A Phragmén–Lindelöf Theorem via Proximate Orders, and the Propagation of Asymptotics

We prove that, for asymptotically bounded holomorphic functions in a sector in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C},$$\end{document}C, an asymptotic expansion in a single direction towards the vertex with constraints in terms of a logarithmically convex sequence admitting a nonzero proximate order entails asymptotic expansion in the whole sector with control in terms of the same sequence. This generalizes a result by Fruchard and Zhang for Gevrey asymptotic expansions, and the proof strongly rests on a suitably refined version of the classical Phragmén–Lindelöf theorem, here obtained for functions whose growth in a sector is specified by a nonzero proximate order in the sense of Lindelöf and Valiron.


Introduction
In 1999, Fruchard and Zhang [3] proved that, for a holomorphic function in a sector S which is bounded in every proper subsector of S, the existence of an asymptotic expansion following just one direction implies global (non-uniform) asymptotic expansion in the whole of S. Moreover, a Gevrey version of this result is provided with a control on the type as below:

Theorem 1.1 ([3], Theorem 11) Let f be a function analytic and bounded in an open
sector S = S(d, γ, r ) of bisecting direction d ∈ R, opening πγ and radius r , with γ, r > 0. Suppose f has asymptotic expansionf = ∞ n=0 a n z n of Gevrey order 1/k (k > 0) and type (at least) R(θ 0 ) > 0 in some direction θ 0 with |θ 0 − d| < πγ /2, i.e., for every δ > 0, there exists C = C(δ) > 0 such that for every z ∈ S with arg(z) = θ 0 and every nonnegative integer p, we have that Then, in every direction θ of S, f admitsf as its asymptotic expansion of Gevrey order 1/k and type R(θ ) given as follows: Here, α = d − πγ /2 and β = d + πγ /2 are the directions of the radial boundary of S, α = min θ 0 , α + π 2k ∈ (α, θ 0 ], and β = max θ 0 , β − π 2k ∈ [θ 0 , β). We warn the reader that there is no agreement about the terminology in this respect: while most authors adhere, as we will do, to the convention that the asymptotics in (1.1) is Gevrey of order 1/k, others (for example, Fruchard and Zhang or Balser [1]) say this is of order k. Moreover, the notion of type is not standard, as compared with the definition by Canalis-Durand [2] for whom the type in case one has (1.1) is (1/R +δ) k . It should also be mentioned that the factor Γ (1 + p/k) could be changed into ( p!) 1/k without changing the asymptotics, but this would affect the base of the geometric factor providing the type (by Stirling's formula, [2, pp. 3-4]) in any case. As will be explained later, our interest in the type will be limited, and so we will choose a simple approach in this respect-see Definitions 2.2 and 2.11.
The proof of Theorem 1.1 is based on the classical Phragmén-Lindelöf theorem and on the so-called Borel-Ritt-Gevrey theorem. This last statement provides the surjectivity, as long as the opening of the sector is at most π/k, of the Borel map, sending a function with Gevrey asymptotic expansion of order 1/k in that sector to its series of asymptotic expansion (coefficients of which will necessarily satisfy Gevreylike estimates). Also, the injectivity of the Borel map in sectors of opening greater than π/k (known as Watson's lemma) plays an important role when guaranteeing the uniqueness of a function with a prescribed Gevrey asymptotic expansion of order 1/k in a direction.
The main aim of this paper is to extend these results for other types of asymptotic expansions available in the literature. This possibility was already mentioned in [12], where Lastra, Mozo-Fernández, and the second author of this paper generalized the results of Fruchard and Zhang to the several variables setting. They considered holomorphic functions in a polysector (cartesian product of sectors) admitting strong asymptotic expansion in the sense of Majima [14,15], also in the Gevrey case as introduced by Haraoka [5].
We will deal with general ultraholomorphic classes of functions, defined by constraining the growth of their derivatives in a sector in terms of a sequence M = (M p ) p∈N 0 of positive numbers (N 0 = {0, 1, 2, . . .} = {0} ∪ N), see Definition 2.3. This sequence will play the role of (Γ (1 + p/k)) p∈N 0 in (1.1). It will be subject to precise conditions in order to guarantee not only the natural algebraic and analytic properties of the corresponding class, but also the possibility of extending to this more general framework the results on the injectivity or surjectivity of the Borel map and a Phragmén-Lindelöf-like statement. The relation of these classes to those of functions with an asymptotic expansion is extremely close, see Proposition 2.4.
For log-convex sequences M, the considered ultraholomorphic classes are algebras. The injectivity of the Borel map had been characterized in the 1950s by Mandelbrojt [16] for uniform asymptotics (see Theorem 2.19 in this paper) and by Rodríguez-Salinas [17] for uniformly bounded derivatives (see Theorem 2.15 here). However, regarding surjectivity only some partial results were available by Schmets and Valdivia [19] and Thilliez [20] at the very beginning of this century. They rested on results from the ultradifferentiable setting (dealing with classes of smooth functions in open sets of R n with suitably controlled derivatives), and disregarded questions about the optimality of the opening of the sector or the variation of the type along with the direction in the sector. Moreover, the techniques used, of a functional-analytic nature, do not provide any insight into a possible extension of the Phragmén-Lindelöf theorem. However, the second author [18] has recently made intervene the classical concept of proximate order in these concerns, making possible to obtain more precise statements concerning the injectivity and surjectivity of the Borel map. Subsequently, the authors [8,9] have studied the relationship between log-convex sequences, nonzero proximate orders and the property of regular variation. As a result, a deeper understanding has been gained of the property of admissibility of a nonzero proximate order by a log-convex sequence. This is the key for obtaining in this paper an analog of Phragmén-Lindelöf theorem for functions whose growth in a sector is specified in terms of such a sequence M. It is worth mentioning that sequences admitting a nonzero proximate order are strongly regular (in the sense of Thilliez), and that all the instances of strongly regular sequences appearing in applications do admit such a proximate order.
As in the Gevrey case, the study of the type as the direction moves in the sector is possible, although some information is lost in general (see Remark 3.3). This is due to the fact that the classical exponential kernel, appearing in a suitable truncated Laplace transform providing the solution of the Borel-Ritt-Gevrey theorem in the Gevrey case, is now replaced by the exponential of a function whose behavior at infinity is only given by some asymptotic relations. However, in case the sequence M not only admits a nonzero proximate order, but provides one, the type may be better described.
The paper is organized as follows. After fixing some notations, Sect. 2 is devoted to some preliminaries on general asymptotic expansions, ultraholomorphic classes and quasianalyticity results, specially when nonzero proximate orders are available. All this material will be needed in Sect. 3, where several lemmas of a Phragmén-Lindelöf flavor are obtained. A paradigm is Lemma 3.2, where exponential decrease is extended from just one direction to a whole small (in the sense of its opening) sector adjacent to it. Section 4 contains several versions of Watson's lemma on the uniqueness of a function admitting a given asymptotic expansion in a direction, and in the final Sect. 5, we characterize the functions with an asymptotic expansion in a sectorial region as those asymptotically bounded and admitting such expansion in just one direction in the region.
The results presented in this paper are part of the Ph.D. Dissertation of the first author [7], defended at the University of Valladolid (Spain) under the advice of the second author.
A bounded (respectively, unbounded) sector T is said to be a proper subsector of a sectorial region (resp. of an unbounded sector) G, and we write T G (resp. T ≺ G), if T ⊂ G (where the closure of T is taken in R, and so the vertex of the sector is not under consideration).
For an open set U ⊂ R, the set of all holomorphic functions in U will be denoted by H(U ). C[[z]] stands for the set of formal power series in z with complex coefficients.

