if and only if M 1/r has (γ 1 ). In [32, Definition 1.3.5] the growth index γ(M ) has been introduced (for strongly regular sequences and using a definition which is not based on property (γ r ) directly). In , and µ p ≤ Cν p ≤ Cν k for all 1 ≤ p ≤ k.
(ii) Moreover, in (M, N ) γr we can equivalently consider In order to see how these definitions have been motivated, we are describing next the appearance of such (non-)mixed relations in the literature.
Remark 3.3. Condition (γ 1 ) has appeared as (standard) condition (M 3) in [15] and in [21] where it has been used to characterize the validity of Borel's theorem in the ultradifferentiable weight sequence setting. Condition (M, N ) γ1 has appeared in the mixed weight sequence situations in [7] (for the Whitney jet map) and in [31] for the Borel map. More precisely in [31] it has turned out that the characterizing condition is not (M, N ) γ1 directly, but does coincide with this condition whenever M has (mg) (as it has been assumed in [7]), see also Remark 3.8 below.
In [30], condition (γ r ) has appeared (for r ∈ N >0 ) and it has also been used by the authors in [13]. In these works (γ r ) played a key-role proving extension theorems for ultraholomorphic classes defined by weight sequences since one is working with auxiliary ultradifferentiable-like function classes first defined in [30]. In [32, Lemma 2.2.1] this condition has been introduced for r > 0 arbitrary and a connection to the value γ(M ) has been given. Finally, condition (M, N ) γr has appeared in the recent work by the authors [12] (mainly again for r ∈ N >0 ). There we have generalized the results from [31] to the auxiliary ultradifferentiable-like function classes, moreover in [12, Theorem 5.10], we have given a generalization of the ultradifferentiable Whitney extension results from [7] involving a ramification parameter r > 0.
Now we turn to the weight function situation. Let ω be a weight function and r > 0, we write (ω γr ) if with γ(ω) denoting the growth index used and introduced in [14], [11] (by considering a different growth property of ω which is not based on (ω γr )). Note also that 1 γ(ω) does coincide with the socalled upper Matuszewska index, see [1, p. 66]. For a more detailed study of γ(ω) and its connection to the indices studied in [1] we refer to Section 2 in the authors' recent work [10].

Remark 3.4.
(ω γ1 ), which is precisely (ω snq ), has appeared for ω = ω M in [15], and in [3] this condition has been characterized in terms of the validity of the ultradifferentiable Whitney extension theorem in the weight function setting. The mixed condition (σ, ω) γ1 has been treated in [5] for the Borel map and in [25] and [22] for the general Whitney jet map (see also [17] for compact convex sets). In these works, condition (σ, ω) γ1 has been identified as the characterizing property. Finally, in [12, Theorem 5.10] we have introduced (ω M , ω N ) γr in order to prove a generalization of the ultradifferentiable Whitney extension results from [7] (again by involving a ramification parameter r > 0).
Lemma 3.5. Let M, N ∈ LC be given with µ p ≤ ν p and ω, σ be weight functions with σ ω. Then we have Proof. First, if γ(N ) = 0, γ(ω) = 0, then the conclusion is clear. If these values are strictly positive, then for any 0 < r < γ(N ), γ(ω) we get that (γ r ) for N and (ω γr ) for ω hold true and so also (M, N ) γr and (σ, ω) γr are valid (for any M having µ p ≤ ν p , σ having σ ω). We recall the next statement which has been shown in [12,Lemmas 5.8,5.9] in order to see how γ(M, N ) and γ(ω M , ω N ) are related. This result is the generalization of [10, Corollary 4.6 (iii)] to the mixed setting.
Lemma 3.7. Let M, N ∈ LC be given with µ p ≤ ν p (and which is equivalent to µ r p ≤ ν r p for all r > 0 and implies M r ≤ N r ). Assume that (M, N ) γr holds true for r > 0. Then the associated weight functions are satisfying Consequently, for sequences M and N as assumed above, we always get . In [12] we have generalized this condition to We summarize several more properties.
