Optimal Flat Functions in Carleman–Roumieu Ultraholomorphic Classes in Sectors

We construct optimal flat functions in Carleman–Roumieu ultraholomorphic classes associated to general strongly nonquasianalytic weight sequences, and defined on sectors of suitably restricted opening. A general procedure is presented in order to obtain linear continuous extension operators, right inverses of the Borel map, for the case of regular weight sequences in the sense of Dyn’kin. Finally, we discuss some examples (including the well-known q-Gevrey case) where such optimal flat functions can be obtained in a more explicit way.


Introduction
The asymptotic Borel map sends a function, admitting an asymptotic expansion in a sectorial region, into the formal power series providing such expansion. In many instances it is important to decide about the injectivity and surjectivity of this map when considered between so-called Carleman-Roumieu ultraholomorphic classes and the corresponding class of formal series, defined by restricting the growth of some of the characteristic data of their elements (the derivatives of the functions, the remainders in the expansion, or the coefficients of the series) in terms of a given weight sequence M = (M p ) p∈N0 of (iii) M is of, or has, moderate growth (for the sake of brevity, (mg)) if there exists A > 0 such that M p+q ≤ A p+q M p M q , p, q ∈ N 0 . (iv) M satisfies the condition (nq) of non-quasianalyticity if ∞ p=0 M p (p + 1)M p+1 < +∞.
(v) Finally, M satisfies the condition (snq) of strong non-quasianalyticity if there exists B > 0 such that It is convenient to introduce the notation M := (p!M p ) p∈N0 . All these properties are preserved when passing from M to M. In the classical work of Komatsu [16], the properties (lc), 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.
Following Dyn'kin [8], if M is a weight sequence and satisfies (dc), we say M is regular. According to Thilliez [33], 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∈N0 , 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 a strongly regular sequence, M α := M α,0 = (p! α ) p∈N0 , called the Gevrey sequence of order α. (ii) The sequence M 0,β := ( p m=0 log β (e + m)) p∈N0 , with β > 0, satisfies (lc) and (mg), and m tends to infinity, but (snq) is not satisfied. (iii) For q > 1 and 1 < σ ≤ 2, M q,σ := (q p σ ) p∈N0 satisfies (lc), (dc) and (snq), but not (mg). In case σ = 2, we get the well-known q-Gevrey sequence. Two sequences M = (M p ) p∈N0 and L = (L p ) p∈N0 of positive real numbers, with M 0 = L 0 = 1 and with respective quotient sequences m and , are said to be equivalent, and we write M ≈ L, if there exist positive constants A, B such that A p M p ≤ L p ≤ B p M p , p ∈ N 0 . They are said to be strongly Vol. 78 (2023) Optimal flat functions in Carleman-Roumieu Page 5 of 35 98 equivalent, denoted by m , if there exist positive constants a, b such that am p ≤ p ≤ bm p , p ∈ N 0 . Whenever m we have M ≈ L but, in general, not conversely.
In case M 0 or L 0 is not equal to 1, the previous definitions of equivalence are meant to deal with the normalized sequences (M p /M 0 ) p∈N0 or (L p /L 0 ) p∈N0 .

