Surjectivity of the asymptotic borel map in Carleman–Roumieu ultraholomorphic classes defined by regular sequences

RACSAM (2021) 115:181 https://doi.org/10.1007/s13398-021-01119-y ORIGINAL PAPER Surjectivity of the asymptotic Borel map in Carleman–Roumieu ultraholomorphic classes defined by regular sequences Javier Jiménez-Garrido1,2 ·Javier Sanz2,3 ·Gerhard Schindl4 Received: 5 August 2020 / Accepted: 16 August 2021 / Published online: 4 September 2021 © The Author(s) 2021 Abstract We study the surjectivity of, and the existence of right inverses for, the asymptotic Borel map in Carleman–Roumieu ultraholomorphic classes defined by regular sequences in the sense of E. M. Dyn’kin. We extend previous results by J. Schmets and M. Valdivia, by V. Thilliez, and by the authors, and show the prominent role played by an index, associated with the sequence,thatwasintroducedbyV.Thilliez.Thetechniquesinvolveregularvariation,integral transforms and characterization results of A. Debrouwere in a half-plane, stemming from his study of the surjectivity of the moment mapping in general Gelfand–Shilov spaces. Keywords Carleman ultraholomorphic classes ·Asymptotic expansions · Borel–Ritt–Gevrey theorem ·Laplace transform ·regular variation Mathematics Subject Classification 30D60 ·30E05 ·47A57 ·34E05 1 Introduction The concept of asymptotic expansion, introduced by H. Poincaré in 1886, has played an essential role in the understanding of the analytical meaning of the formal power series solutions to large classes of functional equations (ordinary and partial differential equations, BJavier Sanz [email protected]a.es Javier Jiménez-Garrido jesusjavier[email protected] Gerhard Schindl [email protected] 1Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Facultad de Ciencias, Avda. de los Castros, 48, 39005 Santander, Spain 2Instituto de Investigación en Matemáticas IMUVA, Universidad de Valladolid, Valladolid, Spain 3Departamento de Álgebra, Análisis Matemático, Geometría y Topología, Universidad de Valladolid, Facultad de Ciencias, Paseo de Belén 7, 47011 Valladolid, Spain 4Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria 123 181 Page 2 of 18 J. Jiménez-Garrido et al. difference and q-difference equations, and so on). The existence of such an expansion for a complex holomorphic function in a sector Sof the Riemann surface of the logarithm amounts to a precise control on the growth of its derivatives, and this fact gives the link with ultraholomorphic classes, on whose elements’ derivatives are usually imposed local or global bounds in terms of a weight sequence M=(Mp)p∈N0of positive real numbers. See Sect. 2.3 for an account in this respect. The asymptotic Borel map sends a function in one of such classes into its formal power series of asymptotic expansion, and in many instances it is important to decide about its injectivity and surjectivity when considered between suitable spaces. We refer the reader to our previous paper [10], whose introduction contains a non comprehensive historical account of the results in this respect, and where the problem of injectivity in unbounded sectors and for general weight sequences is completely closed, by solving a pending case not covered by the powerful results of S. Mandelbrojt [14]andB. Rodríguez-Salinas [17]. Regarding surjectivity, the classical Borel–Ritt–Gevrey theorem of B. Malgrange and J.