Log-convex Sequences and Ultraholomorphic Classes
In what follows, M = (M p ) p∈N 0 always stands for a sequence of positive real numbers, and we always assume that M 0 = 1.

Definition 2.1
We say a holomorphic function f in a sectorial region G admits the formal power seriesf = ∞ n=0 a n z n ∈ C[[z]] as its M-asymptotic expansion in G (when the variable tends to 0) if for every T G there exist C T , A T > 0 such that for every p ∈ N 0 one has We will write f ∼ Mf in G, andÃ M (G) will stand for the space of functions admitting M-asymptotic expansion in G.

Definition 2.2
Given a sector S, we say f ∈ H(S) admitsf = ∞ n=0 a n z n ∈ C[[z]] as its uniform M-asymptotic expansion in S (of type 1/A for some A > 0) if there exists C > 0 such that for every p ∈ N 0 one has Since the derivatives of f ∈ A M,A (S) are Lipschitzian, for every n ∈ N 0 one may define We recall now the relationship between these classes and the concept of asymptotic expansion. As a consequence of Taylor's formula, we have the following result (see [1,4] Next we specify some conditions on the sequence M that will have important consequences on the previous classes or spaces. Definition 2. 5 We say: (iv) M satisfies the strong non-quasianalyticity condition [for short, (snq)] if there exists C > 0 such that Obviously, (mg) implies (dc).