Index γ(M) and Auxiliary Functions for Weight Sequences and Functions
The index γ(M), introduced by Thilliez [33, Sect. 1.3] for strongly regular sequences M, can be equally defined for (lc) sequences, and it may be equivalently expressed by different conditions: (i) A sequence (c p ) p∈N0 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 [12,13] that for any weight sequence M one has In [10,13] it is proved that for a weight sequence M, (2.2) If we observe that the condition (snq) for M is precisely (γ 1 ) for m, the sequence of quotients for M, and that γ( M) = γ(M) + 1 (this is clear from (2.1)), we deduce from the second statement in (2.2) that M satisfies (snq) if, and only if, γ(M) > 0. (2.3) Given a weight sequence M = (M p ) p∈N0 , we write and ω M (0) = 0. This is the classical (continuous, nondecreasing) function associated with the sequence M, see [16]. Another associated function will play a key-role, namely The functions h M and ω M are related by In [16,Prop. 3.2] we find that, for a weight sequence M, We also introduce the counting function ν m for the sequence m, If M is a weight sequence, then the functions ν m and ω M are related by the following useful integral representation formula, e.g. see [19] and [16, (3.11)]: In [13], the nature of the index γ(M), fundamental in the study of the surjectivity of the asymptotic Borel map, is explained. More precisely, it is shown that γ(M) is the lower Matuszewska index of m. In addition, the relation between ORV-indices of m and ν m is clarified and from this connection we characterized some properties of ν m that will be important for our aim.  We conclude this subsection by introducing the harmonic extension and a particular majorant of a nondecreasing nonquasianalytic function.
A nondecreasing (or even just measurable) function σ : [0, ∞) → [0, ∞) satisfies the nonquasianalyticity property (ω nq ) (and we say σ is nonquasian- Let σ : [0, ∞) → [0, ∞) be a nondecreasing nonquasianalytic function. The harmonic extension P σ of σ to the open upper and lower halfplanes of C is defined by For every z ∈ C one has (see, for example, [ We list some basic properties of the harmonic extension that will be used later: Another important auxiliary function appears in the study of extension results in Braun-Meise-Taylor ultradifferentiable classes, defined in terms of weight functions (see, for example, [4,21] and the references therein).

Asymptotic Expansions, Ultraholomorphic Classes and the Asymptotic Borel Map
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 this paragraph S is an unbounded sector and M a sequence. We start by recalling the concept of uniform asymptotic expansion.
We say a holomorphic function f : S → C admits f = 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 (2.12) In this case we write f ∼ u M,A f in S, and A u M,A (S) denotes the space of functions admitting uniform M-asymptotic expansion of type 1/A in S, endowed with the norm We warn the reader that these notations, while the same as in the paper [15], do not agree with the ones used in [14,30] Since the derivatives of f ∈ A M,A (S) are Lipschitz, for every p ∈ N 0 one may define (2.14) Vol. 78 (2023) Optimal flat functions in Carleman-Roumieu Page 9 of 35 98 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 2.3.
Let M be a sequence and S be a sector. Then,

and so the identity map
One may accordingly define classes of formal power series 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 and S u M in [14].
where I • stands for the interior of the interval I.

Optimal Flat Functions and Surjectivity of the Borel Map
The following result appeared, in a slightly different form, in [14,Lemma 4.5].
Subsequently, a converse, more precise statement appeared in [15,Th. 3.7] under the additional hypothesis of condition (dc). In So, the surjectivity of the Borel map for regular sequences is governed by the value of the index γ(M).