3.10. Orders of quasianalyticity µ(N ) and µ(ω). In the ultraholomorphic weight sequence setting another important growth index is known and related to the injectivity of the asymptotic Borel map, the so-called order of quasianalyticity. It has been introduced in [26, Def. 3.3, Thm. 3.4], see also [9], [13] and [10]. We use the notation from [10] to avoid confusion in the weight function case below and to have a unified notation (coming from [1, p. 73]). For given N ∈ LC we set (3.2) µ(N ) := sup{r ∈ R >0 :  [13, p. 145]. If none (nq r ) holds true, then we put µ(N ) A first immediate consequence is the following: Proof. Note that for r > µ(N ) property (M, N ) γr cannot be valid for any choice M (see (iii) in Remark 3.9). According to this observation one can ask now the following question: Is it possible to get extension results for values γ > 0 with γ(N ) ≤ γ < µ(N )? As we will see in Section 5, for values γ < γ(M, N ) we can prove extension results in a mixed setting between M and N but it is still not clear how large the gap between γ(M, N ) and µ(N ) can be in general.
Given N ∈ LC and r > 0 we consider If for none r > 0 (3.3) holds true, then the sup in (3.4) equals 0. As commented in Remark 3.2, given N ∈ LC and r > 0 with having (3.3) for some choice M ∈ LC, µ p ≤ Cν p , then this M is sufficient to guarantee (3.3) for all 0 < r ′ < r as well.
In [26,Theorem 3.4] (see also (3.2)) it has been shown that for any N ∈ LC we have Thus for given N ∈ LC and 0 < r < µ(N ) we see that ν p ≥ p r for all p ∈ N sufficiently large and Cν p ≥ p r for all p ∈ N by choosing C large enough. Consequently, in (3.3) the choice M ≡ G r , i.e. the Gevrey sequence with index r > 0, does always make sense and the next result is becoming immediate: Proposition 3.13. Let N ∈ LC be given, then Proof. If µ(N ) = 0, then we have obviously equality. So let now µ(N ) > 0. First, if 0 < r < sup{r ∈ R >0 : (3.3) is satisfied}, then the choice p = 1 in (3.3) immediately implies (nq r ) for N , hence r ≤ µ(N ) and so the first half is shown. Conversely, let r < µ(N ) be given, then (3.3) is satisfied for µ p = p r and which can be taken as seen above. Hence µ(N ) ≤ sup{r ∈ R >0 : (3.3) is satisfied} is also shown and we are done. A disadvantage of taking directly M ≡ G r is that it is not clear that this precise choice is optimal in the sense that it is the largest sequence µ p ≤ Cν p admitting (3.3). To obtain this optimal sequence we recall the following construction: In [24, Section 4.1], and which is based on an idea arising in the proof of [21, Proposition 1.1], it has been shown that to each N ∈ LC satisfying (nq) we can associate a sequence S N with good regularity properties and which has been denoted by descendant.
For the reader's convenience we recall now the construction in the following observation and are involving a ramification parameter r > 0 as well. We have that s N,r ≤ Cs N,r ′ ⇔ S N,r ≤ CS N,r ′ for all 0 < r ′ ≤ r (since r → τ r k is increasing for all k ∈ N fixed).
Hence L N,r ∈ LC and moreover → 0 as k → +∞ and so L N,r is strictly larger than G r .
As mentioned above, condition (mg) for N does always imply this property for the descendant S N,1 =: S. However we can obtain a precise characterization of this growth behavior.
Proof. Since S ∈ LC, this sequence has (mg) if and only if sup k∈N σ 2k σ k < ∞, e.g. see [24,Lemma 2.2]. By the definitions given in Remark 3.14 we get that σ 2k ≤ Dσ k ⇔ τ k ≤ D 2 τ 2k (with τ k ≡ τ 1 k ) and which is equivalent to k ν k + j≥k νj and so to having νj and so finally (mg) is equivalent to The sum on the left-hand side above is estimated by below by k ν 2k and by above by k ν k (since (ν k ) k is increasing). Hence (3.7) implies ν 2k In this case S is equivalent to N and so S has (mg) if and only if N has this property.
(iii) Instead of having (3.6) one can study the more "compact and easy to handle" requirement νj ≥ ε for some ε > 0 and all k ∈ N we immediately get that (3.8) implies (3.6). However and which tends to infinity as k → ∞. Finally let us show that (3.8) holds (and so (3.6)). Let now k ∈ N be given with p! ≤ 2k < (p + 1)!, p ≥ 2. We split the sum ν k k j≥2k and remark that both summands are nonnegative for all k ∈ N under consideration. We study the second summand and distinguish between two cases. If k < p!, then we have k ≥ p!/2 ≥ (p − 1)! and so ν k k If p! ≤ k, then we can estimate by ν k k Thus the descendant S does have (mg). But since N violates this property, N cannot be equivalent to S and so N does not satisfy (γ 1 ).
Similarly, there does exist also an inverse construction concerning the descendant, called the predecessor. However, this does not provide any new insight, see Remark 6.6.
This growth index likely will have an interpretation for the quasianalyticity of classes of ultraholomorphic functions defined in terms of weight functions, more precisely in order to prove analogous results to [13], see also [26] and [8].
Let ω and r > 0 be given and assume that (3.10) holds with some σ, then the same choice is sufficient to have (3.10) for all 0 < r ′ < r as well. Lemma 3.11 turns now into: Lemma 3.17. Let σ and ω be weight functions (in the sense of Section 2.2) with σ ω. Then we get γ(σ, ω) ≤ µ(ω).
Proof. For any r > µ(ω) we see that (ω nq r ) is violated and so (σ, ω) γr cannot be valid for any choice σ (see (iii) in Remark 3.9).
The next result is analogous to Proposition 3.13 and showing that µ(ω) is the upper value for our considerations. Proof. If µ(ω) = 0, then we have obviously equality. So let now µ(ω) > 0.
We are closing this section by establishing now the connection between µ(ω) and µ(W l ), with W l ∈ Ω and Ω denoting the matrix associated with ω.
Proof. First, given r > 0, by the formula on p. 7 in [11] we know that the matrix associated with the weight ω r coincides with the set {V l,r := (W l/r ) 1/r : l > 0}, i.e taking the r-th root of the sequences belonging to Ω and making a re-parametrization of the index (in terms of r).  Construction of outer functions. The aim of this paragraph is to obtain holomorphic functions in the right half-plane of C whose growth is accurately controlled by two given weight functions ω and σ. Since in the forthcoming sections we want to treat the weight sequence and weight function case simultaneously we will transfer the general proofs from [14, Section 6] to a mixed setting. First we translate (σ, ω) γ1 into a property for σ ι and ω ι (recall ω ι (t) = ω(1/t) In particular (4.1) holds true for each ω satisfying (σ, ω) γ1 (and σ some other possibly quasianalytic weight).
In the next step we are generalizing [14,Lemma 6.3] to a mixed setting, the idea of the construction in the proof is coming from [32, Lemma 2.1.3].

Lemma 4.4.
