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Summability in a monomial for some classes of singularly perturbed partial differential equation

The aim of this paper is to complete the study of asymptotic expansions and summability in a monomial in any number of variables. In particular we characterize these expansions in terms of bounded derivatives and we develop tauberian theorems for the summability processes involved. Furthermore, we develop and apply the Borel-Laplace analysis in this framework to prove the monomial summability of solutions of a specific class of singularly perturbed PDEs.


Introduction
The theory of asymptotic expansions in one complex variable is a well established and widely used branch of analysis. It provides the adequate setting to treat solutions of analytic problems at singular points and opens naturally the way to study (Borel) summability of formal solutions, as power series or exponential series. It finds its applications in some classes of ordinary, partial and singularly perturbed differential equations, analytic classification of formal objects and some other classes of functional equations. In this framework we have at our disposal two main tools to approach such problems: the Ramis-Sibuya theorem and the Borel-Laplace analysis.
To treat problems in several variables, there are different approaches to asymptotic expansions available in the literature. We can mention Majima's notion of strong asymptotic expansions which allows to work with each variable independently, although the problem of identifying singular directions for summability persists, see [16]. In the Gevrey case this notion also can be approached through a Borel-Laplace analysis. Let us mention that in [9] it is used as a regularization process to prove the existence and "summability" of solutions for classes of n−th order nonlinear quasilinear partial differential equations.
Our goal in this paper is to follow the sketch of the theory of asymptotic expansions in one variable presented in Section 2 to provide analogous results for the monomial case. This concept fits between the theory of one variable and that of Majima in several variables. It was introduced in [4] for the case of two variables and then in [14] for any number of them, to treat for instance doubly singular equations. The monomial involved helps us to identify the possible singular directions for summability and in this aspect the theory can be reduced to the case of one variable.
The main theoretical results we obtain are: a characterization of having a monomial asymptotic expansion in terms of bounded derivatives (Theorem 3.6), equivalent methods to establish monomial summability based on integral transformations (Theorem 4.7) and Tauberian theorems comparing such summability processes (Theorem 5.5).
For the case of two variables the last two items were treated in [6], [7]. The key idea was to weigh the variables adequately to obtain such integral transformations (as the ones introduced in [3]) and then to have at hand a Borel-Laplace analysis. One of the improvements we obtain here is that some of the weights can be zero. At a first look it might seem that the zero weighted variables act as parameters. The main difference with the parametric case is that the domains where the asymptotic expansions take place also depend on them. In fact, those are incompatible summability methods as we prove in Theorem 5.8.
Asymptotic expansions and summability have been recently generalized by J. Mozo and R. Schäfke in [14] from monomials to germs of analytic functions and in particular for polynomials. Tauberian theorems and integral transformations for the corresponding summability methods are still lacking and it is an interesting problem to determine whether it is possible to extend our results to these cases.
The plan for the paper is the following: in Section 2 and 3 we recall the basic results on asymptotic expansions and summability in one variable and for monomials, respectively. Section 4 is devoted to introducing and developing integral transformations to characterize monomial summability and then in Section 5 these tools are applied to prove tauberian theorems for these summability methods. Finally Section 6 contains the proof of the monomial summability of the formal solution of the partial differential equation mentioned above.
To check that f ∼f on S it is actually sufficient to have inequalities (1) only for the values N = M p, where p ∈ N >0 is fixed. The asymptotic expansion also holds if instead of the partial sums off we consider a sequence (f N ) N ∈N ⊂ O b (D R , E) such that for all S ′ ⋐ S and N ∈ N there are constants C N (S ′ ) > 0 such that The seriesf is completely determined by f since a n = lim x→0 , for any S ′ ⋐ S, or it is given by the limit of the Taylor series at the origin of the f n , in the m−topology of E[[x]], m = (x). On the other hand, a map has an asymptotic expansion on S if and only if it has bounded derivatives of all orders in every S ′ ⋐ S, a fact that follows using Taylor's formula.