Construction of Optimal Flat Functions
Our aim is to relate the surjectivity of the Borel map in a sector to the existence of optimal flat functions in it, which we now define and construct in this subsection. (i) There exist K 1 , K 2 > 0 such that for all x > 0, Besides the symmetry imposed by condition (i) (observe that G(x) > 0 for x > 0, and so G(z) = G(z), z ∈ S), we note that the estimate in (3.2) amounts to the fact that in (3.1) makes the function optimal in a sense, as its rate of decrease on the positive real axis when x tends to 0 is accurately specified by the function h M . Note that, in previous instances where such optimal flat functions appear [11,18,33], the estimates from below in (3.1) are imposed and/or obtained in the whole sector S, and not just on its bisecting direction. We think the present definition is more convenient, since it is easier to check for concrete functions, and for our purposes it provides all the necessary information in order to work with such functions. As a first step for the construction of such flat functions, we need to estimate the harmonic extension P σ in terms of the majorant κ σ . The righthand side estimate in the next result is a slight refinement of the one in [4, Lemma 3.3], which was not precise enough for our purposes. We include the whole proof for the sake of completeness.
Proof. If y = 0 all the values are equal to σ(0) and so the inequalities hold true. Now, for y > 0 we have Since s 2 + 1 ≤ 2s 2 for s ≥ 1, we deduce that In order to prove the right inequality, we start by splitting the integral into two parts: As σ is nondecreasing, we may write The key condition for weight sequences that will allow us to construct optimal flat functions appeared in a work of Langenbruch [17]. Definition 3.5. Let M be a weight sequence such that q M satisfies (nq), so that P ω M is well-defined. We say that the sequence satisfies the Langenbruch's condition if there exists a constant C > 0 such that for all y ≥ 0 we have We can characterize the previous condition in terms of the index γ(M). This connection has very recently appeared for the first time in a work of Nenning et al. [22]. Although the additional hypothesis of (dc) appears in their (indirect) arguments, it can be removed as long as the sequence satisfies (snq), as we now show. Observe that, by Lemma 3.1, the condition (snq) (equivalently, γ(M) > 0) is necessary for surjectivity, so it is not a restriction for our aim. (3.8) The last inequality is a consequence of the monotonicity of ν m .
Thanks to (3.8) and the monotonicity of the harmonic extension with respect to the argument function we get from above Next, by using the integral expression (2.7) and the monotonicity of ν m we have that Finally, by Lemma 2.2, we deduce that for suitable D > 0. This is condition (ω snq ) for ν m and, by Lemma 2.2, we may conclude that γ(M) > 1.
(ii)⇒(i) Condition γ(M) > 1 implies that γ(M) > 0, and amounts to condition (γ 1 ) for m (see (2.2)), so that q M clearly satisfies (nq). By Lemma 2.2, the condition γ(M) > 1 is equivalent to the existence of a constant C > 0 such that Then, from (3.3), (2.11) and the above inequality we deduce that By (3.8), we have from above that which completes the proof.
Remark 3.7. The condition γ(M) > 1 is the same as γ( q M) > 0, or equivalently, (snq) for q M (even if q M might not satisfy (lc), we can apply [13,Coro. 3.13] to obtain this equivalence). So, for a weight sequence M satisfying (snq), Langenbruch's condition allows to pass from (nq) to (snq) for q M. Observe also that, by [13,Lemma 3.20], the condition (nq) for q M implies that the index ω( q M), introduced in [30] and studied in detail in [13], is nonnegative, and so ω(M) = ω( q M) + 1 ≥ 1. As one only knows that γ(M) ≤ ω(M) in general, and these indices can perfectly be different, one may better understand the effect of Langenbruch's condition.
Remark 3.8. On the one hand, as said before, for a weight sequence M the condition γ(M) > 1 amounts to the condition (γ 1 ) for m, and it is well-known (see [16,Prop. 4.4]) that then ω M satisfies (ω snq ). As it can be deduced from [21,Prop. 1.7], this last fact is, in its turn, equivalent to the existence of a constant C > 0 such that On the other hand, in [16,Prop. 3.6] the condition (mg) for a weight sequence M is shown to be equivalent to the fact that 2ω M (y) ≤ ω M (Dy) + D for all y ≥ 0 and suitable D > 0. Gathering these estimates, we conclude that if M is strongly regular then γ(M) > 1 if, and only if, M satisfies Langenbruch's condition. This was basically the reasoning that allowed V. Thilliez to obtain optimal {M}-flat functions, in the very same way as we are doing in the next result, but dropping now the moderate growth condition by means of Proposition 3.6.
Thanks to the previous result, we will construct optimal {M}-flat functions in the right half plane as long as γ(M) > 1.

Proposition 3.9. Let M be a weight sequence. If γ(M) > 1, then the function
is an optimal {M}-flat function in the halfplane S 1 , where Q ω M is the harmonic conjugate of P ω M in the upper half plane.
Proof. It is clear that the function G is holomorphic in S 1 . On the one hand, by taking into account (2.9), for z ∈ S 1 we have that On the other hand, the condition γ(M) > 1 implies, by Proposition 3.6, that there exists C > 0 such that P ω M (ix) ≤ ω M (Cx) + C for every x > 0. Since one can easily check that Q ω M (i/x) = 0, we have that By a ramification of the variable we can extend this method to an arbitrary weight sequence with γ(M) > 0 and any sector whose opening is less than πγ(M).
Now, let us prove that the function F (z) = (G(z s )) 1/s , z ∈ S γ , is an optimal {M}-flat function in S γ . From the fact that G is an optimal {M s }-flat function, (2.4) and (3.10), we get

Surjectivity of the Borel Map for Regular Sequences
We will describe next how, by means of an optimal flat function, one can obtain extension operators, right inverses for the Borel map, for ultraholomorphic classes defined by regular sequences. If G is an optimal {M}-flat function in A u {M} (S), we define the kernel function e : S → C given by It is obvious that e(x) > 0 for all x > 0, and there exist K 1 , K 2 , K 3 , K 4 > 0 such that Note that the positive value m(0) need not be equal to 1.
The following result is crucial for our aim. Proof. We only need to reason for p ∈ N. On the one hand, because of the right-hand inequalities in (3.11) and Lemma 2.1.(ii), for every p ∈ N and s > 0 we may write

Proposition 3.11. Suppose M is a regular sequence and G is an optimal {M}flat function in
Note that in the last equality we have used (2.5), and then we have applied (dc) with a suitable constant D > 0. Since s > 0 was arbitrary, we finally get Then, again by (2.5), we deduce that for a suitable constant B 1 > 0 (note that M is eventually nondecreasing).
We can already state the following main result. The forthcoming implication (ii) ⇒ (v) for strongly regular sequences M was first obtained by Thilliez [33,Th. 3.2.1], and the proof heavily rested on the moderate growth condition, both for the construction [33, Th. 2.3.1] of optimal {M}-flat functions in sectors S γ for every γ > 0 such that γ < γ(M), and for the subsequent use of Whitney extension results in the ultradifferentiable setting. In [18] the implication (ii) ⇒ (iii) was proved again for strongly regular sequences, but with a completely different technique, and it is this approach which allows here for the weakening of condition (mg) into (dc).  From the left-hand inequalities in (3.12), we deduce that Hence, the formal Borel-like transform of f , is convergent in the disc D(0, R) for R = B 1 /A > 0, and it defines a holomorphic function g there. Choose R 0 := B 1 /(2A) < R, and define which is a truncated Laplace-like transform of g with kernel e. By virtue of Leibniz's theorem for parametric integrals and the properties of e, we deduce that this function, denoted by f for the sake of brevity, is holomorphic in S γ .
We will prove that f ∼ u {M}f uniformly in S γ , and that the map f → f , which is obviously linear, is also continuous from Let p ∈ N 0 and z ∈ S γ . We have After a change of variable u = zv in the last integral, one may use Cauchy's residue theorem and the right-hand estimates in (3.11) in order to rotate the path of integration and obtain So, we can write the preceding difference as Then, we have where We now estimate f 1 (z) and f 2 (z). Observe that for every u ∈ (0, R 0 ] we have 0 < Au/B 1 ≤ 1/2. So, from (3.13) we get Hence, Regarding f 2 (z), for u ≥ R 0 and 0 ≤ n ≤ p − 1 we have (u/R 0 ) n ≤ (u/R 0 ) p , so u n ≤ R n 0 u p /R p 0 . Again by (3.13), and taking into account the value of R 0 , we may write Then, we get In order to conclude, note that the second inequality in (3.11), followed by the first one, and the fact that e(x) > 0 for x > 0, together imply that for every z ∈ S γ and every u > 0 we have We use this fact, a simple change of variable and the right-hand estimates in (3.12), and obtain that This estimate can be taken into both (3.15) and (3.16), and from (3.14) we easily get that for every p ∈ N 0 , and so f admits f as its uniform {M}-asymptotic expansion in S γ . Moreover, recalling the definition (2.13) of the norm in these spaces with uniform asymptotics and fixed type, if we put c := 2K 4 B 2 /(K 2 B 1 ) > 0, we see that f ∈ A u M,cA (S γ ) and The following three corollaries become now clear.
Corollary 3.14. Let M be a regular sequence, and γ > 0. The following are equivalent: As a consequence of (2.3) and Theorem 3.12 we get the following result.

Optimal Flat Functions and Strongly Regular Sequences
Under the moderate growth condition, the implication (ii) ⇒ (i) in the version of Corollary 3.15 for the space A { M} (S γ ) can be shown independently by using a result from Bruna [5], where a precise formula for nontrivial flat functions in Carleman-Roumieu ultradifferentiable classes, appearing in a work of Bang [2], is exploited. For the sake of completeness, we will present this proof below. The proof requires two auxiliary results which we state and prove now. First, given a weight sequence M, the sequence of quotients m = (m p ) p∈N0 is nondecreasing and tends to infinity, but it can happen that it remains constant on large intervals [p 0 , p 1 ] of indices, so that the counting function ν m defined in (2.6) yields ν m (m p0 ) = ν m (m p1 ) = p 1 + 1. However, in some applications or proofs it would be convenient to have ν m (m p ) = p + 1 for all p ≥ 0. This can be assumed without loss of generality by the following result.
Lemma 3.18. Let a = (a p ) p≥1 be a nondecreasing sequence of positive real numbers satisfying lim p→+∞ a p = +∞ (it suffices that a p−1 < a p holds true for infinitely many indices p). Then there exists a sequence b = (b p ) p≥1 of positive real numbers such that p → b p is strictly increasing and satisfies So, in the language of weight sequences, we prove that for any weight sequence M there exists a strongly equivalent weight sequence L (and so M ≈ L) such that ν ( p ) = p+1 for all p ∈ N 0 . Note that equivalent weight sequences define the same Carleman-Roumieu ultraholomorphic classes and associated weighted classes of formal power series.
By definition (3.18) we have b q = a q for all q = p j , j ≥ 0, and b q > a q otherwise. We conclude if we show that b q ≤ Aa q for all q with 1 + p j ≤ q ≤ p j+1 − 1, j ≥ 0. For this, since q → b q is strictly increasing, it suffices to observe that, thanks to the second half in (3.17) The second result is the following.
as desired. We mention that an alternative, more abstract proof can be based in the theory of O-regular variation and Matuszewska indices for functions. By [13,Th. 4.4] we have that the lower Matuszewska indices of ν m and ω M agree, that is, β(ν m ) = β(ω M ), and by [ This shows that the Carleman-Roumieu ultradifferentiable class E { M} ((−ε, +∞)), consisting of all smooth complex-valued functions g defined on the interval (−ε, ∞) for some ε > 0, and such that 1 ) + C 1 (n + 1) + C 1 , since m is strictly increasing. Hence, h(B/m n ) ≤ C 2 (n + 1) for some C 2 ∈ N and all n ∈ N 0 . By definition of h, we get R C2(n+1)+1 ≤ B/m n , i.e.,

Construction of Optimal Flat Functions for a Family of Non strongly Regular Sequences
As deduced in Theorem 3.12, the construction of optimal {M}-flat functions in sectors within an ultraholomorphic class, given by a regular sequence M, provides extension operators and surjectivity results. Although such general construction has been shown in Proposition 3.10, we wish to present here a family of (non strongly) regular sequences for which an alternative, more explicit technique works. We recall that, for logarithmically convex sequences (M p ) p∈N0 , the condition (dc) is equivalent to the condition log(M p ) = O(p 2 ), p → ∞ (see [19,Ch. 6]). On the other hand, the condition (mg) implies that the sequence is below some Gevrey order (there exists α > 0 such that M p = O(p! α ) as p → ∞; see e.g. [20,33]).
The case σ = 2 is well-known, as it corresponds to the so-called q-Gevrey sequences, appearing in the study of formal and analytic solutions for q-difference equations, see for example [3,7] and the references therein.
First, we will construct a holomorphic function on C \ (−∞, 0] which will provide, by restriction, an optimal {M q,σ }-flat function in any unbounded sector S γ with 0 < γ < 2. Subsequently, we will obtain such functions on general sectors of the Riemann surface R of the logarithm by ramification. This, according to Theorems 3.2 and 3.12, agrees with the fact that γ(M q,σ ) = ∞.

Flatness in the Class Given by M q,σ
It will be convenient to note that for a fixed σ ∈ (1, 2], there exists a unique s ≥ 2 such that σ = s/(s − 1).