- P. Ramis [16], solving the case of Gevrey asymptotics, was extended to different more general situations by J. Schmets and M. Valdivia [19], V. Thilliez [20,21] and the authors [10,18]. For a weight sequence M, our main satisfactory results have been the following: (i) The Borel map is never bijective [10, Theorem 3.17]. (ii) The strong nonquasianalyticity condition is equivalent to the fact that the index γ(M)of V. Thilliez is positive, and this condition is necessary for surjectivity [10, Lemma 4.5]. (iii) For a sector Sγof opening πγ (γ>0) and under the hypothesis of strong regularity for M, V. Thilliez [21] proved surjectivity of the Borel map for γ<γ(M).Conversely, we have proved [10, Corollaries 4.18 and 4.19] that surjectivity implies γ≤γ(M) and, in case γ(M)is a rational number, even γ<γ(M)is obtained whenever uniform asymptotics are considered. (iv) Surjectivity was completely characterized whenever Madmits a nonzero proximate order [18, Theorem 6.1]. The present paper intends to go one step further and complete the partial information given in [10, Theorem 4.14] concerning the case of regular weight sequences in the sense of E.M.Dyn’kin[6],whichinsteadofmoderategrowthsatisfythemilderconditionofderivation closedness (see Sect. 2.2 for the precise definitions). Moreover, the existence of extension operators, continuous linear right inverses for the Borel map, is studied in this general case. It is interesting to note that the condition (β2), introduced by H.-J. Petzsche [15] in a similar study for ultradifferentiable classes, plays again a prominent role here, and its relationship with other conditions of rapid variation is elucidated. In particular, the condition γ(M)=∞, stronger than (β2), guarantees the surjectivity of the Borel map and the existence of global extension operators for any sector in the Riemann surface of the logarithm. We have not considered in this paper the closely related case of Beurling ultraholomorphic classes.ThesurjectivityoftheBorelmapinthissetting,forγ<γ(M)andunderthemoderate growth condition, was established by V. Thilliez [21, Cor. 3.4.1], and A. Debrouwere [4]has very recently proved the existence of extension operators under the same hypotheses by using results from the splitting theory of Fréchet spaces. We think our techniques apply, with really slight modifications, to the Beurling framework, and results similar to the ones presented here could be established for regular sequences, so extending the results in [4]. We note that, in this context, the condition (β2)(see Sect. 4) is no longer needed for global extension operators to exist, and so this condition will not appear in the corresponding Beurling version of Theorem 4.2 in this paper. 123 Surjectivity of the asymptotic Borel map… Page 3 of 18 181 2 Preliminaries 2.1 Notation We set N:= {1,2,...},N0:= N∪{0}.Rstands for the Riemann surface of the logarithm, where the notation z=|z|eiθrefers to the element (|z|,θ)∈(0,∞)×R.C[[z]] is the space of formal power series in zwith complex coefficients. For γ>0, we consider unbounded sectors bisected by direction 0, Sγ:= z∈R:|arg(z)|<γπ 2 or, in general, bounded or unbounded sectors S(d,α,r):= z∈R:|arg(z)−d|<απ 2,|z|<r,S(d,α):= z∈R:|arg(z)−d|<απ 2 with bisecting direction d∈R, opening απ and (in the first case) radius r∈(0,∞). A sector Tis said to be a proper subsector of a sector Sif T⊂S(where the closure of Tis taken in R, and so the vertex of the sector is not under consideration). In case such Tis also bounded, we say it is a bounded proper subsector of S. 2.2 Weight sequences and their properties In what follows, M=(Mp)p∈N0will always stand for a sequence of positive real numbers, and we will always assume that M0=1. We define its sequence of quotients m=(mp)p∈N0 by mp:= Mp+1 Mp,p∈N0; clearly, the knowledge of Mamounts to that of m,sinceMp= m0···mp−1,p∈N. We will denote by small letters the quotients of a sequence given by the corresponding capital letters. The following properties for a sequence will play a role in this paper: (i) Mis logarithmically convex (for short, (lc)) if M2 p≤Mp−1Mp+1,p∈N. (ii) Mis stable under differential operators or satisfies the derivation closedness condition (briefly, (dc)) if there exists D>0 such that Mp+1≤Dp+1Mp,p∈N0. (iii) Mis of, or has, moderate growth (briefly, (mg)) whenever there exists A>0 such that Mp+q≤Ap+qMpMq,p,q∈N0. (iv) Msatisfies the condition (snq) if there exists B>0 such that ∞  q=p Mq (q+1)Mq+1≤BMp Mp+1,p∈N0. It will be convenient to introduce the notation  M:= (p!Mp)p∈N0. All these properties are preserved when passing from Mto  M. In the classical work of H. Komatsu [11], the properties (lc), (dc) and (mg) are denoted by (M.1),(M.2)and (M.2), respectively, while (snq) for Mis the same as property (M.3)for  M. Obviously, (mg) implies (dc). 123 181 Page 4 of 18 J. Jiménez-Garrido et al. Thesequenceofquotients misnondecreasingifandonlyifMis(lc).Inthiscase, it is wellknown that (Mp)1/p≤mp−1for every p∈N, the sequence ((Mp)1/p)p∈Nis nondecreasing, and limp→∞(Mp)1/p=∞if and only if limp→∞ mp=∞. In order to avoid trivial situations, we will restrict from now on to (lc) sequences Msuch that limp→∞ mp=∞, which will be called weight sequences. Following E. M. Dyn’kin [6], if Mis a weight sequence and satisfies (dc), we say  Mis regular. According to V. Thilliez [21], if Msatisfies (lc), (mg) and (snq), we say Mis strongly regular; in this case Mis a weight sequence, and the corresponding  Mis regular. We mention some interesting examples. In particular, those in (i) and (iii) appear in the applications of summability theory to the study of formal power series solutions for different kinds of equations. (i) The sequences Mα,β := p!αp m=0logβ(e+m)p∈N0,whereα>0andβ∈R,are strongly regular (in case β<0, the first terms of the sequence have to be suitably modified in order to ensure (lc)). In case β=0, we have the best known example of a strongly regular sequence, Mα:= Mα,0=(p!α)p∈N0, called the Gevrey sequence of order α. (ii) The sequence M0,β := (p m=0logβ(e+m))p∈N0, with β>0, satisfies (lc) and (mg), and mtends to infinity, but (snq) is not satisfied. (iii) For q>1, Mq:= (qp2)p∈N0satisfies (lc), (dc) and (snq), but not (mg). Two sequences M=(Mp)p∈N0and L=(Lp)p∈N0of positive real numbers, with respective quotients mand , are said to be: (i) equivalent, and we write M≈L, if there exist positive constants A,Bsuch that ApMp≤Lp≤BpMp,p∈N0. (ii) strongly equivalent, and we write m≃, if there exist positive constants a,bsuch that amp≤p≤bmp,p∈N0. Whenever m≃we have M≈L, but not conversely. As an example, for α>0wesetLα:= ((1+αp))p∈N0,wheredenotes the Eulerian Gamma function; it is well-known that α≃((p+1)α)p∈N0and so Lα≈Mα, the Gevrey sequence of order α. Conditions (dc) and (mg) are clearly preserved by ≈, and so also by ≃, for general sequences; (snq) is obviously preserved for weight sequences by ≃, but also by ≈(see the work of H.-J. Petzsche [15, Cor. 3.2] for an indirect argument, and our paper [9, Cor. 3.14] for a direct proof of a more general statement). Given two sequences Mand L, we use the notation M·L=(MpLp)p∈N0and M/L= (Mp/Lp)p∈N0. We will use the fact that Msatisfies (mg), respectively (dc), if and only if M·Lαor M/Lαsatisfy (mg), resp. (dc), for some α>0. 2.3 Asymptotic expansions, ultraholomorphic classes and the asymptotic Borel map In this paragraph Sis a sector and Ma sequence. We start by recalling the concept of asymptotic expansion. We say a holomorphic function fin Sadmits the formal power series  f=∞ p=0apzp∈ C[[z]] asits{M}-asymptotic expansion in S(whenthevariabletendsto 0)if foreverybounded 123 Surjectivity of the asymptotic Borel map… Page 5 of 18 181 proper subsector Tof Sthere exist CT,AT>0 such that for every p∈N0, one has f(z)− p−1  n=0 anzn≤CTAp TMp|z|p,z∈T. If the expansion exists, it is unique, and we will write f∼{M} fin S. A{M}(S)stands for the space of functions admitting {M}-asymptotic expansion in S. We say a holomorphic function f:S→Cadmits  fas its uniform {M}-asymptotic expansion in G (of type 1/AforsomeA>0) if there exists C>0 such that for every p∈N0, one has f(z)− p−1  n=0 anzn≤CApMp|z|p,z∈S.(2.1) In this case we write f∼u {M},A fin S,and  Au {M},A(S)denotes the space of functions admitting uniform {M}-asymptotic expansion of type 1/Ain S, endowed with the norm fM,A,∼ u:= sup z∈S,p∈N0f(z)−p−1 k=0akzk ApMp|z|p, which makes it a Banach space.  Au {M}(S)stands for the (LB)space of functions admitting a uniform {M}-asymptotic expansion in S, obtained as the union of the previous classes when Aruns over (0,∞). When the type needs not be specified, we simply write f∼u {M} fin S. Note that, taking p=0in(2.1), we deduce that every function in  Au {M}(S)is a bounded function. Finally,wedefineforeveryA>0 the class A{M},A(S)consisting of the functions holomorphic in Ssuch that fM,A:= sup z∈S,p∈N0 |f(p)(z)| ApMp <∞. (A{M},A(S), ·M,A) is a Banach space, and A{M}(S):= ∪A>0A{M},A(S)is called a Carleman–Roumieu ultraholomorphic class inthe sector S, whose natural inductive topology makes it an (LB)space. We warn the reader that these notations do not agree with the ones used in [10,18], where  A{M}(S)was denoted by  AM(S), Au {M}(S)by  Au M(S),A{M},A(S)by AM/L1,A(S), and A{M}(S)by AM/L1(S). If Mis (lc), the spaces A{M}(S), Au {M}(S)and  A{M}(S)are algebras, and if Mis (dc) they are stable under taking derivatives. Moreover, if M≈Lthe corresponding classes coincide. Since the derivatives of f∈A{M},A(S)are Lipschitz, for every p∈N0one may define f(p)(0):= lim z∈S,z→0f(p)(z)∈C.(2.2) As a consequence of Taylor’s formula and Cauchy’s integral formula for the derivatives, there is a close relation between Carleman–Roumieu ultraholomorphic classes and the concept of asymptotic expansion (the proof may be easily adapted from [1]). Proposition 2.1 Let Mbe a sequence and S be a sector. Then, 123 181 Page 6 of 18 J. Jiménez-Garrido et al. (i) If f ∈A{ M},A(S)then f admits  f:= p∈N0 1 p!f(p)(0)zpas its uniform {M}-asymptotic expansion in S of type 1/A,where(f(p)(0))p∈N0is given by (2.2). Moreover, fM,A,∼ u≤ f M,A, and so the identity A{ M},A(S)→ Au {M},A(S)is continuous. Consequently, we also have that A{ M}(S)⊆ Au {M}(S)⊆ A{M}(S), and A{ M}(S)→ Au {M}(S)is continuous. (ii) f∈ A{M}(S)if and only if for every (bounded or, if possible, unbounded) proper subsector T of S there exists AT>0such that f |T∈A{ M},AT(T). In case any of the previous holds and f ∼{M}∞ p=0apzp, then for every such T and every p ∈N0one has ap=lim z→0 z∈T f(p)(z) p!,(2.3) and