When no restrictions on the coefficients C N (S ′ ) or on the sectors are imposed, the map f →f is not injective. In the applications to differential equations the type of asymptotic expansions that appear are of s−Gevrey type (denoted by f ∼ sf on S) for some s > 0. This means that we can choose C n (S ′ ) = C(S ′ )A(S ′ ) n n! s , for some C(S ′ ), A(S ′ ) independent of n. From (1) it follows that a n ≤ C n (S ′ ) for all n ∈ N. Then in the s−Gevrey case we conclude thatf is a s−Gevrey series, space that we will be denote as ], k > 0 and a direction θ we say that: We also say thatf is k−summable in the direction θ. The corresponding space is denoted as E{x} 1/k,θ . (2) The seriesf is k−summable if it is k−summable in all directions up to a finite number of them mod. 2π (the singular directions). The corresponding space is denoted as E{x} 1/k .
Due to Watson's lemma the k−sum of a k−summable series is unique. We have at our disposal integral transformations to compute these sums. Among the moment summability through kernels of order k, see e.g. [1,Sec. 6.5], it is common to consider: , S = S(θ, π/k + 2ǫ, R 0 ), 0 < 2ǫ < π/k and γ is the boundary, oriented positively, of a subsector of S of opening larger than π/k. Its formal counterpart B k is given on monomials by where g is continuous and with exponential growth of order at most k on its domain of integration. If g is defined on a sector, changing θ leads to an analytic continuation that will be simply written as L k (g).
Using these transformations, and due to their compatibility with asymptotic expansions, a 1/k-Gevrey seriesf = ∞ n=0 a n x n is called k−Borel-summable in direction θ if B k f − n≤k a n x n can be analytically continued and has exponential growth of order at most k in an unbounded sector S ′ containing θ. If ϕ is this analytic continuation, the Borel sum off is defined by f (x) = n≤k a n x n + L k (ϕ)(x). It is by now well-known e.g. [15] that a power seriesf ∈ E[[x]] is k−Borel summable in a direction θ if and only if it is k−summable in the direction θ and both sums coincide. Using this equivalence, Tauberian theorems for k−summability are available. In particular a series that is k−summable for two different values of the parameter k is convergent.
The Borel-Laplace analysis has been applied as a regularization process in differential equations to prove the summability of formal solutions in generic situations. It exploits the isomorphism between the following structures where × denotes the usual product and * k stands for the k−convolution product, that for holomorphic maps is defined as For more isomorphisms of this nature see e.g. [13].

Asymptotic expansions in a monomial
We will recall the concept of asymptotic expansions and k−summability in a monomial, in any number of variables, that was introduced in [4] for the case of two variables and then carried on in [14] for the general case. We introduce the remaining notation we will use along the text. For a fixed d ∈ for the standard d−simplex they generate and σ d for its topological closure. We will denote the standard inner product on R d as < · >. We will also use complex coordinates x = (x 1 , . . . , x d ) ∈ C d and we will write x ′ = (x 2 , . . . , x d ) to indicate that we remove the first one.
we use the common multi-index notation Along the text we work with the partial order on N d defined by β ≤ α if and only if β j ≤ α j for all j ∈ [1, d]. Note that β ≤ α if and only if there For a complex Banach space E = (E, · ), E[[x]] (resp. E{x}) will stand for the space of formal (resp. convergent) power series in the variables x with coefficients in E. For s = (s 1 , . . . , s d ) ∈ R d Furthermore, if α ∈ N d >0 is given and we consider the monomial x α ,f can be also written uniquely as To ensure that eachf α,n = f α,n gives rise to a holomorphic map defined in a common polydisc at the origin it is necessary and sufficient thatf If this is the case then alsof J,β J = f J,β J ∈ E{x J c } and they have a common radius of convergence for all β J ∈ N J , J [1, d]. Besides f α,n ∈ E α = lim r→0 E α r , where E α r is the space of holomorphic maps g ∈ O b (D d r , E) with ∂ β g ∂x β (0) = 0 for all α ≤ β. All f α,n have a common radius of convergence as well. Note that each E α r becomes a Banach space with the norm g = sup |x1|,...,|x d |≤r g(x) .
For anyf ∈Ô ′ d (E) and γ ∈ N d we will use the notation for the formal approximate off order γ. In particular for γ = N α we have given byT α (f ) = ∞ n=0 f α,n t n , using decomposition (4). We will sayf is a s−Gevrey series in the monomial x α if for some r > 0, and it is a s−Gevrey series in t. The set of s−Gevrey series in the monomial x α will be denoted by E[[x]] α s i.e., there are constants B, D > 0 such that f α,n ≤ DB n n! s for all n ∈ N. ( Proof. (1) Assume there are constants B, D > 0 such that f α,n ≤ DB n M n for all n ∈ N. Given γ ∈ N d , let n = min 1≤j≤d [γ j /α j ] and thus γ = nα + β with β l < α l for some l. Then by Cauchy's inequalities we see that what yields one implication. The converse follows by the same argument as in (2) below.