Definition 2.7
For a sequence M, we define the sequence of quotients m = (m p ) p∈N 0 by It is obvious that M is (lc) if, and only if, m is nondecreasing.

Definition 2.8
Let M and L be sequences, we say that M is equivalent to L, and we write M ≈ L, if there exist positive constants A, B > 0 such that Example 2. 9 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.

Definition 2.11
Let f be a function defined in a sectorial region G = G(d, γ ), and θ be a direction in G, i.e. |θ − d| < πγ /2. We say f has M-asymptotic expansion f = ∞ n=0 a n z n in direction θ if there exist r θ , C θ , A θ > 0 such that the segment (0, r θ e iθ ] is contained in G, and for every z ∈ (0, r θ e iθ ] and every p ∈ N 0 one has In this case, we say the type is 1/A θ . Of course, the definition makes sense as long as the function is defined only in direction θ near the origin, i.e. in a segment (0, r e iθ ] for suitable r > 0.
One may accordingly define classes of formal power series it is plain to check that for every bounded proper subsector T of G and every p ∈ N 0 , one has and we can set

Classical Quasianalyticity Results
We introduce first the notions of flatness and quasianalyticity.

Definition 2.13 A function f in any of the previous classes is said to be flat ifB( f )
is the null formal power series (denoted0), or in other words, f ∼ M0 .
Definition 2.14 Let S be a sector, G a sectorial region and M = (M p ) p∈N 0 be a sequence of positive numbers. We say that is quasianalytic if it does not contain nontrivial flat functions (in other words, the Borel map is injective in this class).
In order to simplify some statements or to avoid trivial situations, from now on in this paper, we will assume the standard property that The sequence M is logarithmically convex with lim p→∞ m p = ∞.
The following result characterizes quasianalyticity for the classes of functions with uniformly bounded derivatives in an unbounded sector. It first appeared in Rodríguez-Salinas [17], although it is frequently attributed to Korenbljum [10].

diverges.
This result can be rewritten in terms of the classical notion of exponent of convergence of a sequence. ([6], p. 65) Let (c n ) n∈N 0 be a nondecreasing sequence of positive real numbers tending to infinity. The exponent of convergence of (c n ) n is defined as

Proposition 2.16
According to this last formula, we may define the index in such a way that So, Theorem 2.15 may be stated as Corollary 2.17 Let M and γ > 0 be given. The following statements are equivalent: diverges.

Remark 2.18
The problem of quasianalyticity for classes of functions with uniformly bounded derivatives in bounded regions has also been treated. In the works of Trunov and Yulmukhametov [23,25], a characterization is given, for a convex bounded region containing 0 in its boundary, in terms of the sequence M and of the way the boundary approaches 0. In particular, for bounded sectors, if γ ≤ 1, d ∈ R and r > 0, it turns out that the class A M (S(d, γ, r )) is quasianalytic precisely when condition (ii) above is satisfied.
The study of quasianalyticity for the classes of functions with uniform Masymptotic expansion in an unbounded sector rests on the following statement by Mandelbrojt.
Observe that a function f is holomorphic in H and verifies the estimates (2.2) if, and only if, the function g given by g(z) and is flat. From this fact and the first equality in (2.1), it is immediate to deduce the next characterization.
Corollary 2.20 (Generalized Watson's lemma for uniform asymptotics) Let M and γ > 0 be given. The following are equivalent: diverges.
Regarding the class of functions with (non-uniform) asymptotic expansion in a sectorial region G, we first express flatness inÃ M (G) by means of an auxiliary function: for t > 0, we define which is a non-decreasing continuous map in [0, ∞) with lim t→∞ M(t) = ∞. Then, we have the following result.

Remark 2.23
In the conditions of Definition 2.11, iff is the null series, then we say that f is M-flat in direction θ. As in the previous statement, this amounts to the existence of r θ , C θ , A θ > 0 such that the segment (0, r θ e iθ ] is contained in G, and for every Suppose moreover that f is bounded throughout the (bounded or not) sectorial region G. Since the function t → e −M(t) is non-increasing in [0, ∞), it is obvious that f is M-flat in direction θ if, and only if, there existC θ > 0 and the same constant A θ > 0 as before, such that for every z ∈ G with arg z = θ one has This fact will be used later on.

