Surjectivity of the asymptotic Borel map in Carleman-Roumieu ultraholomorphic classes defined by regular sequences

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 and introduced by Thilliez. The techniques involve regular variation, integral transforms and characterization results of A. Debrouwere in a half-plane, steming from his study of the surjectivity of the moment mapping in general Gelfand-Shilov spaces.


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, difference and qdifference equations, and so on). The existence of such an expansion for a complex holomorphic function in a sector S of 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 = (M n ) n∈N 0 of positive real numbers. See Subsection 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] and B. 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 [18,10]. 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 (equivalent to the fact that the index γ(M) of Thilliez is positive) is necessary for surjectivity [10,Lemma 4.5].
(iii) For a sector S γ of opening πγ (γ > 0) and under the hypothesis of moderate growth for M, surjectivity has been characterized (at least for uniform asymptotics, and except for the limiting case in some situations) by the condition γ < γ(M) [10,Theorem 4.17].
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], which instead of moderate growth satisfy the milder condition of derivation closedness (see Subsection 2.2 for the precise definitions). Moreover, the existence of extension operators, right inverses for the Borel map, is studied in this general case. It is interesting to note that the condition (β 2 ), 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 (β 2 ), 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. The surjectivity of the Borel map in this setting, for γ < γ(M) and under the moderate 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.
For γ > 0, we consider unbounded sectors bisected by direction 0, or, in general, bounded or unbounded sectors with bisecting direction d ∈ R, opening α π and (in the first case) radius r ∈ (0, ∞). A sector T is said to be a proper subsector of a sector S if T ⊂ S (where the closure of T is taken in R, and so the vertex of the sector is not under consideration). In case such T is also bounded, we say it is a bounded proper subsector of S.

Weight sequences and their properties
In what follows, M = (M p ) p∈N 0 will always stand for a sequence of positive real numbers, and we will always assume that M 0 = 1. We define its sequence of quotients m = (m p ) p∈N 0 by Mp , p ∈ N 0 ; clearly, the knowledge of M amounts to that of m, since M p = m 0 · · · m p−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: (ii) M is stable under differential operators or satisfies the derivation closedness condition (briefly, (dc)) if there exists D > 0 such that (iii) M is of, or has, moderate growth (briefly, (mg)) whenever there exists A > 0 such that It will be convenient to introduce the notation M := (p!M p ) p∈N 0 . All these properties are preserved when passing from M to M. In the classical work of H. Komatsu [11], the properties The sequence of quotients m is nondecreasing if and only if M is (lc). In this case, it is well-known that (M p ) 1/p ≤ m p−1 for every p ∈ N, the sequence ((M p ) 1/p ) p∈N is nondecreasing, and lim p→∞ (M p ) 1/p = ∞ if and only if lim p→∞ m p = ∞. In order to avoid trivial situations, we will restrict from now on to (lc) sequences M such that lim p→∞ m p = ∞, which will be called weight sequences. It is immediate that if M satisfies (lc) and M satisfies (snq), then M is a weight sequence.
Following E. M. Dyn'kin [6], if M is a weight sequence and satisfies (dc), we say M is regular. According to V. Thilliez [21], if M satisfies (lc), (mg) and (snq), we say M is strongly regular ; in this case M is a weight sequence, and the corresponding M is 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 M α,β := p! α p m=0 log β (e + m) p∈N 0 , where α > 0 and β ∈ 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 strongly regular sequence, M α := M α,0 = (p! α ) p∈N 0 , called the Gevrey sequence of order α.
Two sequences M = (M p ) p∈N 0 and L = (L p ) p∈N 0 of positive real numbers, with respective quotients m and ℓ, are said to be: (i) equivalent, and we write M ≈ L, if there exist positive constants A, B such that (ii) strongly equivalent, and we write m ≃ ℓ, if there exist positive constants a, b such that Whenever m ≃ ℓ we have M ≈ L, but not conversely.
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 M and L, we use the notation M · L = (M n L n ) n∈N 0 and M/L = (M n /L n ) n∈N 0 . We will use the fact that M satisfies (mg), respectively (dc), if and only if M · L α or M/L α satisfy (mg), resp. (dc), for some α > 0.

Asymptotic expansions, ultraholomorphic classes and the asymptotic Borel map
In this paragraph S is a sector and M a sequence. We start recalling the concept of asymptotic expansion.
We say a holomorphic function f in S admits the formal power series f = ∞ p=0 a p z p ∈ C[[z]] as its {M}-asymptotic expansion in S (when the variable tends to 0) if for every bounded proper subsector T of S there exist C T , A T > 0 such that for every p ∈ N 0 , one has If the expansion exists, it is unique, and we will write f ∼ {M} f in S. A {M} (S) stands for the space of functions admitting {M}-asymptotic expansion in S. We say a holomorphic function f : S → C admits f as its uniform {M}-asymptotic expansion in G (of type 1/A for some A > 0) if there exists C > 0 such that for every p ∈ N 0 , one has In this case we write f ∼ u is called a Carleman-Roumieu ultraholomorphic class in the 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 [18,10], where Since the derivatives of f ∈ A {M},A (S) are Lipschitzian, for every n ∈ N 0 one may define 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 M be a sequence and S be a sector. Then, In case any of the previous holds and f ∼ {M} ∞ p=0 a p z p , then for every such T and every p ∈ N 0 one has and we can set f (p) (0) := p!a p . One may accordingly define classes of formal power series 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 f in the classes, as defined in (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. Since the problem under study is invariant under rotation, we will focus on the surjectivity of the Borel map in unbounded sectors S γ . So, we define We again note that these intervals were respectively denoted by S M , S u M and S M in [10]. It is clear that S { M} , S u {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 where I • stands for the interior of the interval I.

Surjectivity results for regular sequences
In the study of the surjectivity the index γ(M), introduced in this regard by V. Thilliez  (i) A sequence (c p ) p∈N 0 is almost increasing if there exists a > 0 such that for every p ∈ N 0 we have that c p ≤ ac q for every q ≥ p. It was proved in [8,9] that for any weight sequence M one has (ii) For any β > 0 we say that m satisfies the condition (γ β ) if there exists A > 0 such that Using this condition, which was introduced for β = 1 by H. Komatsu [11] (and named (γ 1 ) after H.-J. Petzsche [15]), and generalized for β ∈ N by J. Schmets and M. Valdivia [19], we can obtain (see [7,9]) that  A straightforward verification shows that for any sequence M and for every s > 0 one has 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]. Our aim in this section is to solve (except for some limiting cases) the problem of surjectivity whenever M is a weight sequence satisfying (dc) or, in other words, M is 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.   At that moment and to the best of our knowledge, no general surjectivity result had been proved for regular M, except for the special case of the q-Gevrey sequences M q = (q p 2 ) p∈N 0 , q > 1, see C. Zhang [22]. In a recent collaboration of the first two authors with 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 by using only functionalanalytic methods. 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 M to be (lc), it is enough that M is; secondly, the condition γ(M) > 1 amounts, in view of (7) and (6) 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).
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, L α , and Borel, B α , 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]. We recall that, for 0 < α < 2, one considers the Laplace kernel function e α (z) : whose moment function is and the corresponding Borel kernel function which is the classical Mittag-Leffler function of order α.
Subsequently, given a function f holomorphic in a sector S = S(d, α) and with suitable growth, for any direction τ in S the α-Laplace transform in direction τ of f is defined as where the integral is taken along the half-line parameterized by t ∈ (0, ∞) → te iτ . The family {L α,τ f } τ in S defines a holomorphic function L α f named the α-Laplace transform of f . Secondly, let S = S(d, β, r) be a sector with β > α, and f : S → C be holomorphic in S and continuous at 0 (i.e. the limit of f at 0 exists when z tends to 0 in every proper subsector of S). For τ ∈ R such that |τ − d| < (β − α)π/2 we may consider a path δ α (τ ) in S like the ones used in the classical Borel transform, consisting of a segment from the origin to a point z 0 with arg(z 0 ) = τ + α(π + ε)/2 (for some suitably small ε ∈ (0, π)), then the circular arc |z| = |z 0 | from z 0 to the point z 1 on the ray arg(z) = τ − α(π + ε)/2 (traversed clockwise), and finally the segment from z 1 to the origin.
The α-Borel transform in direction τ of f is then defined as The following result, involving two sequences, can be found in a slightly different form in [1, Thms. 27 and 28], where only the case of two Gevrey sequences is considered, and in [13,Thm. 3.16], where a general sequence and a sequence admitting a nonzero proximate order intervene. Here, we consider an intermediate situation.

Global extension operators
One may ask about the existence of extension operators, right inverses for the asymptotic Borel map. This can be done, in principle, in the Banach spaces A u {M},A (S) and A {M},A (S), which we call the local case, or in the (LB) spaces A u {M} (S) and A {M} (S), which we refer to as the global one. The first situation was studied by V. Thilliez, see [21, Thm. 3.2.1], who obtained local extension operators with an scaling of the type for strongly regular sequences in sectors S γ as long as γ < γ(M).
In the global situation and in the ultradifferentiable setting, H.-J. Petzsche introduced the condition which again appeared in the results of J. Schmets and M. Valdivia [19] and A. Debrouwere [3] about the existence of global extension operators in the ultraholomorphic framework. Please note that the sequence of quotients considered in these two previously cited papers results from our sequence m after an index shift by 1, what explains the slightly different expression given here to condition (β 2 ). We subsequently mention a version of the result by A. Debrouwere adapted to our needs, in a similar way as in Theorem 3.4. (ii) γ(M) > 1, and M satisfies (β 2 ).
The use of Laplace and Borel transforms of arbitrary positive order allows us to generalize this statement. We will also take into account that condition (β 2 ) is evidently stable under strong equivalence ≃ and, as a consequence of Stirling's formula (see [   (a.2) If r ≤ 1, consider α such that α + r > 1, and take r ′ with r < r ′ < γ(M). The sequence M · L α satisfies (β 2 ) and γ(M · L α ) > r ′ + α > {M} (S (r+r ′ )/2 ), and the restriction of the elements of this space to S r provides the desired extension operator as before.
(ii) =⇒ (iii) Obvious from Proposition 2.1.(i). (iii) =⇒ (iv) We consider again two cases: (b.1) Suppose r > 1, and take a real number r ′ with 1 < r ′ < r. The existence of V M,r implies that the corresponding Borel map is surjective in S r , and by Theorem 3.7 we have γ(M) ≥ r. So, repeating the argument in (a.1), there exists a weight sequence P such that p ≃ m/ℓ r ′ −1 , satisfies (dc) and γ(P) = γ(M) + 1 − r ′ > 1. Since the classes associated with M and P·L r ′ −1 agree, we have an extension operator V M,r : . Note that 1+(r−r ′ )/2 < r−(r ′ −1), and so the mapping . Then, Theorem 4.1 guarantees that P satisfies (β 2 ), and so M will also do according to the stability properties of (β 2 ). Moreover, γ(P) > 1, from where γ(M) > r ′ . Since r ′ was arbitrarily close to r, we deduce that γ(M) ≥ r, as desired.

✷
Our conjecture is that (i), (ii) and (iii) in Theorem 4.2 are equivalent, but we are not able to fill the gap at this moment.
Observe that if M is a weight sequence satisfying (β 2 ), we may apply Lemma 2.4 in [19] to the sequence M and deduce that γ(M) > 0. So, if M is regular and satisfies (β 2 ), one can always obtain extension operators for 0 < r < γ(M) thanks to the previous theorem.
In the last part of our study, we want to determine the weight sequences for which extension operators exist for sectors of arbitrary opening. In this respect, J. Schmets and V. Valdivia state the following result for sequences with fast growth. Please recall that the sequence of quotients considered by these authors results from our sequence m after an index shift by 1. , Thm. 5.6). Let M be a weight sequence such that for every r ∈ N, (m n−1 /n r ) n∈N is increasing from some term on.
However, it turns out that the conditions (9) and (β 2 ) are related to each other. The connection among these and other conditions of fast growth, usually appearing in the literature, can be inferred from the theory of rapid variation (see the classical book of Bingham et al. [2]) and our study of the indices and orders of regular variation associated with weight sequences [9].
We recall that in the study of the injectivity of the Borel map for ultraholomorphic classes in unbounded sectors, completed in [10], the growth index (introduced in [18], see also [8]) played a prominent role. Moreover, the moderate growth condition (mg) is satisfied by M precisely when the upper Matuszewska index associated with its sequence of quotients, α(m), is finite (see [9,Cor. 3.17]), and we recall that, for a general weight sequence,   (iii) For every k ∈ N, k ≥ 2, one has lim n→∞ m kn m n = ∞.
(v) M satisfies (β 2 ).   Proof. (i) =⇒ (ii) The condition (9) clearly implies that the sequence (m n−1 /n r ) n∈N is almost increasing for every r ∈ N. As indicated in [9,Rem. 3.8], this entails the same for the sequence (m n /n r ) n∈N , and we only need to recall (5) in order to deduce γ(M) = ∞. On the contrary, consider the sequence M whose quotients (m n ) n∈N 0 are given by m n−1 = q 2n+1 if n = 2 k + 1 for every k ∈ N 0 , q 2n−1 if n = 2 k + 1 for some k ∈ N 0 , where q > 1 and n ≥ 1. It is not difficult to check that the sequence (m n−1 /n) n∈N is not eventually increasing, while (m n /n r ) n∈N is almost increasing for every r ∈ N, and so γ(M) = ∞.
(vii) =⇒ (viii) The implication comes from (11). However, from the theory of rapid variation we learn that strict inequalities are possible in every case in (11). 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 an scaling in the type (see [21, Thm. 3.2.1]), but no global extension operator is possible, since condition (β 2 ) avoids moderate growth.
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).  (iv) All the surjectivity intervals are (0, ∞).