(2) Iff = a β x β and a β ≤ CA |β| min 1≤j≤d {β j ! s/α ′ j } for all β ∈ N d , we can directly estimate the growth of the f α,n by means of formula (4): if |x j | < r, j = 1, . . . , d and rA < 1 we get The result follows from the limit lim n→∞ (nl)! 1/l /l n n! = 0, valid for any integer l ≥ 2. In the analytic setting we use sectors in the monomial x α , i.e. sets of the form Here any convenient branch of arg may be used. The number r denotes the radius, b−a the opening and θ = (b + a)/2 the bisecting direction of the monomial sector. We will also use the notation Π α (a, b, r) = S α (θ, b − a, r) = S α . In the case r = +∞ we will simply write Π α (a, b) = S α (θ, b − a) and we will refer to them as unbounded sectors. The definition of subsector in a monomial is clear.
we can always cover the first one by polysectors, i.e. cartesian product of sectors, of constant opening contained in the second one. In particular, we can check that Π ′ α ⊂ U ⊂ Π ′′ α , where U is given by Indeed, if x 0 = (x 1,0 , . . . , x d,0 ) ∈ Π ′ α then x 0 belongs for instance to the polysector with the µ j given by If f ∈ O b (Π α (a, b, r), E) then we can construct an operator T α (f ) ρ : V = V (a, b, ρ d ) → E α ρ , 0 < ρ < r, as it is done for the formal case, such that We recall this construction following [14], proceeding first when α = 1 = (1, . . . , 1). Define the map g(t, x 2 , . . . , x d ) := f t x2···x d , x 2 , . . . , x d for |x 2 |, . . . , |x d | < r and |t|/r < |x 2 · · · x d |. The map g admits a Laurent series expansion g(t, x 2 , . . . , where M d is the set of all (n 1 , . . . , n d ) ∈ N d such that at least one of the n j vanishes. Then by definition With these considerations we guarantee that all the exponents in x are non-negative. To check that this expression is well-defined and satisfies what it is required note that since f is bounded, say by some constant C, then by Cauchy's inequalities g m (t) ≤ Cr −m2 If m l = µ(m) choose r j = r for all j = l and r l such that r 2 · · · r d = |t|/r. Then we deduce and thus each t µ(m) g m (t) is holomorphic and bounded on V . It is also clear that x φ(m) ∈ E 1 ρ since φ(m) ∈ M d and thus the map defined through the previous series also belongs to the same space.

More generally if there is a function
We conclude that In the general case for an arbitrary monomial, we can write In fact, if ω j is a α j -th primitive root of unity then Then we define To see that T α (f ) ρ is well-defined and thus holomorphic we can actually show that if f satisfies Indeed, by (10) we get We can thus apply (8) to each summand in (11) to finally obtain Then (12) follows as |x j | αj < ρ < r and also using the identity 0≤βj <αj a β1 2 for |t|/r < |x 2 | < r and then we have As an application we can prove the following proposition regarding the dependence and growing of a holomorphic map in a monomial.
We will say that f hasf as asymptotic expansion at the origin in The asymptotic expansion is said to be of s−Gevrey type (denoted by From the very definition of f ∼ αf = a β x β on Π α we can deduce using (13) for N = 1 that (14) a 0 = lim for all subsector Π ′ α ⊂ Π α . As mentioned in the introduction, monomial asymptotic expansions can be reduced to the case of one variable, using the operators T α andT α . Indeed, direct estimates using (12) show that if The following assertions are equivalent: For In any case,f is given by the limit of the Taylor series at the origin of F n , in the m−topology Proof. If f ∼ αf on Π α andT α (f )(t) = f α,n t n then F N (x) = n<N f α,n (x)x nα satisfies the requirements. Conversely, suppose we have such a family. Note that each T α (F N ) R is holomorphic on D R and hasT α (F N ) as Taylor series at the origin. Let g N +1 = T α (F N ) R . Applying inequalities (12) for N + 1 to (15) in the corresponding sector where 0 < |t| < ρ ′ < min{ρ, R}. Thus we obtain that T α (f ) ρ ′ has as asymptotic expansionT α (f ) given by the limit of the seriesT α (F N ) in the m−topology of E α r [[t]], m = (t), and thus f ∼ αf on Π α as we wanted to show. The s−Gervey case also follows since the sequence (g N +1 ) N ∈N has bounds of s−Gevrey type if the sequence (F N ) N ∈N does.