Quasianalyticity Results via Proximate Orders
An easy characterization of quasianalyticity in the classesÃ M (G) may be given thanks to the notion of proximate order, appearing in the theory of growth of entire functions and developed, initially, by Lindelöf and Valiron. We will focus our discussion mainly on the results given by Maergoiz (see [13]).

Definition 2.24
We say a real function ρ, defined on (c, ∞) for some c ≥ 0, is a proximate order, if the following hold: (A) ρ is continuous and piecewise continuously differentiable in (c, ∞) (meaning that it is differentiable except possibly at a sequence of points, tending to infinity, at any of which it is continuous and has distinct finite lateral derivatives), In case the value ρ ∞ in (C) is positive (respectively, is 0), we say ρ is a nonzero (resp. zero) proximate order.

Remark 2.25
If ρ is a proximate order with limit ρ ∞ at infinity and α > ρ ∞ , then there exists r (α) > 1 such that ρ(r ) < α for r > r (α) and, consequently, We now associate to a nonzero proximate order a class of functions with nice properties, which will play a prominent role in our Phragmén-Lindelöf result. Before returning to the study of quasianalyticity, we indicate how to go from sequences to proximate orders (for more information on this relation and its reversion, see [9]). Given M and its associated function M, for t large enough, we can consider d M (t) := log(M(t))/ log(t).
The following result characterizes those sequences for which d M is a proximate order. A less restrictive condition on the sequence M, namely the admissibility of a proximate order, is indeed sufficient for our purposes.
From this result, we deduce that whenever a classÃ M (G) (orÃ u M (S) or A M (S)) is defined in terms of a sequence M admitting a nonzero proximate order, we can exchange M by another equivalent (lc) sequence L, whose sequence of quotients is regularly varying. Then, we can briefly say that the M-asymptotic expansion of a function f ∈Ã M (G) =Ã L (G) has log-convex regularly varying constraints.

Remark 2.30
If M admits a nonzero proximate order ρ, then for every γ > 0, we know that for the function V ∈ MF(γ , ρ) given by Theorem 2.26 there exist positive constants A, B, t 0 such that (2.5)

Example 2.31
We provide an example showing that the results in this paper are indeed generalizations of the ones by Fruchard and Zhang [3]. Consider, for α > 0 and β = 0, the sequence M α,β introduced in Example 2.9(i). It is not equivalent to any Gevrey sequence and, as indicated in [9,Remark 4.2], it admits the nonzero proximate order ρ α,β given by , t large enough.
In [9,Remark 4.15], it has been shown that sequences admitting a nonzero proximate order are indeed strongly regular. So, as indicated in [18,Remark 4.11(iii)], for such sequences M one may construct nontrivial flat functions inÃ M (G ω(M) ), what allows us to state the following version of Watson's Lemma for non-uniform asymptotics.

M-Flatness Extension
From this point onward, we will assume not only that the sequence M is logarithmically convex with lim p→∞ m p = ∞, but also that