we can set f (p)(0):= p!ap. (iii) If S is unbounded and T is a proper subsector of S, then there exists a constant c = c(T,S)>0such that the restriction to T , f |T, of functions f defined on S and admitting a uniform {M}-asymptotic expansion in S of type 1/A>0, belongs to A{ M},cA(T), and f|T M,cA ≤fM,A,∼ u. So, the restriction map from  Au {M},A(S)to A{ M},cA(T)is continuous, and it is also continuous from  Au {M}(S)to A{ M}(T). One may accordingly define classes of formal power series C[[z]]{M},A=⎧ ⎨ ⎩ f= ∞  p=0 apzp∈C[[z]] :  fM,A:= sup p∈N0 |ap| ApMp <∞⎫ ⎬ ⎭. (C[[z]]{M},A,|·|M,A)is a Banach space and we put C[[z]]{M}:= ∪A>0C[[z]]{M},A,again an (LB)space. Given f∈ A{M}(S)with f∼{M} f, and taking into account (2.3), it is straightforward that  f∈C[[z]]{M}, so it is natural to consider the asymptotic Borel map  B: A{M}(S)−→ C[[z]]{M} sending a function f∈ A{M}(S)into its {M}-asymptotic expansion  f. By Proposition 2.1.(i) the asymptotic Borel map may be defined in  Au {M}(S),A{ M}(S)and A{ M},A(S)(in the last case, with target space C[[z]]{M},A). We would like to highlight that, alternatively, the target space for the Borel map could be considered to be a space of sequences comprising the derivatives at 0 of a function fin the classes, as defined in (2.2), and subject to the corresponding control on the growth of their terms. This equivalent approach has been followed by many authors, and in particular in the works of J. Schmets and M. Valdivia [19] and A. Debrouwere [3]. Note that their results, stated in this paper as Theorems 3.4,4.1 and 4.3, have been adapted to our setting. If Mis (lc),  Bis a homomorphism of algebras; if Mis also (dc), differentiation commutes with  B. Moreover, it is continuous when considered between the corresponding Banach or (LB)spaces previously introduced. Finally, note that if M≈L,thenC[[z]]{M}=C[[z]]{L}, and the corresponding Borel maps are in all cases identical. 123 Surjectivity of the asymptotic Borel map… Page 7 of 18 181 Since the problem under study is invariant under rotation, we will focus on the surjectivity of the Borel map in unbounded sectors Sγ. So, we define S{ M}:={γ>0; B:A{ M}(Sγ)−→ C[[z]]{M}is surjective},  Su {M}:={γ>0; B: Au {M}(Sγ)−→ C[[z]]{M}is surjective},  S{M}:={γ>0; B: A{M}(Sγ)−→ C[[z]]{M}is surjective}. We again note that these intervals were respectively denoted by SM, Su Mand  SMin [10]. It is clear that S{ M}, Su {M}and  S{M}are either empty or left-open intervals having 0 as endpoint, called surjectivity intervals. Using Proposition 2.1, items (i) and (iii), we easily see that ( Su {M})◦⊆S{ M}⊆ Su {M}⊆ S{M},(2.4) where I◦stands for the interior of the interval I. 3 Surjectivity results for regular sequences In the study of the surjectivity of the Borel map the index γ(M), introduced in this regard by V. Thilliez [21, Sect. 1.3] for strongly regular sequences M, will play a central role. His definition makes sense for (lc) sequences, in this case γ(M)∈[0,∞],anditmaybe equivalently expressed by different conditions: (i) A sequence (cp)p∈N0is almost increasing if there exists a>0 such that for every p∈N0we have that cp≤acqfor every q≥p. It was proved in [8,9] that for any weight sequence Mone has γ(M)=sup{γ>0:(mp/(p+1)γ)p∈N0is almost increasing}.(3.1) (ii) For any β>0 we say that msatisfies the condition (γβ)if there exists A>0 such that ∞  =p 1 (m)1/β ≤A(p+1) (mp)1/β ,p∈N0.(γβ) For β=1, this condition was introduced by H. Komatsu [11], and named (γ1)after H.-J. Petzsche [15]). Subsequently, it was considered for β∈Nby J. Schmets and M. Valdivia [19]. We have obtained (see [7,9]) that for a weight sequence M, γ(M)=sup{β>0;msatisfies (γβ)}; γ(M)>β ⇐⇒ msatisfies (γβ). (3.2) Note that the definition of this index can be coherently extended for an arbitrary sequence Mof positive real numbers as γ(M)=sup{γ∈R:(mp/(p+1)γ)p∈N0is almost increasing} (with values ∞,resp.−∞, in case the previous set is R, resp. empty). We consider in the sequel this extension. Whenever  M=(p!Mp)p∈N0is (lc) we have (see [7, Ch. 2] and [9, Cor. 3.13]) that γ(M)>0 if and only if Mis (snq). We recall also the following result for later use. Lemma 3.1 ([9], Remark 3.15). For an arbitrary sequence M1such that γ(M1)>1,there exists a weight sequence M2such that  m2≃m1, and so γ( M2)=γ(M1). 123 181 Page 8 of 18 J. Jiménez-Garrido et al. A straightforward verification shows that for any sequence Mand for every s>0 one has γ((p!sMp)p∈N0)=γ (((1+sp)Mp)p∈N0)=γ(M)+s,(3.3) γ((Mp/p!s)p∈N0)=γ(Mp/((1+sp))p∈N0)=γ(M)−s.(3.4) As a consequence of the characterization of the surjectivity of the Borel map in the ultradifferentiable setting given by H.-J. Petzsche [15, Thm. 3.5], we proved the following result, already announced by V. Thilliez in [21]. Lemma 3.2 ([10], Lemma 4.5). Let Mbe a weight sequence. If  S{M}=∅,thenMhas (snq) or, equivalently, γ(M)>0. Our aim in this section is to solve (except for some limiting cases) the problem of surjectivity whenever Mis a weight sequence satisfying (dc) or, in other words,  Mis a regular sequence in the sense of Dyn’kin. Our previous main result is the following. We denote by xthe greatest integer not exceeding x. Theorem 3.3 ([10], Thm. 4.14 and Cor. 4.15). Let Mbe a weight sequence satisfying (dc). (i) Let α>0be such that  B: Au {M}(Sα)→C[[z]]{M}is surjective. Then, γ(M)>α. (ii) If we have that  Su {M}=(0,∞),thenS { M}= Su {M}= S{M}=(0,∞)and γ(M)=∞. One has S{ M}⊆ Su {M}⊆(0,γ(M)+1); if moreover γ(M)∈N,thenS { M}⊆ Su {M}⊆ (0,γ(M)). At that moment and to the best of our knowledge, no general surjectivity result had been proved for regular  M,exceptforthe specialcaseof theq-GevreysequencesMq=(qp2)p∈N0, q>1,see C.Zhang [22].Ina recentcollaboration ofthefirsttwoauthorswith A.Debrouwere [5] we have studied the existence and uniqueness of solutions for the Stieltjes moment problem in Gelfand–Shilov spaces, subspaces of the Schwartz space of rapidly decreasing smooth functions for which the growth of the products of monomials times the derivatives of their elements is controlled in terms of weight sequences. By a suitable application of the Fourier transform, there exists a close connection between this problem and the surjectivity or injectivity of the asymptotic Borel map in ultraholomorphic classes in a half-plane, and so our results in [10] could be transferred, providing a complete solution for the surjectivity of the moment map whenever strongly regular sequences are considered, and only a partial one for regular sequences. The key point for our coming results is a new work by A. Debrouwere [3], where the surjectivity of the Stieltjes moment problem for regular sequences has been characterized. Again thanks to the Fourier transform (but in the opposite direction) he has taken this information into the asymptotic framework. We state next a version adapted to our needs: firstly, while we ask for Mto be (lc), it is enough that  Mis; secondly, the condition γ(M)>1 amounts, in view of (3.3)and(3.2), to the fact that  Msatisfies (γ2), which is the condition appearing in [3,Thm.7.4.(b)]. Theorem 3.4 ([3]). Let  Mbe regular. The following are equivalent: (i)  B:A{ M}(S1)→C[[z]]{M}is surjective. (ii) γ(M)>1. We highlight that (i)⇒(ii) is slightly weaker than part (i) of Theorem 3.3 when α=1; on the other hand, the implication (ii)⇒(i) provides the first general surjectivity result for weight sequences not subject to condition (mg) (apart from a result of J. Schmets and M. Valdivia for rapidly varying sequences which we will comment on later). 123 Surjectivity of the asymptotic Borel map… Page 9 of 18 181 However, the previous method seems to be valid only for a half-plane. We will be able to carry the information to the case of a general sector by applying general Laplace, Lα,and Borel, Bα, transforms of order α>0, which basically arise from the classical transforms (inverse of each other) combined with ramifications of exponent α. Namely, we will follow the approach in Sections 5.5 and 5.6 of the book of W. Balser [1], where details can be found. We recall that, for 0 <α<2, one considers the Laplace kernel function eα(z):= 1 αz1/α exp(−z1/α), z∈Sα, whose moment function is mα(λ) := ∞ 0 tλ−1eα(t)dt =(1+αλ), (λ) ≥0, and the corresponding Borel kernel function Eα(z):= ∞  p=0 zp mα(p)= ∞  p=0 zp (1+αp),z∈C, which is the classical Mittag-Leffler function of order α. Subsequently, given a function fholomorphic in a sector S=S(d,β)(for some β>0) and with suitable growth, for any direction τin Sthe α-Laplace transform in direction τof fis defined as (Lα,τ f)(z):= ∞(τ ) 0 eα(u/z)f(u)du u,|arg(z)−τ|<απ/2,|z|small enough, where the integral is taken along the half-line parameterized by t∈(0,∞)→ teiτ.The family {Lα,τ f}τin Sdefines a function Lαf, named the α-Laplace transform of f,whichis holomorphic in a sectorial region bisected by dof opening π(β +α). Secondly, let S=S(d,β,r)be a sector with β>α,and f:S→Cbe holomorphic in S and continuous at 0 (i.e. the limit of fat 0 exists when ztends to 0 in every proper subsector of S). For τ∈Rsuch that |τ−d|<(β−α)π/2 we may consider a path δα(τ ) in Slike the ones used in the classical Borel transform, consisting of a segment from the origin to a point z0with arg(z0)=τ+α(π +ε)/2 (for some suitably small ε∈(0,π)), then the circular arc |z|=|z0|from z0to the point z1on the ray arg(z)=τ−α(π +ε)/2 (traversed clockwise), and finally the segment from z1to the origin. The α-Borel transform in direction τof fis then defined as (Bα,τ f)(u):= −1 2πiδα(τ) Eα(u/z)f(z)dz z,u∈S(τ, ε0), ε0small enough. The family {Bα,τ f}τdefines the α-Borel transform of f, holomorphic in the sector S(d,β− α) and denoted by Bαf. In case α≥2, the integral transforms Lαfand Bαfare introduced by the combination of the previous ones with suitable ramification operators, see [1] for details. The formal α-Laplace and α-Borel transforms, defined from C[[z]] into C[[z]], are respectively given by  Lα⎛ ⎝ ∞  p=0 apzp⎞ ⎠:= ∞  p=0 (1+αp)apzp, Bα⎛ ⎝ ∞  p=0 apzp⎞ ⎠:= ∞  p=0 ap (1+αp)zp. 123 181 Page 16 of 18 J. Jiménez-Garrido et al. Note that one has qk<pk<qk+1for every k∈N. We define the sequence Mwhose quotients are given by mp−1=exp ⎛ ⎝ p  j=1 δj⎞ ⎠,p≥1, where (δj)j≥1is the sequence with δ1=δ2=0and δj=0,if qk+1≤j≤pkfor some k≥1, 1,if pk+1≤j≤qk+1for some k≥1. Since the δjare nonnegative, Mautomatically satisfies (lc). It is plain to show that for every p∈None has mp (Mp)1/p=exp ⎛ ⎝1 p p  j=1 jδj+1⎞ ⎠. Then, log mpk (Mpk)1/pk=1 pk pk  j=1 jδj+1≤1 pk qk  j=1 j=qk qk−1 1 2≤1, and thus (vi) is violated. On the other hand, note that for every k≥1 one has log(mqk+1−1) log(qk+1)=1 log(qk+1) qk+1  j=1 δj≥qk+1−pk log(qk+1)≥k, where the last inequality stems from (4.4). From here, log(mpk+1) log(pk+1+1)=1 log(pk+1+1) pk+1+1  j=1 δj=1 log(q2 k+1) qk+1  j=1 δj≥k 2. A simple study of the monotonicity of the sequence (log(mp) log(p+1))p∈Nimplies that limp→∞ log(mp) log(p+1)=+∞(in particular, Mis a weight sequence), and so ω(M)=∞. (vii) ⇒ (viii)The implicationcomes from (4.3).However,fromthe theoryof rapidvariation we learn that strict inequalities are possible in every case in (4.3). A particular example showing that α(m)=∞and ω(M)<∞may simultaneously hold can be found in [7,p. 106], resting on another example by M. Langenbruch [12].  As a first consequence, note that for strongly regular sequences surjectivity does hold for small openings and local extension operators exist with a scaling in the type (see [21, Thm. 3.2.1]), but no global extension operator is possible, since condition (β2)implies that moderate growth cannot hold. Secondly, the next result clarifies the situation for rapidly growing sequences and avoids to impose the condition (dc). Note that γ(M)=∞guarantees that (snq) is satisfied, but is independent from condition (dc). Theorem 4.5 Let Mbe a weight sequence. The following are equivalent: (i) γ(M)=∞. 123 Surjectivity of the asymptotic Borel map… Page 17 of 18 181 (ii) For every r >0, there exists a global extension operator UM,r:C[[z]]{M}→A{ M}(Sr). (iii) For every r >0, there exists a global extension operator VM,r:C[[z]]{M}→ Au {M}(Sr). (iv) All the surjectivity intervals are (0,∞). Proof (i) ⇒ (ii) Given r>0, consider r0:= r+1>r.From(3.3) it is clear that γ( M)=∞, and by Proposition 4.4 we have that  Msatisfies (β2). Moreover, (3.2) implies that  msatisfies (γr0+1). We can apply Theorem 5.4 in [19] for the sequence  Mand the positive integer r0+1, and subsequently Theorem 5.5 in [19] for the value α=r, in order to obtain an extension operator UM,r:C[[z]]{M}→A{ M}(Sr). (ii) ⇒ (iii) It is clear by Proposition 2.1.(i). (iii) ⇒ (iv) By the definition of global extension operators as right inverses for the Borel map, we obviously have  Su {M}=(0,∞). Then, (2.4) leads to the statement. (iv) ⇒ (i) It suffices to apply Theorem 4.10 in [10].  We note that pathological situations are possible. For example, if  Mis regular, γ(M)∈ (0,∞)and (β2) holds, we have surjectivity in Sγ, with global right inverses, for every γ<γ(M), but surjectivity fails for γ>γ(M);since(β2) implies ω(M)=∞, injectivity will not hold in any (narrow or wide) sector. Author Contributions All the results in this publication have been obtained in joint work by the authors. Funding The first two authors are partially supported by the Spanish Ministry of Economy, Industry and Competitiveness under the project MTM2016-77642-C2-1-P, and by the Spanish Ministry of Science and Innovation under the projectPID2019-105621GB-I00.Thethirdauthor is supported by FWF-ProjectsP32905- N and P33417-N. Availability of data and materials Full availability. Code Availability Not applicable. Declarations Conflict of interest The authors declare that they have no conflict of interest. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Balser, W.: Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations. Springer, Berlin (2000) 2. 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