These characterizations allow us to prove that monomial asymptotic expansions are stable under sums, products and partial derivatives. In particular it follows from the relations (14) and thusf is completely determined by f .
We can finally give a new but expected characterization of monomial asymptotic expansion in terms of bounded derivatives. The proof follows the same lines as in [11,Prop. 3].
More precisely, if s > 0 then the following assertions are equivalent: Proof. We only consider the statement for the s−Gevrey case. It is clear that (3) implies (2). To prove that (2) implies (1) we use Taylor's formula joint with equations (5) and (16) to write and then conclude the desired bounds. Note that the paths of integration are the segments w j = ρ j x j , 0 < ρ j ≤ 1 and w = ( Using Remark 3.2 we can find a polysector P = d j=1 V j containing x such that P ⊂ Π ′′ α . Moreover V j has opening independent of x since it is given by (b ′ − a ′ )/(d − 1)α j for j = 1, . . . , d − 1 and b ′′ − a ′′ − (b ′ − a ′ ) for j = d. We then can find numbers σ 1 , . . . , σ d depending only on Π ′ α and Π ′′ α such that the circles C j given by |w j − x j | = σ j |x j | are contained in V j for all j = 1, . . . , d. Then using Cauchy's formula and Remark 3.5 we see that for some large enough constants C(Π ′′ α ), A(Π ′′ α ) independent of β. Then (3) follows.
When we fix some of the variables in the monomial asymptotic expansion of a map, the expansion is valid in the remaining ones. We state this result when we only fix one of the variables. The proof is an immediate consequence of Proposition 3.4.
It is straightforward to characterize maps with null s−Gevrey asymptotic expansion in a monomial: for f ∈ O(Π α , E), f ∼ α s0 on Π α if and only if for every subsector Π ′ α ⊂ Π α there are constants C, A such that x ∈ Π ′ α . In this framework Watson's lemma holds: if f ∼ α s0 on Π α (a, b, r) and b − a > sπ then f ≡ 0. As in the case of one variable it is natural to consider the following definition. (1) The seriesf is called k−summable in x α on S α = S α (θ, b − a, r) with sum f ∈ O(Π α , E) if b − a > π/k and f ∼ α 1/kf on S α . We also say thatf is k−summable in x α in the direction θ. The space of k−summable series in x α in the direction θ will be denoted by E{x} α to a finite number of them mod. 2π (the singular directions). The corresponding space is denoted as E{x} α 1/k .
Note that both E{x} α 1/k,θ and E{x} α 1/k are vector spaces stable by partial derivatives and they inherit naturally a structure of algebra when E is a Banach algebra.

Borel-Laplace analysis for monomial summability
The goal of this section is to generalize the Borel and Laplace transformations for monomial asymptotic expansions from [6] in any number of variables and develop their main properties to prove that monomial summability is equivalent to Borel-summability in this framework. We point out that in contrast with [6] we have generalized the results as we can allow some of the weights to be zero. This will be crucial in the applications presented in the last section.
For any c ∈ R d ≥0 we will denote from now on J c := {j ∈ [1, d]|c j = 0}, the set of indexes where c has nonzero entries.
Definition 4.1. The k−Borel transform w.r.t. x α and a weight s ∈ σ d , of a map f is defined by the formula where γ denotes a Hankel path as explained below.
From now on, we will write λ = s1 α1k , . . . , s d α d k and λ ′ = α1k s1 , . . . , α d k s d . If s j = 0 for some j we interpret the jth entry of λ ′ as 0, and the variable x j remains unchanged. In this situation, to avoid cumbersome notation, we call it ξ j nevertheless. This convention will be used further on without explicit mention. Note also that the monomial factor outside the integral only includes the variables ξ j such that s j = 0. The ambient space C d with coordinates ξ will be referred to the ξ−Borel plane.