The sequence M admits a nonzero proximate order.
This is not a strong assumption for strongly regular sequences, since it is satisfied by every such sequence appearing in applications (the Gevrey ones, or the one associated to the 1 + -level asymptotics). However, note that there are strongly regular sequences which do not satisfy it, see [9].
We are ready for proving an important lemma about the extension of M-flatness from a boundary direction into a whole small sector for functions bounded there and admitting a continuous extension to the boundary (considered in R, i.e., disregarding the origin). First, we recall a classical version of Phragmén-Lindelöf theorem needed in the proof.
According to Remark 2.23, and specifically to (2.3), there exist c 1 , It is clear that ε < 1, so we have that for every z ∈ S γ . We observe that arg a/z ∈ [ωα, ωβ] ⊆ (−πω/2, πω) for every z ∈ S γ . Taking into account Remark 2.29 and using property (I) of the functions in MF(2ω, ρ) we see that for |z| < s 1 small enough and arg z ∈ [−πγ /2, πγ /2]. For convenience, we choose s 1 < 1/(t 0 c 2 ). Consider the function The function z → V (a/z) is holomorphic in S(arg a, 2ω) ⊃ S γ , so F is holomorphic in S γ and continuous up to ∂ S γ . Our aim is to apply the Phragmén-Lindelöf Theorem 3.1 to this function in a suitable bounded sector S(0, γ, s 3 ).
If arg z = −πγ /2, we have that arg a − arg z = βω. Then, since f is bounded in S γ by a constant K > 0, by using (3.4), we see that for |z| < s 1 , Using property (I) of the functions in MF(2ω, ρ) we have that Then, for |z| < s 2 ≤ s 1 small enough, we have that V (1/(c 2 |z|)) ≥ d 2 V (1/|z|), and we conclude that Since |a| has been chosen small enough in order that −Ad 2 + 2|a| 1/ω < 0, we deduce that |F(z)| ≤ c 1 for every |z| < s 2 and arg z = πγ /2. For z ∈ S γ with |z| < s 1 , by using (3.2) and (3.4), we have that As γ < ω, there exists μ > 0 such that γ < μ < ω. By property (VI), we know that ρ V : t → log(V (t))/ log(t) is a proximate order equivalent to ρ, hence tending to 1/ω at infinity. Then, we can apply Remark 2.25: there exists 0 < s 3 ≤ s 2 small enough such that for every z ∈ S γ , |z| ≤ s 3 , and, in particular, By applying Phragmén-Lindelöf Theorem 3.1 to the function F in S(0, γ, s 3 ), we obtain that Assuming that arg z ∈ [−πγ /2 + δ, πγ /2], we deduce that Then, for r 2 := η|a| 1/ω > 0, we find that for every z with arg z ∈ [−πγ /2 + δ, πγ /2] and |z| < s 3 , we have that Choose k 2 > 0 such that (1/k 2 ) 1/ω < r 2 /B. Property (I) of the functions in MF(2ω, ρ) implies that, for z with |z| < s 4 < min(s 3 , 1/(t 0 k 2 )), small enough, and arg z ∈ [−πγ /2 + δ, πγ /2], we have Indeed, we may give a more precise information about the type. Following the previous proof, one notes that and k 2 may be made arbitrarily close to the last expression at the price of enlarging the constant k 1 = k 1 (δ). So, the original type c 2 is basically affected by a precise factor when moving to a direction θ = −πγ /2 + δ with 0 < δ < πγ. It is obvious that k 2 (δ) explodes at least like 1/ sin ω δ as δ → 0. This means that the type of the null asymptotic expansion tends to 0 as the direction in the sector approaches the boundary d = −πγ /2, in the same way as in the Gevrey case (see Theorem 1.1). Moreover, the constant 2 in δ/(2ω) could be any number greater than 1 and, by suitably choosing the value ε in the proof, the constant 2B/A appearing before can be made as close to B/A as desired, so that the only indeterminacy in the previous factor is caused by the values A, B involved in (2.5). In the common situation that the function d M is indeed a proximate order, the constants A and B can also be taken as near to 1 as wanted, which makes the expression even more explicit.
Finally, note that, by using Theorem 2.28 one may change M by an equivalent sequence L such that d L is a proximate order. However, this fact does not improve the proof, since again Theorem 2.26 will be applied to obtain a function V ∈ MF(2ω, d L ), and we will work with the same type of estimate that we have in (2.5).

By recursively reasoning in the sectors
and finally in the sector It is clear then that for k 1 := max j k 1, j and k 2 := max j k 2, j , we have that In the next result, we impose M-flatness in both boundary directions of the sector, and conclude uniform M-flatness throughout the sector. We conclude taking k 1 := max{k 1,1 , k 1,2 } and k 2 := max{k 2,1 , k 2,2 }.

Remark 3.6
By carefully inspecting its proof, we see that Lemma 3.2 holds true in any bounded sector S(d, γ, r ) and, consequently, Lemmas 3.4 and 3.5 are also valid in bounded sectors.
We show next that, as Remark 3.6 suggests, it is also possible to work in sectorial regions.
Proof By suitably enlarging the opening of the subsector, we can assume that θ is one of the directions in T . There exist R, c 1 , If θ 1 < θ 2 are the (radial) boundary directions of T , we consider δ > 0 such that −πγ /2 < θ 1 − δ and θ 2 + δ < πγ /2. There exists 0 < r < R such that the sectors Taking into account (3.7) and Remark 3.6, we can apply Lemma 3.4 to the restriction of f to each sector, and we conclude that f is M-flat for arg z ∈ [θ 1 , θ 2 ] and |z| ≤ r . Since M is nondecreasing, by suitably enlarging the constant k 1 we obtain (3.6).