Let ω and σ be two weight functions satisfying (σ, ω) γ1 . Then for all a > 0 there exists a function F a which is holomorphic on the right half-plane H 1 := {w ∈ C : ℜ(w) > 0} ⊆ C and constants A, B ≥ 1 (large) depending only on the weights ω and σ such that Proof. We are following the lines of the proof of [14,Lemma 6.3], see also [32, Lemma 2.1.3] for the single weight sequence case. Since (σ, ω) γ1 is valid, the weight ω has to satisfy (ω nq ). Hence for w ∈ H 1 we can put we need only consider in the proof a = 1 and put for simplicity F := F 1 .
where f (t) := ω ι (|t|), g u (t) := u/(t 2 + u 2 ). f and g u are symmetrically nonincreasing functions, hence the convolution too, and as argued in [14,Lemma 6.3] this means that for w → log(|F (w)|) the minimum is attained on the positive real axis and we have for all w ∈ H 1 : For the left-hand side in (4.2) consider K > 0 (small) and get The first integral is estimated by since t → −ω ι (t) is nondecreasing and since by (σ, ω) γ1 we have σ ω. Thus −ω ι (t) ≥ −D(σ ι (t)+1) for some D ≥ 1 and all t > 0 follows.
(I) Let M, N ∈ LC be given such that µ p ≤ ν p and γ(M, N ) > 0 holds true. Then for any 0 < γ < γ(M, N ) there exist constants K 1 , K 2 , K 3 , K 4 > 0 depending only on M , N and γ such that for all a > 0 there exists a function G a holomorphic on S γ and satisfying Moreover, if N has in addition (mg), then G a ∈ A { N } (S γ ) with N := (p!N p ) p∈N (and G a is flat at 0). If M has in addition (mg), then there exists K 5 > 0 depending also on given a > 0 such that Let ω and σ be weight functions such that γ(σ, ω) > 0 holds true. Then for any 0 < γ < γ(σ, ω) there exist constants K 1 , K 2 , K 3 > 0 depending only on σ, ω and γ such that for all a > 0 there exists a function G a holomorphic on S γ and satisfying Moreover, if ω is normalized and satisfies (ω 3 ), then G a ∈ A { Ω} (S γ ) (and G a is flat at 0), more precisely where Ω = {W x : x > 0} shall denote the matrix associated with ω and Ω := { W x = (p!W x p ) p∈N : x > 0}. If σ ∈ W 0 , then there exist an index x > 0 and a constant K 4 > 0 depending also on a such that where S x ∈ Σ, Σ the matrix associated with σ.
Proof. We will give some more details for the proof of (I) (following the lines of [32, Theorem 2.3.1]).
So we can use Lemma 4.4 for ω M s ≡ σ and ω N s ≡ ω and obtain a function F a holomorphic on the right half-plane and satisfying (A|w|) . Then put G a (ξ) = F a (ξ s ), ξ ∈ S δ . Note that, as sδ < 1, the ramification ξ → ξ s maps holomorphically S δ into S δs ⊆ S 1 = H 1 , and so G a is well-defined. We show that the restriction of G a to S γ ⊆ S δ satisfies the desired properties by proving that (4.3) holds indeed on the whole S δ . First we consider the lower estimate. Let ξ ∈ S δ be given, then ℜ(ξ s ) ≥ cos(sδπ/2)|ξ| s (since sδπ/2 < π/2). If B ≥ 1 denotes the constant coming from the left-hand side in (4.2) applied to the weight ω M s (or see (4.8)), then for all t, s > 0, see (2.4), and finally (2.6). Now we consider the right-hand side in (4.2) respectively in (4.8) and proceed as before. Let A be the constant coming from the right-hand side of (4.2) applied to ω N s , so and (4.3) has been proved for every ξ ∈ S δ . In order to show G a ∈ A { N } (S γ ) we put in the estimate above A 1 := A 1/s and see If we can show then by applying [14, Lemma 6.4 (i.1)] we see that G a ∈ A { N } (S γ ) (and it is a flat function at 0). Since h N ≤ 1, (4.9) holds true whenever sa 2 ≥ 1 ⇔ sa ≥ 2. But in general we have to use (mg) for N and iterate (2.7) (applied for N ) l-times, l ∈ N chosen minimal to ensure sa 2 ≥ 1 2 l . The proof of (4.4) follows analogously by iterating (mg) for M (if necessary) in order to get rid of the exponent 2aK 2 .
The remaining statements, in particular the estimate (4.7), follow analogously as in the proof of [14,Theorem 6.7], replacing τ by ω or σ in the arguments.

Right inverses for the asymptotic Borel map in ultraholomorphic classes in sectors
The aim of this section is to obtain an extension result in the ultraholomorphic classes considered in a mixed setting for both the weight sequence and the weight function approach following the proofs and techniques in [14,Section 7]. The existence of the optimal flat functions G a obtained in Theorem 4.6 will be the main ingredient in the proof which is inspired by the same technique as in previous works of A. Lastra, S. Malek and the second author [18,19] in the single weight sequence approach. Although for the general construction the weight functions σ and ω need not be normalized, we are interested in working with the weight matrices associated with them, which will be standard log-convex if we ask for normalization and (ω 3 ) to hold. Note that any weight function may be substituted by a normalized equivalent one (e.g. see [3, Remark 1.2 (b)]) and equivalence preserves the property (ω 3 ), so it is no restriction to ask for normalization from the very beginning.
1. An important difference to the complete approach in [14] is, see also the comments given in the introduction in Section 7 there, that condition γ(ω) > 0 and which amounts to (ω 1 ) as shown in [14, Lemma 4.2] will not be valid in general anymore in the mixed situation. In the following we are only requiring γ(σ, ω) > 0 and recall that γ(σ, ω) ≥ γ(ω) as shown above. An explicit example of this situation, having γ(σ, ω) > 0 (as large as desired) and γ(ω) = γ(σ) = 0 will be provided in the Appendix A below. We are able to treat this situation by recognizing that in [14] we have worked in a very general framework for weight functions and the assumption γ(ω) > 0 can be replaced by γ(σ, ω) > 0 without causing problems.
Recall that (ω 1 ) is standard in the ultradifferentiable setting and thus our techniques make it possible to treat "exotic" weight function situations as well. Moreover (ω 1 ) has also been used to have that the class defined by ω admits a representation by using the associated weight matrix Ω, see Section 2.4. Thus the warranty that the ultraholomorphic (and also the ultradifferentiable) spaces associated with ω and its corresponding weight matrix Ω coincide is not clear anymore, see the comments preceding (2.9). Therefore the main and most general ultraholomorphic extension result Theorem 5.7 deals with a mixed situation between classes defined by (associated) weight matrices. If one imposes (ω 1 ) on the weights one is able to prove a mixed version of classes defined by weight functions, see Corollary 5.10. Finally, in Theorem 5.12 we will treat the mixed weight sequence case as well.
The function e a enjoys the following properties: (i) z −1 e a (z) is uniformly integrable at the origin, it is to say, for any t 0 > 0 we have |e a (te iτ )|dt < ∞.
(ii) There exist constants K > 0, independent from a, and C > 0, depending on a, such that (iii) For ξ ∈ R, ξ > 0, the values of e a (ξ) are positive real.
Proof. The proof is completely the same as for [14, Lemma 7.1], for (i) we apply the right-hand side in (4.5), for (ii) we use (4.6) and (2.8) together with the definition given in (2.5).
Analogously as in [14,Definition 7.2] we introduce now the moment function associated with e a .
Definition 5.4. We define the moment function associated with the function e a (introduced in the previous Lemma) as From Lemma 5.3 and the definition of h W x in (2.5) we see that for every p ∈ N, So, we easily deduce that the function m a is well defined and continuous in {λ : ℜ(λ) ≥ 0}, and holomorphic in {λ : ℜ(λ) > 0}. Moreover, m a (ξ) is positive for every ξ ≥ 0, and the sequence (m a (p)) p∈N is called the sequence of moments of e a . The next result is generalizing [14,Proposition 7.3], which is similar to Proposition 3.6 in [18], to a mixed setting.