Since we admit that some weights can be zero it is necessary to consider monomial sectors where some of the variables are bounded.
If all entries of s are different from zero then S s α (θ, 2ǫ, R 0 ) will be simply denoted by S α (θ, 2ǫ). We will use the same notation for every c ∈ R d ≥0 other than s. If ξ ∈ S α (θ, 2ǫ ′ ), 0 < ǫ ′ < ǫ, we take γ oriented positively and given by the arc of a circle centered at 0 and radius R > max with endpoints on the directions −π/2 − k(ǫ − ǫ ′ ) and π/2 + k(ǫ − ǫ ′ ) and the half-lines with those directions from this arc to ∞. If u goes along this path the integrand is evaluated on S α and the integral converges absolutely, since f is bounded and the exponential term tends to 0 on those directions. The result is independent of ǫ ′ and R due to Cauchy's theorem.
Using Hankel's formula for the Gamma function we obtain the formula and thusB λ , the formal k−Borel transform w.r.t. x α and weight s, can be defined on formal power series by linearity.
By looking at the derivative w.r.t. u of the integrand defining B λ it is natural to consider the vector field X λ and its flow (at time z) φ λ z given by For f ∈ O b (S α , E) as before it follows that Both formulas are naturally related since the first one is the linearization of the second one at z = 0. In the variable t = x α the vector field X λ reduces to t k+1 /k∂/∂t, and the first formula is just the one contained in the isomorphism (2).
We will say that f has exponential growth of order at most c = (c 1 , . . . , , we can use any of the terms R c (ξ) or j∈Jc |ξ j | cj by changing M above.
If f is holomorphic on C d ∩ {ξ ∈ C d ||ξ j | < r j , j ∈ J c } for some fixed r j > 0 (resp. entire if c ∈ R d >0 ) and β∈N d a β x β is its Taylor series at the origin then the above condition is equivalent to the existence of constants D, B 1 , . . . , B d > 0 such that This standard assertion can be deduced using Cauchy's integral formulas for the coefficients, Stirling's formula and the inequalities satisfied by the Gamma function.
Jsφ λ ) = ϕ α,n τ n then f α,n and ϕ α,n are related by ξ nα ξ −kαJ s Js ϕ α,n = B λ (x nα f α,n ), n ≥ k. Since the f α,n are holomorphic in a common polydisc D d ρ we see that the ϕ α,n are all holomorphic maps on where L, M > 0 are some constants independent of n but depending on ρ.
The behavior of the Borel transform w.r.t. monomial asymptotic expansions is presented in the next proposition. It is based on estimates included in [16] for the Borel transform in several variables.
We may assume that ifT α (f ) = f α,n t n , the f α,n are holomorphic on D d R0 , reducing R 0 if necessary. By hypothesis inequality (13) holds with C N = CA N Γ(1 + N s) on S α . Setting a = sin k 2 (ǫ ′′ − ǫ ′ ) and with R > 1 to be chosen, a direct estimate using Remark 4.1 shows that for all N ∈ N we have For N = 0 we are denoting C = sup x∈Sα f (x) (note that f is bounded here, as we have reduced the radius and the opening of the sector).
To prove the statement we divide our sector in two parts. First of all, consider ξ ∈ S α (θ, 2ǫ ′ , r), with r > R 0 /2 fixed. In this case choose R ≥ max j∈Js (2r/R 0 ) k s j > 1. Since it is enough to establish the bounds for large N we can suppose N is large enough and take R = N/k. Then the bound follows using Stirling's formula, since asymptotically and also because Γ(1 + N s) ≤ 2 sN Γ(1 + N/k)Γ (1 + N (s − 1/k)) if s ≥ 1/k, due to the second inequality in (18).
To finish we establish the bound in the complementary region, i.e., for ξ ∈ S ′ α \ S α (θ, 2ǫ ′ , r). For each j ∈ J s choose R j < R 0 /2 and let us take (7)). Then we can Let M = max j∈Js (4/R 0 ) k/sj . For each j ∈ J s we consider two cases: choose R j = |ξ j | αj /(s j N/k) sj /k < R 0 /2 as long as this inequality holds. Then In the second case choose R j = R 0 /4 < R 0 /2 ≤ |ξ j | αj /(s j N/k) sj/k and then In both cases we conclude that Using Stirling's formula we conclude that there are large enough constants L, K > 0 such that on S ′ α \S α (θ, 2ǫ ′ , r). We can use Stirling's formula again or the second inequality in (18) to conclude the proof.