Example 3.8
Boundedness of the considered function is necessary in any of the previous results in this section. The next example shows that having an M-asymptotic expansion in a direction d does not guarantee its validity in any sector containing that direction. Our inspiration comes from a similar example in Wasow's book [24, p. 38], which concerned the function f (z) = sin(e 1/z )e −1/z . Given M, by Remark 2.30 for every γ > 0, there exists V ∈ MF(γ , ρ) such that we have (2.5). We consider the function Since sin(e V (1/z) ) is bounded for real z > 0, we see that f is M-flat in direction 0. If we compute the derivative of f in S γ , we see that

Remark 3.9
At this point it is worth saying a few words about a situation which, although not usually considered in the theory of asymptotic expansions, plays an important role in the general framework of ultradifferentiable or ultraholomorphic classes, namely that of the so-called Carleman classes of Beurling type. We will not give full details here, but let us say that a function f , holomorphic in a sectorial region G, has Beurling M-asymptotic expansionf = ∞ n=0 a n z n in a direction θ in G if there exists r θ > 0 such that the segment (0, r θ e iθ ] is contained in G, and for every A θ > 0 (small) there exists C θ > 0 (large) such that for every z ∈ (0, r θ e iθ ] and every p ∈ N 0 one has Following the idea in Remark 2.23, one can prove that f , bounded throughout G, is Beurling M-flat in direction θ if, and only if, for every c 2 > 0 (small) there exist c 1 > 0 (large) such that for every z ∈ G with arg z = θ one has | f (z)| ≤ c 1 e −M(1/(c 2 |z|)) . (3.8) Then, the following analog of Lemma 3.2 is valid: given M and 0 < γ < ω(M), suppose f is a bounded holomorphic function in S γ that admits a continuous extension to the boundary ∂ S γ , and that is Beurling M-flat in direction d = πγ /2. Then for every 0 < δ < πγ and every k 2 > 0, there exists a constant k 1 = k 1 (δ, k 2 ) > 0 such that The proof of this statement follows the same lines as that of the original lemma, by carefully tracing the dependence of the different constants involved in the estimates. Indeed, the constants A, B, α, β, ε, η are determined in the same way. Choose r 2 > 0 such that r 2 /B > k −1/ω 2 , and a point a with the specified argument and modulus (r 2 /η) ω . Take a positive d 2 such that d 2 > 2|a| 1/ω /A, and then c 2 > 0 such that By definition of Beurling M-flatness in direction γ π/2, there exists c 1 > 0 such that (3.8) holds for arg z = γ π/2. Then, the desired estimates hold for the same k 1 > 0 obtained in the proof of that lemma.
Note that also Lemmas 3.4, 3.5 and Proposition 3.7 will be valid in this Beurling setting.

Watson's Lemmas
We will now obtain several quasianalyticity results by combining those in Sects. 2.2 and 2.3 with the results on the propagation of null asymptotics in Sect. 3.

Remark 4.1
In a similar way as in the proof of Theorem 2.22 (see [21]), it is easy to deduce that, given a bounded holomorphic function f in a sector S γ that admits a continuous extension to the boundary ∂ S γ , the fact that f ∈Ã u M (S γ ) and f is M-flat amounts to the existence of constants k 1 , k 2 > 0 such that (3.5) holds.
In the first version, an immediate consequence of previous information, we assume the function is flat at both boundary directions. Proof By Lemma 3.5, we know that (3.5) holds for suitable k 1 , k 2 > 0. The previous remark implies that f ∈Ã u M (S γ ) and f ∼ M0 , and by Corollary 2.20 we deduce that f ≡ 0.
In the proof of Lemma 4.3, we need to distinguish two situations: in case γ > ω(M), we have been given an M-flat function f in a wide enough sector (what entails uniqueness), while in case γ = ω(M) an M-flat function F in a sector of opening πω(M) has to be constructed in order to apply Corollary 2.20, what is possible thanks to the additional assumption on the series ∞ p=0 (m p ) −1/ω(M) . It is interesting to note that in the Gevrey case the aforementioned series diverges, so that the previous result extends Lemma 5 in [3]. Indeed, in that instance the very divergence of the series allows one to treat the case γ > ω(M) by restricting the function to a sector with γ = ω(M), an argument which is not available in our situation.

Asymptotic Expansion Extension
The next result (see [18,Theorem 6.1]) was stated for strongly regular sequences M such that d M is a proximate order. However, as it is deduced from [18,Remark 4.11(iii)] and [9,Remark 4.15], it is enough to ask for the sequence to satisfy our two general assumptions (see Sect. 3). From this result, we may generalize Theorem 1 in [3].
Proof We distinguish two cases: