Pascal Remy

GEVREY PROPERTIES AND SUMMABILITY OF FORMAL POWER SERIES SOLUTIONS OF SOME INHOMOGENEOUS LINEAR CAUCHY-GOURSAT PROBLEMS

Keywords: 2000 Mathematics Subject Classification. 35C10, 35C20, 40B05 Linear partial differential equation, linear integro-differential equation, divergent power series, Newton polygon, Gevrey order, Gevrey asymptotic, summability

In this article, we investigate the Gevrey and summability properties of the formal power series solutions of some inhomogeneous linear Cauchy-Goursat problems with analytic coefficients in a neighborhood of p0, 0q P C 2 . In particular, we give necessary and sufficient conditions under which these solutions are convergent or are k-summable, for a convenient positive rational number k, in a given direction.

For several years, various works have been done on the divergent solutions of some classes of linear (see [1, 3-6, 8, 11, 17, 18, 27-34, 36, 42-45] etc.), nonlinear (see [START_REF] Lastra | On parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems[END_REF][START_REF] Lastra | On parametric multisummable formal solutions to some nonlinear initial value Cauchy problems[END_REF][START_REF] Lastra | On Gevrey solutions of threefold singular nonlinear partial differential equations[END_REF][START_REF] Malek | On singularly perturbed partial integro-differential equations with irregular singularity[END_REF][START_REF] Malek | On the summability of formal solutions of nonlinear partial differential equations with shrinkings[END_REF][START_REF] Malek | On Gevrey asymptotic for some nonlinear integro-differential equations[END_REF][START_REF] Malek | On the summability of formal solutions for doubly singular nonlinear partial differential equations[END_REF][START_REF] Pliś | Borel resummation of formal solutions to nonlinear Laplace equations in 2 variables[END_REF] etc.) or singular (see [START_REF] Hibino | Borel summability of divergence solutions for singular first-order partial differential equations with variable coefficients[END_REF][START_REF] Hibino | On the summability of divergent power series solutions for certain first-order linear PDEs[END_REF][START_REF] Malek | On the Stokes phenomenon for holomorphic solutions of integrodifferential equations with irregular singularity[END_REF][START_REF] Malek | On Gevrey functions solutions of partial differential equations with fuchsian and irregular singularities[END_REF][START_REF] Ouchi | Borel summability of formal solutions of some first order singular partial differential equations and normal forms of vector fields[END_REF] etc.) partial differential equations or integro-differential equations in two variables or more, allowing thus to formulate many results on Gevrey properties, summability or multisummability.

In this paper, we are interested in the formal power series solutions of linear Cauchy-Goursat problems of the form U pt, xq ´wpt, xq " Opt κ x p q, where ' the partial differential operator L satisfies conditions: pC 1 q: κ ě 1 is a positive integer and p ě 0 is a nonnegative integer, pC 2 q: K is a subset of t0, ..., κu which contains at least one positive element and which does not contain 0 if p " 0, pC 3 q: Q 0 is a non-empty subset of t0, ..., p ´1u if 0 P K, pC 4 q: Q i is a non-empty finite subset of N (= the set of nonnegative integers) for all i P K, i ‰ 0, pC 5 q: v i,q ě 0 is a nonnegative integer and the coefficients a pi,qq pt, xq are holomorphic in the two variables t and x in a polydisc D ρ1 ˆDρ2 centered at the origin p0, 0q P C 2 (D ρj denotes the disc with center 0 and radius ρ j ą 0) for all i P K and q P Q i , pC 6 q: a pi,qq p0, xq ı 0 for all i P K and q P Q i . ' the inhomogeneity r qpt, xq P OpD ρ2 qrrtss 1 is a formal series in t with coefficients in OpD ρ2 q which may be smooth, or not, ' the Cauchy-Goursat data wpt, xq is holomorphic in D ρ1 ˆDρ2 . Under more or less restrictive conditions on valuations v i,q , coefficients a pi,qq pt, xq, degrees Q i , inhomogeneity r qpt, xq and initial data wpt, xq, problem (1.1) was already investigated by many authors (see [1, 3-6, 8, 11, 17, 18, 32-34, 36, 42-45] etc.). Here, we consider the very general problem of the form (1.1), where no generic assumption is made.

For both practical and notational conveniences, we now change the unknown function U to u by U pt, xq " wpt, xq `B´κ t B ´p x upt, xq. Then, problem (1.1) is equivalent to the following integro-differential equation ( ˙" t j` pj ` q! for all ě 1 and j ě 0. It is the same for B ´ x with ě 1.

Notation 1.1. For any series r upt, xq P OpD ρ2 qrrtss, we denote in the sequel r upt, xq "

ÿ jě0 u j,˚p xq t j j! " ÿ ně0 r u ˚,n ptq x n n! .
The organization of the paper is as follows. In Section 2, we prove that the linear integro-differential equation (1.2) admits a unique formal series solution r upt, xq in OpD ρ2 qrrtss (Theorem 2.1) and we give a characterization of its coefficients in OpD ρ2 q. In Section 3, we introduce the Newton polygon N t pDq of the operator D at t " 0 and we give some properties of this. In Section 4, we show that r upt, xq and the inhomogeneity r f pt, xq are together either convergent or 1{k-Gevrey, where k denotes the smallest positive slope of N t pDq (Theorem 4.4). Then, in the latter case, and under four additional conditions on D, we investigate the summability of r upt, xq. In particular, we prove in Section 4 (Theorem 5.4) a necessary and sufficient condition under which r upt, xq is k-summable in a given direction argptq " θ, generalizing thus the results already obtained by the author in [START_REF] Remy | Gevrey order and summability of formal series solutions of some classes of inhomogeneous linear partial differential equations with variable coefficients[END_REF][START_REF] Remy | Gevrey order and summability of formal series solutions of certain classes of inhomogeneous linear integro-differential equations with variable coefficients[END_REF].

Formal series solutions

In this section, we shall be concerned with the formal series solutions in OpD ρ2 qrrtss of the linear integro-differential equation (1.2).

Let us first observe that the operator D is a linear operator acting inside OpD ρ2 qrrtss. Indeed, pOpD ρ2 qrrtss, B t , B x q is a C-differential algebra stable under anti-derivations B ´1 t and B ´1

x and the coefficients a pi,qq pt, xq belong to OpD ρ1 ˆDρ2 q Ă OpD ρ2 qrrtss for all i and q. More precisely, we have the following. where, as usual, the third sum is 0 as soon as j ă v i,q `i. Observe that the index j ´vi,q ´i ´m is ă j when pi, v i,q , mq ‰ p0, 0, 0q and is j otherwise. Thereby, some terms B q´p x u j,˚p xq may occur in the right-hand side of (2.1) and this, only for the q P Q 0 satisfying v 0,q " 0. In particular, when terms B q´p x u j,˚p xq occur, we necessarily have q ´p P t´p, ..., ´1u. Then, Lemma 2.2 below proves that equation Dr u " r f pt, xq admits a unique solution r upt, xq P OpD ρ2 qrrtss. Hence, the bijectivity of D, which completes the proof.

Lemma 2.2. The linear integro-differential equation

(2.2)
y `α1 pxqB ´1 x y `α2 pxqB ´2 x y `... `αp pxqB ´p x y " gpxq, whose coefficients α q pxq and inhomogeneity gpxq are holomorphic in D ρ2 , posseses exactly one solution ypxq. Moreover, this solution is holomorphic in D ρ2 .

Proof. Let z " B ´p x y. Then, ypxq is a solution of equation (2.2) if and only if zpxq is a solution of the Cauchy problem

" B p x z `α1 pxqB p´1 x z `α2 pxqB p´2 x z `... `αp pxqz " gpxq, zp0q " B x zp0q " ... " B p´1 x zp0q " 0.
The result follows then from the Cauchy-Kovalevskaïa theorem for the ordinary differential equations.

As a direct consequence of Theorem 2.1, we deduce in particular that equation (1.2) is uniquely solvable in OpD ρ2 qrrtss.

Corollary 2.3. The linear integro-differential equation (1.2) admits a unique formal series solution r upt, xq P OpD ρ2 qrrtss. Moreover, its coefficients u j,˚p xq P OpD ρ2 q are recursively determined for all j ě 0 by identities (2.1).

Observe that the formal solution r upt, xq is divergent in general. In Section 4, we shall investigate its Gevrey properties. We propose in particular to prove a necessary and sufficient condition under which it is s-Gevrey with a convenient nonnegative rational number s.

Before stating our main result (see Theorem 4.4), let us first introduce the t-Newton polygon of the operator D.

Newton polygon

As definition of the t-Newton polygon of the operator D (or Newton polygon of D with respect to t), we choose the definition of M. Miyake [START_REF] Miyake | Newton polygons and formal Gevrey indices in the Cauchy-Goursat-Fuchs type equations[END_REF] (see also A. Yonemura [START_REF] Yonemura | Newton polygons and formal Gevrey classes[END_REF] or S. Ouchi [START_REF] Ouchi | Multisummability of formal solutions of some linear partial differential equations[END_REF]) which is an analogue to the one given by J.-P. Ramis [START_REF] Ramis | Théorèmes d'indices Gevrey pour les équations différentielles ordinaires[END_REF] for the linear ordinary differential operators. Recall that H. Tahara and H. Yamazawa use in [START_REF] Tahara | Multisummability of formal solutions to the cauchy problem for some linear partial differential equations[END_REF] a slightly different one.

For any pa, bq P R 2 , we denote by Cpa, bq the domain Cpa, bq " tpx, yq P R 2 ; x ď a and y ě bu.

Then, the t-Newton polygon of D is defined as follows.

Definition 3.1. One calls t-Newton polygon of D the convex hull N t pDq of the union of the sets Cp0, 0q and Cpq ´p ´i, v i,q `iq for i P K and q P Q i :

N t pDq " CH » - -Cp0, 0q Y ď iPK qPQi Cpq ´p ´i, v i,q `iq fi ffi fl ,
where CHr¨s denotes the convex hull of the elements in r¨s.

The following lemma specifies the geometric structure of N t pDq.

Lemma 3.2. Let S :" tpi, qq ; i P K, q P Q i and q ´p ´i ą 0u be.

(1) Suppose S " H. Then, N t pDq " Cp0, 0q. In particular, N t pDq has no side with a positive slope. (2) Suppose S ‰ H. Then, N t pDq has (at least) one side with a positive slope.

Moreover, its smallest positive slope k is given by k " min pi,qqPS ˆvi,q `i q ´p ´i ˙.

Proof. Point 1 is straightforward from the fact that condition S " H implies Cpq ṕ ´i, v i,q `iq Ă Cp0, 0q for all i and q. As for point 2, it suffices to remark, on one hand, that Cpq ´p ´i, v i,q `iq Ă Cp0, 0q for all pi, qq R S and, on the other hand, that the segment with two end points p0, 0q and pq ´p ´i, v i,q `iq has, for all pi, qq P S, a positive slope equal to pv i,q `iq{pq ´p ´iq (the positivity stems from the fact that q ´p ă 0 for all q P Q 0 ; hence, pi, qq P S implies i ě 1 and then v i,q `i ą 0). Notation 3.3. When S ‰ H, we choose, and fix once and for all, one of the pairs pi, qq P S such that the side of slope k of N t pDq is the segment with end points p0, 0q and pq ´p ´i, v i,q `iq (see Figure 1 below). In the sequel, we denote this pair by pi ˚, q ˚q. s l o p e k q ˚´p ´iv i ˚,q ˚`i Figure 1. Definition of the pair pi ˚, q ˚q. Remark 3.4. Of course, we have k " v i ˚,q ˚`i q˚´p ´i˚. Moreover, according to the proof of Lemma 3.2, we have besides q ˚´p ą i ˚ě 1.

Let us now turn to the Gevrey properties of r upt, xq.

Gevrey order

The aim of this section is to investigate the Gevrey properties of the unique formal series r upt, xq of equation (1.2) (see Corollary 2.3). In particular, we propose to give necessary and sufficient conditions under which it is s-Gevrey for some s ě 0.

Before stating our main result (see Theorem 4.4 below), let us first recall for the convenience of the reader some definitions and properties about the s-Gevrey formal series.

4.1. s-Gevrey formal series. All along the article, we consider t as the variable and x as a parameter. Thereby, to define the notion of Gevrey classes of formal power series in OpD ρ2 qrrtss, one extends the classical notion of Gevrey classes of elements in Crrtss to families parametrized by x in requiring similar conditions, the estimates being however uniform with respect to x. Doing that, any formal power series of OpD ρ2 qrrtss can be seen as a formal power series in t with coefficients in a convenient Banach space defined as the space of functions that are holomorphic on a disc D ρ (0 ă ρ ď ρ 2 ) and continuous up to its boundary, equipped with the usual supremum norm. For a general study of series with coefficients in a Banach space, we refer for instance to [START_REF] Balser | Formal power series and linear systems of meromorphic ordinary differential equations[END_REF]. hold for all j ě 0.

In other words, Definition 4.1 means that r upt, xq is s-Gevrey in t, uniformly in x on a neighborhood of x " 0.

We denote by OpD ρ2 qrrtss s the set of all the formal series in OpD ρ2 qrrtss which are s-Gevrey. Observe that the set Ctt, xu of germs of analytic functions at the origin p0, 0q P C 2 coincides with the union Ť ρą0 OpD ρ qrrtss 0 ; in particular, any element of OpD ρ2 qrrtss 0 is convergent and Ctt, xu X OpD ρ2 qrrtss " OpD ρ2 qrrtss 0 . Observe also that the sets OpD ρ2 qrrtss s are filtered as follows:

OpD ρ2 qrrtss 0 Ă OpD ρ2 qrrtss s Ă OpD ρ2 qrrtss s 1 Ă OpD ρ2 qrrtss
for all s and s 1 satisfying 0 ă s ă s 1 ă `8. Lemma 4.3. DpOpD ρ2 qrrtss s q Ă OpD ρ2 qrrtss s for all s ě 0. Theorem 4.4 below specifies this statement by showing more especially that the operator D is actually a linear automorphism of OpD ρ2 qrrtss s for some s ě 0. Theorem 4.4. Let S :" tpi, qq ; i P K, q P Q i and q ´p ´i ą 0u and s be the rational number defined by s :"

Following

$ & % 0 if S " H 1 k " q ˚´p ´iv i ˚,q ˚`i ˚if S ‰ H
Then, D is a linear automorphism of OpD ρ2 qrrtss s .

In particular, Theorem 4.4 gives us the Gevrey properties of r upt, xq in view in this section. More precisely, it provides, in the case S " H, necessary and sufficient condition under which r upt, xq is convergent and, in the opposite case S ‰ H, necessary and sufficient condition under which r upt, xq is s-Gevrey with s as above.

Corollary 4.5. Let S :" tpi, qq ; i P K, q P Q i and q ´p ´i ą 0u be.

(1) Assume S " H. Then, r upt, xq is convergent if and only if the inhomogeneity r f pt, xq is convergent.

(2) Assume S ‰ H and set s " q ˚´p ´iv i ˚,q ˚`i ˚. Then, r upt, xq is s-Gevrey if and only if the inhomogeneity r f pt, xq is s-Gevrey.

As a consequence of Corollary 4.5, we deduce in particular a result similar to the Maillet-Ramis theorem for the ordinary linear differential equations [START_REF] Ramis | Dévissage Gevrey[END_REF][START_REF] Ramis | Théorèmes d'indices Gevrey pour les équations différentielles ordinaires[END_REF] (see also [START_REF] Loday-Richaud | Divergent Series, Summability and Resurgence II. Simple and Multiple Summability[END_REF]Thm. 4.2.7]).

Corollary 4.6. Assume that the inhomogeneity r f pt, xq is convergent. Then, r upt, xq is either convergent or s-Gevrey, where k " 1{s is the smallest positive slope of the Newton polygon N t pDq of D with respect to t. 4.3. Proof of Theorem 4.4. According to Theorem 2.1 and Lemma 4.3, the operator D is an injective linear operator acting inside OpD ρ2 qrrtss s . To prove the surjectivity of D, we shall use below an approach based on Nagumo norms [START_REF] Canalis-Durand | Gevrey solutions of singularly perturbed differential equations[END_REF][START_REF] Nagumo | Über das Anfangswertproblem partieller Differentialgleichungen[END_REF] and majorant series; approach which is similar to the ones developed by W. Balser and M. Loday-Richaud in [START_REF] Balser | Summability of solutions of the heat equation with inhomogeneous thermal conductivity in two variables[END_REF] and by the author in [START_REF] Remy | Gevrey order and summability of formal series solutions of some classes of inhomogeneous linear partial differential equations with variable coefficients[END_REF][START_REF] Remy | Gevrey order and summability of formal series solutions of certain classes of inhomogeneous linear integro-differential equations with variable coefficients[END_REF] for some classes of linear integro-differential equations. 4.3.1. Nagumo norms. For the convenience of the reader, we recall in this section the definition of the Nagumo norms and some of their properties which are needed in the sequel. Proposition 4.8 (Properties of Nagumo norms). Let f, g P OpD ρ q, n, n 1 ě 0 and 0 ă r ă ρ be. Then,

(1) }¨} n,r is a norm on OpD ρ q.

(2) For all x P D r , |f pxq| ď }f } n,r d r pxq ´n.

(3) }f } 0,r " sup |x|ăr |f pxq| is the usual sup-norm on D r .

(4) }f g} n`n 1 ,r ď }f } n,r }g} n 1 ,r .

(5) }Bf } n`1,r ď epn `1q }f } n,r .

› › B ´1f › › n,r ď r }f } n,r . Proof. (6)
Properties 1-4 are straightforward and are left to the reader.

To prove Property 5, we proceed as follows. Let x P D r and 0 ă R ă d r pxq be. Using the Cauchy integral formula, we have

|Bf pxq| " 1 2π ˇˇˇˇż |x 1 ´x|"R f px 1 q px 1 ´xq 2 dx 1 ˇˇˇˇď 1 R max |x 1 ´x|"R |f px 1 q|
and then Hence, the result:

|Bf pxq| ď }f } n,r 1 R max |x 1 ´x|"R d r px 1 q ´n " }f } n,r 1
› › B ´1f › › 1,r " sup |x|ăr ˇˇB ´1f pxqd r pxq ˇˇď r }f } 1,r . ' Case n ě 2. Since ż |x| 0 du pr ´uq n " 1 pn ´1qd r pxq n´1 ´1 rpn ´1q ď 1 d r pxq n´1 , inequality (4.1) implies ˇˇB ´1f pxqd r pxq n ˇˇď }f } n,r d r pxq ď r }f } n,r .
Hence, the result again:

› › B ´1f › › n,r " sup |x|ăr ˇˇB ´1f pxqd r pxq n ˇˇď r }f } n,r .
This achieves the proof of Proposition 4.8.

Remark 4.9. Inequalities 4-6 are the most important properties. Observe that the same index r occurs on their both sides, allowing thus to get estimates for the product f g in terms of f and g, for the derivative Bf in terms of f and for the anti-derivative B ´1f in terms of f without having to shrink the disc D r .

Let us now turn to the proof of Theorem 4.4.

Proof of Theorem 4.4.

Ÿ First step: a fundamental technical lemma. Before starting the calculations, let us first begin with the following technical lemma which will play a central role in our proof. Lemma 4.10. Assume S ‰ H. Then, the inequalities

(4.2)
ps `1qpv i,q `iq ě q ´p `vi,q hold for all i P K and q P Q i .

Proof. Inequalities (4.2) are clear when i " v i,q " 0 (indeed, q ´p ă 0 for all q P Q 0 ). When pi, v i,q q ‰ p0, 0q, we have v i,q `i ě 1 and the inequality

(4.3) s " 1 k ě q ´p ´i v i,q `i
which stems, on one hand, from the definition of k when pi, qq P S and, on the other hand, from the fact that q ´p ´i ď 0 when pi, qq R S. Lemma 4.10 follows then by adding "`1" to both sides of (4.3).

Remark 4.11. In fact, inequalities (4.2) still hold when S " H. Indeed, we have s " 0 and q ´p ď i for all i P K and q P Q i . Nevertheless, we shall only use subsequently these inequalities in the case where S ‰ H. Hence, the statement of Lemma 4.10 as it is written.

We are now able to prove Theorem 4.4.

Ÿ Second step: preliminaries.

As we said at the beginning of Section 4.3, we are left to prove the surjectivity of the linear integro-differential operator D. To do that, let us fix

r f pt, xq " ÿ jě0 f j,˚p xq t j j! P OpD ρ2 qrrtss s
and let us write the solution r upt, xq P OpD ρ2 qrrtss of equation (1.2) in the same form:

r upt, xq " ÿ jě0 u j,˚p xq t j j! .
By assumption, the coefficients f j,˚p xq satisfy the following two conditions ' f j,˚p xq P OpD ρ2 q for all j ě 0, ' there exist three positive constants 0 ă r 2 ă ρ 2 , C ą 0 and K ą 0 such that |f j,˚p xq| ď CK j Γp1 `ps `1qjq for all j ě 0 and |x| ď r 2 .

We shall now prove that the coefficients u j,˚p xq satisfy similar conditions. The calculations below are analogous to those detailed in [START_REF] Balser | Summability of solutions of the heat equation with inhomogeneous thermal conductivity in two variables[END_REF][START_REF] Remy | Gevrey order and summability of formal series solutions of some classes of inhomogeneous linear partial differential equations with variable coefficients[END_REF][START_REF] Remy | Gevrey order and summability of formal series solutions of certain classes of inhomogeneous linear integro-differential equations with variable coefficients[END_REF], but are much more complicated because of the terms B ´i t B q´p x with q ´p P Z.

Ÿ Third step: some inequalities.

From identities (2.1), we obtain the relations

u j,˚p xq Γp1 `ps `1qjq " f j,˚p xq Γp1 `ps `1qjq ÿ iPK ÿ qPQi j´vi,q´i ÿ m"0 j! pj ´vi,q ´mq! a pi,qq m,˚p xq m! B q´p x u j´vi,q´i´m,˚p xq Γp1 `ps `1qjq
for all j ě 0 (as before, we use the classical convention that the third sum is 0 if j ă v i,q `i).

Notation 4.12. In the sequel, we denote by σ the positive integer2 defined by

(4.4) σ :" " v `κ if S " H ps `1qpv i ˚,q ˚`i ˚q if S ‰ H
where v is the nonnegative integer v :" maxtv i,q ; i P K and q P Q i u.

Let us now apply the Nagumo norm of indices pσj, r 2 q. From Property 4 of Proposition 4.8, we first obtain

}u j,˚p xq} σj,r2 Γp1 `ps `1qjq ď }f j,˚p xq} σj,r2 Γp1 `ps `1qjq `ÿ iPK ÿ qPQi j´vi,q´i ÿ m"0 A j,i,q,m pxq with A j,i,q,m pxq :" j! pj ´vi,q ´mq! › › ›a pi,qq m,˚p xq › › › σpvi,q`i`mq´δq,r2 m! › › B q´p x u j´vi,q´i´m,˚p xq › › σpj´vi,q´i´mq`δq,r2
Γp1 `ps `1qjq , where δ q is the nonnegative integer defined by δ q :" " 0 if q ´p ď 0, q ´p if q ´p ą 0.

Then, Properties 5-6 of Proposition 4.8 imply the inequality

}u j,˚p xq} σj,r2 Γp1 `ps `1qjq ď }f j,˚p xq} σj,r2 Γp1 `ps `1qjq `ÿ iPK ÿ qPQi j´vi,q´i ÿ m"0 B j,i,q,m pxq with B j,i,q,m pxq :" β j,i,q,m › › ›a pi,qq m,˚p xq › › › σpvi,q`i`mq´δq,r2 m! › › u j´vi,q´i´m,˚p xq › › σpj´vi,q´i´mq,r2 ,
where β j,i,q,m is the nonnegative integer defined by

β j,i,q,m :" $ ' ' ' ' ' ' ' & ' ' ' ' ' ' ' % j!r p´q 2 pj ´vi,q ´mq!Γp1 `ps `1qjq if q ´p ď 0, j! ˜q´p´1 ź "0
pσpj ´vi,q ´i ´mq `q ´p ´ q ¸eq´p pj ´vi,q ´mq!Γp1 `ps `1qjq if q ´p ą 0.

Remark 4.13. Norms

› › ›a pi,qq m,˚p xq › › › σpvi,q`i`mq,r2 and › › u j´vi,q´i´m,˚p xq › › σpj´vi,q´i´mq,r2
are both clearly well-defined. Norms

› › ›a pi,qq m,˚p xq › › › σpvi,q`i`mq´pq´pq,r2
are well-defined too when q ´p ą 0. Indeed, in the case S " H, conditions κ ě 1, i ě 0, v i,q ě 0 and q ´p ď i imply σpv i,q `i `mq ´pq ´pq ě σi ´pq ´pq ě κi ´pq ´pq ě κi ´i " ipκ ´1q ě 0 and, in the opposite case S ‰ H, Lemma 4.10 and conditions i ˚ě 1 (see Remark 3.4) and v i,q ě 0 imply σpv i,q `i `mq ´pq ´pq ě σpv i,q `iq ´pq ´pq " ps `1qpv i ˚,q ˚`i ˚qpv i,q `iq ´pq ´pq ě pq ´p `vi,q qpv i ˚,q ˚`i ˚q ´pq ´pq " pq ´pqpv i ˚,q ˚`i ˚´1q `vi,q pv i ˚,q ˚`i ˚q ě 0

Following proposition allows us to bound the β j,i,q,m 's.

Proposition 4.14. Let i P K, q P Q i , j ě v i,q `i and m P t0, ..., j ´vi,q ´iu be. Then, j! pj ´vi,q ´mq!Γp1 `ps `1qjq ď 1 Γp1 `ps `1qpj ´vi,q ´i ´mqq .

Moreover, if q ´p ą 0, we have j! ˜q´p´1 ź "0 pσpj ´vi,q ´i ´mq `q ´p ´ q pj ´vi,q ´mq!Γp1 `ps `1qjq ď pv `κq q´p Γp1 `ps `1qpj ´vi,q ´i ´mqq .

Proof. The first inequality stems from Lemma 4.16 (inequality (4.5)) and Lemma 4.17 and the second one from Lemma 4.16 (inequality (4.6)) and Lemma 4.18.

Remark 4.15. Observe that the second inequality of Proposition 4.14 may occur only when i ě 1. Indeed, we have q ă p for all q P Q 0 (see Condition pC 3 q). Lemma 4.16. Let i P K, q P Q i , j ě v i,q `i and m P t0, ..., j ´vi,q ´iu be. Then,

(4.5) j! pj ´vi,q ´mq! 1 Γp1 `ps `1qjq ď 1 Γp1 `ps `1qpj ´vi,q ´mqq .
Moreover, if i ě 1, we also have

(4.6) j! pj ´vi,q ´mq! 1 Γp1 `ps `1qjq ď 1 Γp1 `ps `1qpj ´mq ´vi,q q .
Proof. Lemma 4.16 is clear when v i,q `m " 0. Let us now suppose v i,q `m ě 1.

‹ Proof of inequality (4.5). For pi, mq ‰ p0, j ´v0,q q, let us write the two factors of the left-hand side of inequality (4.5) as follows: j! pj ´vi,q ´mq! "

vi,q`m´1 ź "0 pj ´ q, Γp1 `ps `1qjq " Γp1 `ps `1qj ´vi,q ´mq vi,q`m´1 ź "0 pps `1qj ´ q.

Then,

j! pj ´vi,q ´mq! 1 Γp1 `ps `1qjq " vi,q`m´1 ź "0 j ´ ps `1qj ´ Γp1 `ps `1qj ´vi,q ´mq ď 1 Γp1 `ps `1qj ´vi,q ´mq .
Observe that these relations make sense since the following inequalities ps `1qj ´ ě 1 `ps `1qj ´vi,q ´m ě 1 `sj `i ě 1 hold for all P t0, ..., v i,q `m ´1u. Inequality (4.5) follows then from the increase of the Gamma function on r2, `8r. Indeed, we have the inequalities 1 `ps `1qj ´vi,q ´m ě 1 `ps `1qpj ´vi,q ´mq ě 1 `ps `1qi ě 2 for i ě 1 and the inequalities 1 `ps `1qj ´v0,q ´m ě 1 `ps `1qpj ´v0,q ´mq ě 2 `s ě 2 for i " 0 and m P t0, ..., j ´v0,q ´1u. We are left to prove inequality (4.5) for pi, mq " p0, j ´v0,q q, that is the inequality

j! Γp1 `ps `1qjq " Γp1 `jq Γp1 `ps `1qjq ď 1.
This latter is clear for j " 0 and stems from the inequalities 1 `ps `1qj ě 1 `j ě 2 and from the increase of the Gamma function on r2, `8r for j ě 1.

‹ Proof of inequality (4.6). Let i ě 1 be. From calculations above, we have j! pj ´vi,q ´mq! 1 Γp1 `ps `1qjq ď 1 Γp1 `ps `1qj ´vi,q ´mq .

Then, inequality (4.6) stems as previously from the increase of the Gamma function on r2, `8r applied to the inequalities 1 `ps `1qj ´vi,q ´m ě 1 `ps `1qpj ´mq ´vi,q ě 1 `ps `1qi `sv i,q ě 2.

This ends the proof of Lemma 4.16.

Lemma 4.17. Let i P K, q P Q i , j ě v i,q `i and m P t0, ..., j ´vi,q ´iu be. Then, 1 Γp1 `ps `1qpj ´vi,q ´mqq ď 1 Γp1 `ps `1qpj ´vi,q ´i ´mqq .

Proof. For m ď j ´vi,q ´i ´1, we have 1 `ps `1qpj ´vi,q ´mq ě 1 `ps `1qpj ´vi,q ´i ´mq ě 2 `s ě 2 and Lemma 4.17 follows from the increase of the Gamma function on r2, `8r. For m " j ´vi,q ´i, we must prove the inequality 1 Γp1 `ps `1qiq ď 1. This latter is clear for i " 0 and stems again from the increase of the Gamma function on r2, `8r for i ě 1. Indeed, we have the inequalities 1`ps`1qi ě 2`s ě 2; hence, Γp1 `ps `1qiq ě Γp2q " 1. This achieves the proof.

Lemma 4.18. Let i P K, i ‰ 03 , q P Q i , j ě v i,q `i and m P t0, ..., j ´vi,q ´iu be. Assume q ´p ą 0. Then, q´p´1 ź "0 pσpj ´vi,q ´i ´mq `q ´p ´ q Γp1 `ps `1qpj ´mq ´vi,q q ď pv `κq q´p Γp1 `ps `1qpj ´vi,q ´i ´mqq .

Proof. ‹ Let us first assume S " H (hence, σ " v `κ and s " 0). From the relations 0 ă q ´p ď i ď κ ď v `κ and from the identities q´p´1 ź "0 pσpj ´vi,q ´i ´mq `q ´p ´ q " pv `κq q´p q´p´1 ź "0 ˆj ´vi,q ´i ´m `q ´p ´ v `κ ȧnd Γp1 `ps `1qpj ´mq ´vi,q q " Γp1 `j ´m ´vi,q q " Γp1 `j ´vi,q ´i ´mq i´1 ź "0 pj ´vi,q ´m ´ q we deduce the inequality q´p´1 ź "0 pσpj ´vi,q ´i ´mq `q ´p ´ q Γp1 `ps `1qpj ´mq ´vi,q q ď pv `κq q´p Γp1 `j ´vi,q ´i ´mq ˆq´p´1 ź "0 j ´vi,q ´i ´m `q ´p ´ v `κ j ´vi,q ´m ´ i´1 ź "q´p pj ´vi,q ´m ´ q with the convention that the product i´1 ź "q´p pj ´vi,q ´m ´ q is 1 when q ´p " i.

Observe that j ´vi,q ´m ´ ě 1 for all . Indeed, we have m ď j ´vi,q ´i and ď i ´1. In particular, we obtain i´1 ź "q´p pj ´vi,q ´m ´ q ě 1.

On the other hand, inequalities 0 ď ď q ´p ´1 ď i ´1 and q ´p ď v `κ imply

ˆj ´vi,q ´i ´m `q ´p ´ v `κ ˙´pj ´vi,q ´m ´ q " ´i `q ´p ´ v `κ ` ď ´i `q ´p v `κ `i ´1 ď 0.
Thereby, the following inequality

q´p´1 ź "0 j ´vi,q ´i ´m `q ´p ´ v `κ j ´vi,q ´m ´ ď 1 holds; hence, q´p´1 ź "0
pσpj ´vi,q ´i ´mq `q ´p ´ q Γp1 `ps `1qpj ´mq ´vi,q q ď pv `κq q´p Γp1 `j ´vi,q ´i ´mq " pv `κq q´p Γp1 `ps `1qpj ´vi,q ´i ´mqq , which proves Lemma 4.18 for S " H.

‹ Let us now assume S ‰ H. Thanks to the relation s`1 " σ v i ˚,q ˚`i ˚ě σ v `κ , we have the following inequality:

(4.7) q´p´1 ź "0 pσpj ´vi,q ´i ´mq `q ´p ´ q ď pv `κq q´p q´p´1 ź "0 ˆps `1qpj ´vi,q ´i ´mq `q ´p ´ v `κ ˙.
Let us now write Γp1 `ps `1qpj ´mq ´vi,q q in the form (4.8) Γp1 `ps `1qpj ´mq ´vi,q q " Γp1 `ps `1qpj ´mq ´vi,q ´pq ´pqq ˆq´p´1 ź "0 pps `1qpj ´mq ´vi,q ´ q.

Observe that the term Γp1 `ps `1qpj ´mq ´vi,q ´pq ´pqq is already well-defined. Indeed, condition m ď j ´vi,q ´i and Lemma 4.10 imply 1 `ps `1qpj ´mq ´vi,q ´pq ´pq ě 1 `ps `1qpv i,q `iq ´pq ´p `vi,q q ě 1.

From relations (4.7) and (4.8), we obtain q´p´1 ź "0 pσpj ´vi,q ´i ´mq `q ´p ´ q Γp1 `ps `1qpj ´mq ´vi,q q ď pv `κq q´p Γp1 `ps `1qpj ´mq ´vi,q ´pq ´pqq ˆq´p´1 ź "0 ps `1qpj ´vi,q ´i ´mq `q ´p ´ v `κ ps `1qpj ´mq ´vi,q ´ where the product on the right-hand side is ď 1. Indeed, Lemma 4.10 and the conditions ă q ´p and v `κ ě 1 imply relations ˆps `1qpj ´vi,q ´i ´mq `q ´p ´ v `κ ˙´pps `1qpj ´mq ´vi,q ´ q " ´ps `1qpv i,q `iq `q ´p ´ v `κ `vi,q ` ď ´pq ´p `vi,q q `q ´p ´ v `κ `vi,q `

" pq ´p ´ q ˆ1 v `κ ´1ď 0
Let us now assume m ă j ´vi,q ´i. Then, Lemma 4.18 follows from inequalities 1 `ps `1qpj ´mq ´vi,q ´pq ´pq " 1 `ps `1qpj ´mq ´pq ´p `vi,q q ě 1 `ps `1qpj ´mq ´ps `1qpv i,q `iq " 1 `ps `1qpj ´vi,q ´i ´mq ě 2 `s ě 2

and from the increase of the Gamma function on r2, `8r. Observe that the first inequality stems from Lemma 4.10 and that the second inequality stems from the condition m ă j ´vi,q ´i. In particular, this latter inequality shows that the calculations above do not allow to prove Lemma 4.18 when m " j ´vi,q ´i, since it fails in this case.

To get around this problem, we shall proceed as follows. Let us first recall we must prove the inequality q´p´1 ź "0 pq ´p ´ q Γp1 `ps `1qpv i,q `iq ´vi,q q ď pv `κq q´p Γp1q " pv `κq q´p .

From Lemma 4.10 and the condition q ´p ą 0, we obtain 1 `ps `1qpv i,q `iq ´vi,q ě 1 `q ´p ě 2;

hence, applying the increase of the Gamma function on r2, `8r, the relation Γp1 `ps `1qpv i,q `iq ´vi,q q ě Γp1 `q ´pq " q´p´1 ź "0 pq ´p ´ q and, consequently, the following inequality q´p´1 ź "0 pq ´p ´ q Γp1 `ps `1qpv i,q `iq ´vi,q q ď 1.

This achieves the proof since v `κ ě 1.

Let us now apply Proposition 4.14. We get

β j,i,q,m :" $ ' ' ' ' & ' ' ' ' % r p´q 2
Γp1 `ps `1qpj ´vi,q ´i ´mqq if q ´p ď 0, pepv `κqq q´p Γp1 `ps `1qpj ´vi,q ´i ´mqq if q ´p ą 0.

Then, the following inequalities (4.9)

}u j,˚p xq} σj,r2 Γp1 `ps `1qjq ď g j ÿ iPK ÿ qPQi j´vi,q´i ÿ m"0 γ i,q,m › › u j´vi,q´i´m,˚p xq › › σpj´vi,q´i´mq,r2
Γp1 `ps `1qpj ´vi,q ´i ´mqq hold for all j ě 0 with g j :" }f j,˚p xq} σj,r2 Γp1 `ps `1qjq and γ i,q,m :"

$ ' ' ' ' ' ' & ' ' ' ' ' ' % r p´q 2 › › ›a pi,qq m,˚p xq › › › σpvi,q`i`mq,r2 m! if q ´p ď 0, pepv `κqq q´p › › ›a pi,qq m,˚p xq › › › σpvi,q`i`mq´pq´pq,r2
m! if q ´p ą 0.

We now shall bound the Nagumo norms }u j,˚p xq} σj,r2 . To do that, we shall proceed as in [START_REF] Balser | Summability of solutions of the heat equation with inhomogeneous thermal conductivity in two variables[END_REF][START_REF] Remy | Gevrey order and summability of formal series solutions of some classes of inhomogeneous linear partial differential equations with variable coefficients[END_REF][START_REF] Remy | Gevrey order and summability of formal series solutions of certain classes of inhomogeneous linear integro-differential equations with variable coefficients[END_REF] Γp1 `ps `1qjq may occur in the right-hand side of inequalities (4.9). More precisely such terms exist only for the q P Q 0 such that v 0,q " 0 and are obtained when pi, v i,q , mq " p0, 0, 0q. Consequently, we suppose in the sequel that the positive number r 2 Ps0, ρ 2 r has been chosen so that ÿ qPQ0 v0,q"0 γ 0,q,0 "

ÿ qPQ0 v0,q"0 r p´q 2 › › ›a p0,qq 0,˚p xq › › › 0,r2 ă 1.
Observe that such a choice is already possible since p ´q ą 0 for all q P Q 0 (see Condition pC 3 q). Ÿ Fourth step: majorant series.

Let us consider the nonnegative numerical sequence pw j q defined for all j ě 0 by the recurrence relations

w j " g j `ÿ iPK ÿ qPQi j´vi,q´i ÿ m"0 γ i,q,m w j´vi,q´i´m
where, as previously, the third sum is 0 when j ă v i,q `i. Observe that the fact that w j ě 0 for all j stems from the choice of r 2 (see Remark 4.19). Observe also we have

0 ď }u j,˚p xq} σj,r2
Γp1 `ps `1qjq ď w j for all j ě 0 by construction (proceed by induction on j). Let us now bound the w j 's. To this end, we proceed as follows.

By assumption on the f j,˚' s (see the beginning of section 4.3.2), we have 0 ď g j ď CK j Γp1 `ps `1qjq Γp1 `ps `1qjq r σj 2 " CpKr σ 2 q j for all j ě 0 and the series gpXq :" ÿ jě0 g j X j is thereby convergent.

On the other hand, all the terms a pi,qq pt, xq belong to OpD ρ2 qttu. Then, there exist two positive constants C 1 , K 1 ą 0 such that |a pi,qq m,˚p xq| ď C 1 K 1m m! for all i P t0, ..., κu, q P Q i , m ě 0 and x P D r2 . Hence,

0 ď γ i,q,m ď $ & % C 1 1 pK 1 r σ 2 q m if q ´p ď 0, C 1 2 pK 1 r σ 2 q m if q ´p ą 0, with C 1 1 " C 1 r σpvi,q`iq´pq´pq 2 " C 1
2
pepv `κqq q´p and, thereby, the series A i,q pXq :" ÿ mě0 γ i,q,m X m are convergent for all i P t0, ..., κu and q P Q i .

Consequently, since the series wpXq :" ÿ jě0 w j X j satisfies the identity

˜1

´ÿ iPK ÿ qPQi X vi,q`i A i,q pXq ¸wpXq " gpXq, it is convergent too. Indeed, since the constant term 1 ´ÿ qPQ0 v0,q"0 A 0,q p0q " 1 ´ÿ qPQ0 v0,q"0 γ 0,q,0 is not null by construction (see Remark 4.19), the series 1 ´ÿ iPK ÿ qPQi X vi,q`i A i,q pXq is invertible in CtXu. Therefore, there exist two positive constants C 2 , K 2 ą 0 such that w j ď C 2 K 2j for all j ě 0. Hence, the following inequalities

}u j,˚p xq} σj,r2 ď C 2 K 2j Γp1 `ps `1qjq
hold for all j ě 0.

Ÿ Fifth step: conclusion. We are left to prove similar estimates on the sup-norm of the u j,˚p xq's. To this end, we proceed by shrinking the domain D r2 . Let 0 ă r 1 2 ă r 2 be. Then, for all j ě 0 and |x| ď r pr 2 ´r1 2 q σj and, consequently, sup

|x|ďr 1 2 |u j,˚p xq| ď C 2 ˆK2 pr 2 ´r1
2 q σ ˙j Γp1 `ps `1qjq. This achieves the proof of Theorem 4.4.

Summability

In previous Section 4, we have shown that the formal series solution r upt, xq and the inhomogeneity r f pt, xq of equation (1.2) are together s-Gevrey for a convenient s ě 0 (see Theorem 4.4). In particular, when S " H, that is when the Newton polygon N t pDq of the operator D has no side of positive slope, this has allowed us to display a necessary and sufficient condition under which r upt, xq is convergent (see Corollary 4.5).

In the present section, we are interested in the opposite case S ‰ H, that is in the case where N t pDq has at least one side of positive slope. As previously, we denote by k its smallest positive slope and we set s " 1{k. For all i P K, we also denote by p i the maximum of the q P Q i . Moreover, we assume from now on that equation (1.2) satisfies the four following additional conditions: pA 1 q: p " 0; hence, K is a non-empty subset of t1, ..., κu, pA 2 q: v i,pi " 0 for all i P K, pA 3 q: p i ˚ą p i for all i ‰ i ˚, pA 4 q: a pi ˚,p i ˚qp0, 0q ‰ 0. Observe that Assumptions pA 1 q ´pA 2 q imply q ˚" p i ˚and, consequently, (5.1) k " i pi ˚´i ˚and s "

p i i˚´1 .
Indeed, the domains Cpq ´i, v i,q `iq are included in Cpp i ´i, iq for all i P K and q P Q i (see Definition 3.1 for the definition of N t pDq and page 4 for the definition of the domain Cpa, bq).

Observe also that Assumption pA 3 q tells us that k is the unique positive slope of the Newton polygon N t pDq.

The aim of this section is to answer to the following question:

"Under Assumptions pA 1 q ´pA 4 q, how to characterize the k-summability of r upt, xq?" A response to this question has already been done by the author in [START_REF] Remy | Gevrey order and summability of formal series solutions of certain classes of inhomogeneous linear integro-differential equations with variable coefficients[END_REF] when i ˚" κ the maximum of the i P K. In the present paper, we consider a much more general situation, where the smallest slope k ą 0 of N t pDq is given by some i ˚ď κ and, in particular, i ˚ă κ. As we shall see in the sequel, our approach is similar to the one developed in [START_REF] Remy | Gevrey order and summability of formal series solutions of certain classes of inhomogeneous linear integro-differential equations with variable coefficients[END_REF], but the calculations are much more complicated because of i ˚is not necessarily the maximum of the i P K.

Before stating our main result (see Theorem 5.4), let us first begin with some recalls about the k-summability of formal series in OpD ρ2 qrrtss.

5.1. k-summability. Still considering t as the variable and x as a parameter, one extends, in the similar way as the s-Gevrey formal series (see Definition 4.4), the classical notion of k-summability of formal series in Crrtss to the notion of k-summability of formal series in OpD ρ2 qrrtss in requiring similar conditions, the estimates being however uniform with respect to x. Among the many equivalent definitions of the k-summability in a given direction argptq " θ at t " 0, we choose here a generalization of Ramis' definition which states that a formal series r gptq P Crrtss is k-summable in the direction θ if there exists a holomorphic function g which is s-Gevrey asymptotic to r

g in an open sector Σ θ,ąπs bisected by θ and with opening larger than πs [40, Def. 3.1]. To express the s-Gevrey asymptotic, there also exist various equivalent ways. We choose here the one which sets conditions on the successive derivatives of g (see [25, p. 171] or [40, Thm. 2.4] for instance) 4 .

Definition 5.1 (k-summability). A formal series r

upt, xq P OpD ρ2 qrrtss is said to be k-summable in the direction argptq " θ if there exist a sector Σ θ,ąπs , a radius 0 ă r 2 ă ρ 2 and a function upt, xq called k-sum of r upt, xq in the direction θ such that (1) u is defined and holomorphic on Σ θ,ąπs ˆDr2`ε for some ε ą 0;

(2) for any |x| ď r 2 , the map t Þ Ñ upt, xq has r upt, xq "

ÿ jě0 u j,˚p xq t j j!
as Taylor series at 0 on Σ θ,ąπs ; (3) for any proper 5 subsector Σ Ť Σ θ,ąπs , there exist two positive constants C ą 0 and K ą 0 such that, for all ě 0 and all t P Σ, sup |x|ďr2 ˇˇB t upt, xq ˇˇď CK Γp1 `ps `1q q.

We denote by OpD ρ2 qttu k;θ the subset of OpD ρ2 qrrtss made of all the k-summable formal series in the direction argptq " θ. Obviously, OpD ρ2 qttu k;θ is included in OpD ρ2 qrrtss s .

Observe that, for any fixed x, the k-summability of r upt, xq coincides with the classical k-summability. Consequently, Watson's lemma implies the unicity of its k-sum, if any exists.

Observe also that the k-sum of a k-summable formal series r upt, xq P OpD ρ2 qttu k;θ may be analytic with respect to x on a disc smaller than the common disc D ρ2 of analyticity of the coefficients u j,˚p xq of r upt, xq.

Proposition 5.2 ([43, Prop. 2]). pOpD ρ2 qttu k;θ , B t , B x q is a C-differential algebra stable under anti-derivations B ´1 t and B ´1

x . With respect to t, the k-sum upt, xq of a k-summable series r upt, xq P OpD ρ2 qttu k;θ is analytic on an open sector for which there is no control on the angular opening except that it must be larger than πs (hence, it contains a closed sector Σ θ,πs bisected by θ and with opening πs) and no control on the radius except that it 4 In Appendix A page 31, we present various results of the general theory of the Gevrey asymptotic expansions in the framework of the formal power series in OpDρ 2 qrrtss. 5 A subsector Σ of a sector Σ 1 is said to be a proper subsector and one denotes Σ Ť Σ 1 if its

closure in C is contained in Σ 1 Y t0u.
must be positive. Thereby, the k-sum upt, xq is well-defined as a section of the sheaf of analytic functions in pt, xq on a germ of closed sector of opening πs (that is, a closed interval I θ,πs of length πs on the circle S 1 of directions issuing from 0; see [26, 1.1]

x n n!
with a pi,qq ˚,n ptq P OpD ρ1 q for all i P K, q P Q i and n ě 0. Then, an identification of the powers in x in the equation

D ˜ÿ ně0 r u ˚,n ptq x n n! ¸" ÿ ně0 r f ˚,n ptq x n n!
provides for all n ě 0 the recurrence relations where the Q i 's are defined by

Q i ˚" Q i ˚ztp i ˚u and Q i " Q i if i ‰ i ˚.
In particular, these relations tell us that each r u ˚, ptq (hence, r upt, xq too) is uniquely determined from r f pt, xq and from the r u ˚,n ptq with n " 0, ..., p i ˚´1. Indeed, Assumption pA 3 q implies q ă p i ˚for all i P K and q P Q i , and Assumption pA 4 q implies that the quotient 1{a pi ˚,p i ˚q ˚,0 ptq is well-defined in Crrtss.

Main result.

We are now able to state the main result in view in this section.

Theorem 5.4. Let a direction argptq " θ issuing from 0 be given. Then,

(1) The formal series r upt, xq P OpD ρ2 qrrtss is k-summable in the direction θ if and only if the inhomogeneity r f pt, xq and the p i ˚coefficients r u ˚,n ptq P Crrtss with n P t0, ..., p i ˚´1u are k-summable in the direction θ.

(2) Moreover, the k-sum upt, xq in the direction θ, if any exists, satisfies equation (1.2) in which r f pt, xq is replaced by its k-sum f pt, xq in the direction θ.

Remark 5.5. The necessary condition of Point 1 is straigthforward from Proposition 5.2. Indeed, we have r u ˚,n ptq " B n x r upt, xq |x"0 and r f " Dr u. Moreover, Point 2 stems obvious from Corollary 5.3. Thereby, we are left to prove the sufficient condition of Point 1.

Remark 5.6. Theorem 5.4 generalizes the results of summability already proved by W. Balser and M. Loday-Richaud in [START_REF] Balser | Summability of solutions of the heat equation with inhomogeneous thermal conductivity in two variables[END_REF] and by the author in [START_REF] Remy | Gevrey order and summability of formal series solutions of some classes of inhomogeneous linear partial differential equations with variable coefficients[END_REF][START_REF] Remy | Gevrey order and summability of formal series solutions of certain classes of inhomogeneous linear integro-differential equations with variable coefficients[END_REF]. 5.3. Proof of Theorem 5.4. As we said in Remark 5.5 just above, it remains to prove the sufficient condition of Point 1. Consequently, we fix from now on a direction θ and we suppose that the inhomogeneity r f pt, xq and the coefficients r u ˚,n ptq for n P t0, ..., p i ˚´1 u are all k-summable in this direction.

Ÿ First step: the associated equation.

Let us first begin by introducing the functions b pi,qq pt, xq defined, for all i P K and q P Q i , by

b pi,qq pt, xq " $ ' ' & ' ' % 1 a pi ˚,p i ˚q pt, xq if pi, qq " pi ˚, p i ˚q,
t vi,q a pi,qq pt, xq a pi ˚,p i ˚q pt, xq if pi, qq ‰ pi ˚, p i ˚q.