Definition 4.2. The k−Laplace transform w.r.t. x α with weight s ∈ σ d in direction φ, |φ| < π/2, of a map f is defined by the formula As in the case of B λ , if s j = 0 for some j, the variable ξ j is not affected by the transformation, although we will still call it x j .
If the map f ∈ O(S α , E), with S α = S α (θ, b − a), had exponential growth of order k in the monomial ξ α , i.e., f (ξ) ≤ C exp M |ξ kα | on S α , then by Proposition 3.3 f would be a map depending only on ξ α . Instead by Proposition 4.2 we assume that f is defined and of exponential growth of order at most λ ′ on S s α (θ, b − a, R). In such case, L λ,φ (f ) is defined and holomorphic on the domain D s We note that by changing the direction φ by φ ′ we obtain an analytic continuation of L λ,φ (f ) when |φ ′ − φ| < k(b − a), a fact that follows directly from Cauchy's theorem. This process leads to a holomorphic map L λ (f ) defined on the region D s α (θ, b − a + π/k; M, . An adequate choice of the path γ in the definition of B λ , a limiting process and the residue theorem show that if f ∈ O b (S s α (θ, π/k + 2ǫ, R 0 ), E), 0 < 2ǫ < π/k then L λ B λ (f ) = f, on the intersection of their domains.
The operator L λ is also injective as the usual Laplace transform. Thus if g is of exponential growth of order at most λ ′ then B λ L λ (g) = g, on the intersection of their domains.
We then defineL λ , the formal k−Laplace transform w.r.t. x α and weight s, as the inverse of B λ . When we write a series as a series in a monomial it is natural to ask what is the relation between its Laplace transform and the transform of its components. That is the content of next remark. A necessary and sufficient condition onf so thatL λ (f ) is convergent is that ξ kαJ s Jsf defines a holomorphic map f of exponential growth of order at most λ ′ on C d ∩ {ξ ∈ C d ||ξ j | < r j , j ∈ J s }, for some r j > 0 (resp. on C d if s has all nonzero entries). Then L λ (f ) is holomorphic in a polydisc at the origin andL λ (f ) is its Taylor series. Now assume that there are positive constants s, B, D, M such that the family of maps f α,n are holomorphic and satisfy the bounds Then we can conclude that ξ Jsf ) = f α,n t n and s ≥ 0. Assume that: (1) There are constants B, D, K > 0 and 0 < r < R such that f α,n are holomorphic and satisfy bounds of type (2) For every unbounded subsector S ′ α of S s α there are positive constants C, A, M such that for all N ∈ N ξ kαJ s Js In particular ξ kαJ s Js Then Proof. For N = 0 statement (3) is interpreted as f having of exponential growth of order at most λ ′ on S s α . Set h = L λ (f ) andT α (L λ (f )) = h α,n t n . Then using (2) and Remark 4.3 we see that h α,n x nα = L λ (ξ nα ξ −kαJ s Js f n ). For a fixed |φ| < π/2 it is enough to prove the estimates for subsectors S ′′ α contained in D s α (θ − φ/k, b − a; M, r). We can find δ > 0 small enough such that Since Γ(N/k) ≤ N Γ(N/k) = kΓ(1 + N/k), an application of the first inequality in (18) leads us to the result.
Finally we introduce a natural convolution product that in our context shares similar properties with the classical one. Indeed, the convolution between f and g w.r.t. x α and weight s ∈ σ d is defined by As an example we can compute with the aid of the Beta function the convolution between two powers formula valid for µ, η ∈ C d with entries of positive real part.
If f, g have exponential growth of order at most λ ′ (resp. belong to O b (S α , E)) then the same is valid for f * λ g and (20) as in the classical case. This shows in particular that * λ is a bilinear, commutative and associative binary operation just as the usual product.
Remark 4.5. In analogy with the isomorphism explained in (2), for each α ∈ N d >0 , k > 0 and s ∈ σ d we have the following monomorphism between the structures by taking into account (6) for the image and also (17) and (20).
Remark 4.6. We remark that all transformations introduced here reduce to their one variable counterparts for maps depending only on the monomial.
At this point we are ready to define the summation methods based in the above Borel and Laplace transforms and we will see that they turn out to be equivalent to monomial summability. We can be analytically continued, say as ϕ s , to a domain of the form S s α (θ, 2ǫ, R) and having exponential growth of order at most λ ′ there. In this case the k − s−Borel sum off in direction θ is defined as Note that the terms we remove fromf produce an analytic map at the origin sincef is an element ofÔ ′ d (E). Another way to avoid the use of power series with non-integer exponents is to considerB λ (x kαJ s Jsf ), attempt analytic continuation with adequate growth and multiply by x −kαJ s Js the corresponding Laplace transformation. We have not followed this equivalent approach since the introduction of such factor does not adapt well for the non-linear terms in the PDEs we consider in last section. (1)f ∈ E{x} α 1/k,d , i.e.f is k−summable in x α in the direction θ.
(2) There is s ∈ σ d such thatf is k − s−Borel summable in the monomial x α in the direction θ.
(3) For all s ∈ σ d ,f is k − s−Borel summable in the monomial x α in the direction θ.
In all cases the corresponding sums coincide.
Proof. We may restrict out attention to the casef = kαJ s ≤β Js a β x β . To show that (1) implies (3) assumef ∈ E{x} α 1/k,θ and let f ∈ O(S α (θ, π/k + 2ǫ, R 0 ), E) be its k−sum in x α in direction θ. For a fixed s ∈ σ d and λ as usual set ϕ s = B λ (f ) andφ s =B λ (f ), convergent in some D d r . We can apply Proposition 4.2 for s = 1/k to conclude that ξ kαJ s Js Jsφ s on S s α (θ, 2ǫ, R 0 ). These two properties imply that ϕ s coincides with the sum ofφ s on S α (d, 2ǫ) ∩ D d r . In other wordŝ ϕ s can be analytically continued having exponential growth of order at most λ ′ on S s α (θ, 2ǫ, R 0 ) and thereforef is k − s−Borel summable in the monomial x α in direction θ. Since B λ and L λ are inverses of each other the k − s−Borel sum in x α off is f .
To apply Proposition 4.4 we may show that there are constants C, A > 0 such that for all N ∈ N we have Since ξ kαJ s Jsφ s is the convergent Taylor series of ξ kαJ s Js ϕ s at the origin then (22) is satisfied for all |ξ 1 |, . . . , |ξ d | ≤ R for some R > 0. Due to (21) the series of maps ϕ α,n (ξ)ξ nα converges absolutely in compact subsets of {ξ ∈ C d | B|ξ α | < 1}. Then ξ kαJ s Js ϕ s can be analytically continued there through that series. Thus if B|ξ α | < 1/2, inequality (22) is also satisfied. On the other hand, using again (21), the left side of (22) is bounded by and thus (22) is valid in all cases with C, A large enough. Applying Proposition 4.4 to ϕ s ,φ s and In conclusion,f is k−summable in x α in the direction θ and its sum can be found through the k−Laplace transform w.r.t. x α and weight s of ϕ s .
As an immediate corollary we can relate summability w.r.t. to a monomial x α and a power of it x N α , N ∈ N >0 . The proof follows observing that λ = Corollary 4.8. Let k > 0, α ∈ N d >0 , N ∈ N >0 and let θ be a direction. Then E{x} α 1/k,θ = E{x} N α N/k,N θ and in particular E{x} α 1/k = E{x} N α N/k .

Tauberian properties for monomial summability
The goal of this section is to recover some tauberian theorems for monomial summability. For instance, the relation between different levels of summability for different monomials and the comparison of such methods with summability in a monomial in some variables and with holomorphic coefficients in the remaining ones. The main tool we use to treat these situations are blow-ups with centers of codimesion two and the behavior of Borel-Laplace methods under them.
In one variable we have the following two statements that provide tauberian properties for k−summability and that we will recover in the monomial case.
Since for a fixed monomial summability in the monomial is equivalent to summability in the classical sense, we obtain immediately the following statement.
Proposition 5.2. Let 0 < k < l be positive numbers. Then the following statements are true: (1) Iff ∈ E{x} α 1/k has no singular directions thenf is convergent.
To generalize the previous results we consider the monomial transformations where i, j = 1, . . . , d, i = j. Note that the maps π ij , π ji correspond to the usual charts of the blow-up with the center of codimension two given by {x i = x j = 0}. At the formal level we need the following lemma which proof is identical as the one of Lemma 3.6 of [7].