Proposition 5.5. Let σ and ω be normalized weight functions with γ(σ, ω) > 0 and such that both weights satisfy (ω 3 ). Let Σ = {S x : x > 0} and Ω = {W x : x > 0} be the weight matrices associated with σ and ω respectively, and for 0 < γ < γ(σ, ω) and a > 0 let G a , e a , m a be the functions previously constructed. Then, there exist constants C 1 , C 2 > 0, both depending on a, such that for every p ∈ N one has where K 2 and K 3 are the constants, not depending on a, appearing in (4.5).
Proof. The proof follows the lines as in [14,Proposition 7.3] (based on the arguments by O. Blasco in [2]). For the second estimate in (5.2) we use the second inequality in (4.5) (and here also (2.2) is used); the first estimate in (5.2) follows by applying the first inequality in (4.5).

5.6.
Main extension results. Now we are able to formulate and proof the generalization of the main extension result [14,Theorem 7.4].
Theorem 5.7. Let σ and ω be normalized weight functions with γ(σ, ω) > 0 and such that both weights satisfy (ω 3 ) and 0 < γ < γ(σ, ω). Moreover we denote by Σ = {S x : x > 0} and Ω = {W x : x > 0} the weight matrices associated with σ and ω respectively and consider the matrices where S x := (p!S x p ) p∈N and W x := (p!W x p ) p∈N . Then, there exists a constant k 0 > 0 such that for every x > 0 and every h > 0, one can construct a linear and continuous map which allows us to write the preceding difference as Then, we have From (5.4) we deduce that where in the last step we have used that 0 < u < R 0 = K 2 /(4h) we have 1 − 2hu/K 2 > 1/2. In order to estimate f 2 (z), observe that for u ≥ R 0 and 0 ≤ p ≤ N −1 we always have u p ≤ R p 0 u N /R N 0 , and so, using again (5.4) and the value of R 0 , we may write Then, we deduce that In order to conclude, it suffices then to obtain estimates for ∞ 0 |e a (u/z)|u N −1 du. For this, note first that, by the estimates in (4.5), Now, we can follow the first part of the proof of [14,Proposition 7.3] to obtain that Gathering (5.5), (5.6), (5.7) and (5.8), we get A straightforward application of Cauchy's integral formula yields that there exists a constant r, depending only on γ and δ, such that whenever z is restricted to belong to S γ , one has that for every p ∈ N, So, putting k 0 := 4K3r K2 (independent from x and h), we see that f λ ∈ A W 8x ,k0h (S γ ) and f λ W 8x ,k0h ≤ 2C2 C1 |λ| S x ,h . Since the map sending λ to f λ is clearly linear, this last inequality implies that the map is also continuous from Λ S x ,h into A W 8x ,k0h (S γ ). Finally, from (5.9) one may easily deduce that B(f λ ) = λ, and we conclude.  Corollary 5.10. Let σ, ω ∈ W be given, so that γ(σ, ω) ≥ γ(ω) > 0, and let 0 < γ < γ(σ, ω) and Σ = {S x : x > 0} and Ω = {W x : x > 0} be the weight matrices associated with σ and ω respectively and consider the matrices Let τ 1 ∈ W and τ 2 ∈ W be the weight functions coming from Theorem 5.9 applied to σ and ω respectively, so Then, for every l > 0 there exists l 1 > 0 such that there exists a linear and continuous map such that for all λ ∈ Λ τ1,l one has B • E τ1,τ2 l (λ) = B(f λ ) = λ. Thus we have shown that Proof. Let T i := {T i,x : x > 0} be the weight matrix associated with the weight function τ i , i.e. T i,x p := exp 1 x ϕ * τi (xp) for each x > 0 and p ∈ N, i = 1, 2. We may apply (2.9) in order to deduce that and, as it has already been remarked in [14,Corollary 7.6], we get T 1 {≈} Σ, T 2 {≈} Ω. For the rest of the proof we follow [14,Corollary 7.6] and use for the extension Theorem 5.7.

6.
Mixed extension results with only one fixed weight 6.1. Extension results where the weight sequence/function defining the function space is fixed. Using the properties of the index µ(N ) and the construction of the ramified descendant of Section 3.10 we can now prove the following variant of Theorem 5.12.
Theorem 6.2. Let N ∈ LC be given with µ(N ) > 0 and let 0 < r < µ(N ). Assume that (3.6) holds true for N 1/r . Then there does exist L ∈ LC having (mg) such that for each 0 < γ < r we get: There exists a constant k 1 > 0 such that for every h > 0, one can construct a linear and continuous map The sequence L is maximal among those M ∈ LC satisfying µ k ≤ Cν k and (M, N ) γr .
The important difference between Theorem 5.12 and this result is that, of course, L is depending here on given r.
Proof. Let 0 < γ < r < µ(N ) be given according to the requirements above. Then we consider the sequence L N,r defined via the descendant S N,r in (3.5), see Section 3.10.
As seen there we have that (L N,r , N ) γr holds true and which proves γ(L N,r , N ) ≥ r > γ. Moreover λ N,r k ≤ Cν k and since N 1/r has (3.6), Lemma 3.15 yields (mg) for S N,r and so for L N,r , too. Thus we can apply Theorem 5.12 to M ≡ L N,r and N and γ unchanged to obtain: There exists a constant k 1 > 0 such that for every h > 0, one can construct a linear and continuous map p ) p∈N and so (6.1) follows by taking L ≡ L N,r . Remark 6.3. Let N ∈ LC be given with µ(N ) > 0. If N has in addition (mg), then each S N,r and L N,r , 0 < r < µ(N ), share this property, see (iv) in Remark 3.14.
Using µ(ω) we can prove Theorem 6.2 for the weight function setting, so we have the following variant of Theorem 5.7. Theorem 6.4. Let ω be a normalized weight function with (ω 3 ) and µ(ω) > 0. Then for all 0 < r < µ(ω) there does exist a normalized weight function σ satisfying (ω 3 ) such that for each 0 < γ < r we get: There exists a constant k 0 > 0 such that for every x > 0 and every h > 0, one can construct a linear and continuous map Thus we have shown that B(A { Ω} (S γ ) ⊇ Λ { Σ} (by using for Σ and Ω the same notation as in Theorem 5.7). The function σ is chosen minimal among those normalized weight functions τ satisfying (ω 3 ), τ ω (i.e. ω(t) = O(τ (t))) and enjoying (τ, ω) γr .
Proof. According to this value r > 0 given, we consider the weight κ 1/r ω r (see (3.12)) and so (κ 1/r ω r , ω) γr is valid. This weight enjoys all properties like ω except normalization (by definition).
But normalization can be achieved w.l.o.g. by switching to an equivalent weight (redefining κ 1/r ω r near 0, e.g. see [3, Remark 1.2 (b)]) and which will be denoted by σ. Thus γ(σ, ω) ≥ r > γ and we can apply Theorem 5.7 to these weights σ and ω and the value γ and conclude.
Remark 6.5. Due to (3.13) one could try to restate Theorem 6.4 by applying Theorem 6.2 to N ≡ W x , x > 0 arbitrary. However, once chosen γ < µ(ω) = µ(W x ) in Theorem 6.4 we obtain an extension for another weight function σ such that moving the index x we are staying in the same weight matrix associated with σ by the precise choice x → 8x. So here we can take some uniform choice for all sequences in Ω (by obtaining a weight matrix not depending on given x) and which is not following by applying Theorem 6.2.
Remark 6.6. Naturally one might ask what happens in the dual situation, that is, fixing the weight sequence or weight function that controls the derivatives at the origin. However, in this case the inverse construction concerning the descendant, called the predecessor, see [24,Remark 4.3], does not provide any new information, since the bounds for the opening are the same as those that are known for the one level extension theorem.
(i) and (ii) together tell us that M is lying between two Gevrey sequences. By proving (i) one can verify that µ(M ) = lim inf p→∞ log(µp) log(p) = γ holds true. A slight variation of Lemma A.1 yields the following.
Lemma A.2. Let γ > 1 be given, then there exists a sequence M ∈ LC such that (i) M does satisfy (nq), more precisely µ k ≥ k γ for all k ∈ N and so even (nq γ−ε ) holds true for any ε > 0 (small), (ii) µ k ≤ k 2γ 2 for all k ∈ N, (iii) M does not satisfy (β 3 ) or equivalently M = (p!M p ) p∈N does not satisfy (β 1 ) (and consequently M is not strongly nonquasianalytic too), (iv) M does not satisfy (mg).
In fact any choice β > 2γ and α := β − 1 would be working for the following proof.
Again by construction µ(M ) = lim inf k→∞ log(µ k ) log(k) = γ holds true. Using Lemma A.1 we can now underline the importance of Theorem 5.7 and in particular of Theorem 5.12 as follows.
Theorem A.3. There do exist sequences M and N satisfying all requirements from Theorem 5.12 but such that γ(M ) = γ(N ) = γ(ω M ) = γ(ω N ) = 0. Moreover we can achieve γ(M, N ) to be as large as desired.
But γ(M ) = γ(N ) = 0 holds true: (β 1 ) or equivalently (γ 1 ) is violated for both sequences M = (p!M p ) p and N = (p!N p ) p (by property (iii)), and so γ(M ) = γ(N ) = 0. And this is equivalent to having γ(ω M ) = γ(ω N ) = 0, because both sequences have (mg), for a proof see [10,Section 4]. In particular we have seen that neither M ∈ SR nor N ∈ SR and by the characterizations shown in [21], not any to M or N equivalent sequence L can belong to class SR.
Let M and N denote the sequences from Theorem A.3 above with parameters γ ′ > 1 and γ > 1 subject to (A.1). Then by applying Theorem 5.12 for any given 0 < δ < γ(M, N ) there is k 1 > 0 such that for every h > 0 there exists a continuous linear extension map This kind of extension result is not covered by the theory developed by the authors in [14]. More precisely [14,Theorem 7.4] fails since γ(ω M ) = γ(ω N ) = 0 and also the mixed setting from [14, Section 7.1] cannot be applied, neither to M nor to N directly.
Note that both M and N have (mg), thus both matrices associated with ω M and ω N are constant, see (iii) in Lemma 2.7 and Remark 2.5. Now let M and N be the sequences constructed in Lemma A.2 with parameters γ ′ and γ respectively and here we require that Again it is straightforward to check that (M, N ) γr holds true for all 0 < r < γ and which implies γ(M, N ) ≥ γ > 0. Since µ(N ) = γ we again have γ(M, N ) = γ and by having µ p ≤ ν p , Lemma 3.7 yields γ(ω M , ω N ) ≥ γ(M, N ) = γ.
But here neither M nor N does satisfy (mg) and we cannot apply Theorem 5.12 directly. But Theorem 5.7 applied to σ ≡ ω M and ω ≡ ω N with Σ denoting the matrix associated with ω M and Ω the matrix associated with ω N , yields now the following extension result: For any given 0 < δ < γ(ω M , ω N ) there exists a constant k 0 > 0 such that for every x > 0 and every h > 0, one can construct a linear and continuous extension map Hence we have shown B(A { Ω} (S γ )) ⊇ Λ { Σ} .
As mentioned in the introduction and in Remark 5.1 we have that starting directly with a Braun-Meise-Taylor weight function ω with γ(ω) = 0 we do not have (ω 1 ) (as shown in [10, Corollary 2.14]). Hence a basic assumption in the whole theory of ultradifferentiable functions defined in terms of ω, is violated from the very beginning.