] be a formal power series. Then the following assertions are true: (1)f ∈ E{x} if and only iff • π ij ∈ E{x} for some i, j = 1, . . . , d.
( and it has k−sum f • π ij in the direction θ, for all i, j = 1, . . . , d, i = j. Proof. Using Proposition 3.4 we see that if (f N ) N ∈N is a family of bounded analytic functions that provide the monomial asymptotic expansion of f then (f N • π ij ) N ∈N will provide the asymptotic expansion of f • π ij , i, j = 1, . . . , d, i = j.
We are ready to state and prove the third main result so far, comparing summable series in different monomials.
The same idea of proof can be generalized to construct series which are not k−summable in x α for any k > 0 and any monomial. The proof can be done by induction on n and follows the same strategy as in [7,Thm. 3.9] so it will be omitted here.
Givenf ∈ O ′ d (E), we can also consider the case wheref is k−summable in a monomial in some variables with coefficients which are holomorphic maps in the remaining ones and compare the summability phenomena we have at our disposal. As a matter of fact, in this situation the methods are again incompatible and the proofs can be reduced to Theorem 5.5 using monomial transformations. The key statement is the following. In the general case we can assume there are j 0 ∈ J \ I and i 0 ∈ I \ J, otherwise we are in the previous stages. We consider the serieŝ f 2 =f • • i∈I\J π ij0 • • j∈J\I π ji0 . Here • denotes the composition product, that in this case is independent of the order since the monomial transformations involved commute as j 0 = i 0 .f 2 is l−summable in x , E) (resp. E if J ∪ I = [1, d]). Since the i 0 -components (resp. j 0 -components) of these monomials are β i0 and α j0 (resp. β i0 and 2α j0 ) respectively then α j0 /β i0 = 2α j0 /β i0 and thereforef 2 andf are convergent as we wanted to prove.
Having this result at hand it is possible to formulate and prove a statement similar to the one in Theorem 5.6 providing more examples of series that cannot be summed with the methods we have studied along this work.

Monomial summability of a family of singular perturbed PDEs
As mentioned in the introduction, the unique formal solution of the doubly singular equation where p, q ∈ N >0 , F is a C N −valued holomorphic map in some neighborhood of (0, 0, 0) ∈ C × C × C N and ∂F ∂y (0, 0, 0) is an invertible matrix, is 1−summable in x p ε q [4]. The technique applied to prove this result in the case p = q = 1 is to perform directly on the equation the change of variables t = xε to obtain an equation involving t and ε. Then the new equation is solved on large sectors and the differences of such solutions in their common domains are studied to be able to apply Ramis-Sibuya's theorem. The general case follows after rank reduction.
The goal of this section is to generalize the previous result by establishing the 1−summability in x α ε α ′ of the unique formal power series solution of the partial differential equation (23) ε α ′ X λ (y)(x, ε) = x α ε α ′ s 1 α 1 x 1 ∂y ∂x 1 + · · · + s n α n x n ∂y ∂x n = F (x, ε, y), where x = (x 1 , . . . , x n ), ε = (ε 1 , . . . , ε m ) are tuples of complex variables, α = (α 1 , . . . , α n ), α ′ = (α ′ 1 , . . . , α ′ m ) are tuples of positive integers, s 1 , ..., s n are positive real numbers such that s 1 + · · · + s n = 1, F is a C N −valued holomorphic map in some neighborhood of (0, 0, 0) ∈ C n × C m × C N and A 0 = ∂F ∂y (0, 0, 0) is an invertible matrix. We will apply directly the Borel-Laplace analysis developed in Section 4, based on the methods of one variable e.g. [8]. The existence of the unique formal solution is a straightforward result. To determine the Gevrey type of this solution we can use a variant of Nagumo norms as the ones used in [5]. For the summability we will study the convolution equation obtained from (23) after applying the adequate Borel transformation. After introducing suitable spaces of analytic functions and norms, we will solve the convolution equation using the Banach fixed-point theorem.
Proof. We divide the proof in four main steps: existence and Gevrey character of the formal solution, establishment of the associated convolution equation, introduction of adequate Banach algebras and their properties and finally the application of the fixed point theorem.