Thanks to Assumption pA 4 q, all these functions are holomorphic on a common domain D ρ 1 1 ˆDρ 1 2 of p0, 0q P C 2 for two suitable radiuses ρ 1 1 , ρ

Q i ˚" Q i ˚ztp i ˚u and Q i " Q i if i ‰ i ˚.
According to our assumption (see the beginning of Section 5.3) and Proposition 5.2, the inhomogeneity r gpt, xq of equation (5.2) is k-summable in the direction θ. Thereby, to prove our result, it suffices to prove that the formal power series r wpt, xq P OpD ρ2 qrrtss is also k-summable in the direction θ. To do that, we shall proceed similarly as in [START_REF] Balser | Summability of solutions of the heat equation with inhomogeneous thermal conductivity in two variables[END_REF][START_REF] Remy | Gevrey order and summability of formal series solutions of some classes of inhomogeneous linear partial differential equations with variable coefficients[END_REF][START_REF] Remy | Gevrey order and summability of formal series solutions of certain classes of inhomogeneous linear integro-differential equations with variable coefficients[END_REF] by using a standard fixed point procedure.

$ & % r w 0 " r g, r w m`1 " b pi ˚,p i ˚qpt, xqB ´pi x B i t r w m ´ÿ iPK ÿ qPQi b pi,qq pt, xqB q´p i x B i ˚´i t r w m .
Observe that, for all m ě 0, the formal series r w m pt, xq are of order Opx m q in x and, consequently, the series r wpt, xq itself makes sense as a formal series in t and x. Indeed, the definition of the Q i 's and Assumption pA 3 q imply q ´pi ˚ă 0 for all i P K and q P Q i .

Let us now denote by w 0 pt, xq the k-sum of r w 0 " r g in the direction θ and, for all m ě 0, let w m pt, xq be determined as the solution of system (5.3) in which all the r w m are replaced by w m . By construction, all the w m pt, xq are defined and holomorphic on a common domain Σ θ,ąπs ˆDρ 2 2 , where the radius ρ 2 1 of Σ θ,ąπs and the radius ρ 2 2 of D ρ 2 2 can always be chosen so that 0 ă ρ 2 1 ă minp1, ρ 1 1 q and 0 ă ρ 2 2 ă minp1, ρ 2 , ρ 1 2 q. To end the proof, it remains to prove that the series ÿ mě0 w m pt, xq is convergent and that its sum wpt, xq is the k-sum of r wpt, xq in the direction θ.

Ÿ Third step: some estimations on w m pt, xq.

According to Definition 5.1, the k-summability of r w 0 " r g implies that there exists a radius 0 ă r 2 ă ρ 2 2 with the following property: for any proper subsector Σ Ť Σ θ,ąπs , there exist two positive constants C ą 0 and K ą 0 such that, for all ě 0 and all pt, xq P Σ ˆDr2 , the function w 0 satisfies the conditions (5.4) ˇˇB t w 0 pt, xq ˇˇď CK Γp1 `ps `1q q.

Let us now fix a proper subsector Σ Ť Σ θ,ąπs . Let r 1 denote the radius of Σ and let us choose for the constant K of the previous property a constant ě max ˆ1,

1 ρ 2 1 ´r1
˙. Observe that such a choice is already possible since conditions (5.4) still hold for any constant K 1 ě K. Observe also that the quotient 1{pρ 2 1 ´r1 q makes sense since the definition of a proper subsector (see Footnote 5) implies 0 ă r 1 ă ρ 2 1 .

Proposition 5.7. Let us denote by

' I :" Y κ i ˚],
where t¨u denotes the lower integer part of ¨,

' B :" max iPK qPQi ˜max pt,xqPD ρ 2 1 ˆDρ 2 2
ˇˇb pi,qq pt, xq ˇˇ¸the maximum of the functions ˇˇb pi,qq pt, xq ˇǒn

D ρ 2 1 ˆDρ 2 2
, where D ρ denotes the closed disc with center 0 and radius ρ ą 0, ' B 1 :" pκ `1qpIi ˚`1qB.

Let pP m pxqq be the sequence of polynomials in R `rxs recursively determined by

$ ' ' & ' ' % P 0 pxq " 1, P m`1 pxq " ¨B´p i x `ÿ iPK 1 ÿ qPQ 1 i pmp i ˚q! pmp i ˚`p i q! B ´q
x 'P m pxq for m ě 0, with K 1 :" ti P K ; p i ě 1u and Q 1 i :" tmaxpp i ˚´p i , 1q, ..., p i ˚´1u. Then, the following inequalities

(5.5) ˇˇB t w m pt, xq ˇˇď CB 1m K i ˚m` Γp1 `ps `1qpi ˚m ` qqP m p|x|q
hold for all m, ě 0 and all pt, xq P Σ ˆDr2 .

Remark 5.8. Since 1 ď i ˚ď κ, we have 1 ď I ď κ and Ii ˚ě 1. More precisely, and thanks to the definition of the lower interger part, we have Ii ˚ą κ ´i˚.

Remark 5.9. The constant B is well-defined since the functions b pi,qq pt, xq are all holomorphic on D ρ 1 1 ˆDρ 1 2 and the radiuses ρ 2 j satisfy 0 ă ρ 2 j ă ρ 1 j for j " 1, 2.

Remark 5.10. The set K 1 already contains i ˚and, therefore, is never empty. We have indeed the inequalities p i ˚ą i ˚ě 1.

Proof. The proof proceeds by recursion on m ě 0. The case m " 0 is straightforward from inequality (5.4). Let us now suppose that inequalities (5.5) hold for a certain m ě 0.

From identities (5.3) and the Leibniz formula, we first derive the inequalities

ˇˇB t w m`1 pt, xq ˇˇď ÿ j"0 ˆ j ˙ˇˇB ´j t b pi ˚,p i ˚q pt, xq ˇˇˇˇB ´pi x B i ˚`j t w m pt, xq ˇÿ iPK ÿ qPQi ÿ j"0 ˆ j ˙ˇˇB ´j t b pi,qq pt, xq ˇˇˇˇB q´p i x B i ˚´i`j t w m pt, xq ˇf
or all ě 0 and pt, xq P Σ ˆDr2 . On the other hand, for all i P K, q P Q i , k ě 0 and pt, xq P Σ ˆDr2 , the Cauchy integral formula allows us to write the derivative B k t b pi,qq pt, xq on the form

B k t b pi,qq pt, xq " k! p2iπq 2 ż |t 1 ´t|"ρ 2 1 ´r1 |x 1 ´x|"ρ 2 2 ´r2
b pi,qq pt 1 , x 1 q pt 1 ´tq k`1 px 1 ´xq dt 1 dx 1

(we have indeed 0 ă r 1 ă ρ 2 1 and 0 ă r 2 ă ρ 2 2 ), which yields the estimates

ˇˇB k t b pi,qq pt, xq ˇˇď k!B ˆ1 ρ 2 1 ´r1 ˙k ď k!BK k .
Hence, according to the fact that 0 ă r 1 ă 1 and K ě 1, the following inequalities: for all ě 0 and pt, xq P Σ ˆDr2 ,

ˇˇB t w m`1 pt, xq ˇˇď CBB 1m K i ˚pm`1q` ÿ iPKYt0u ˜Si, ,m ÿ qPQi pB q´p i x P m qp|x|q ¸,
where we set Q 0 :" tp 0 " 0u and where S i, ,m is the sum defined by: ' Case i P t0, ..., i ˚u:

S i, ,m :"

ÿ j"0 ! j!
Γp1 `ps `1qpi ˚m `i˚´i `jqq.

' Case i P ti ˚`1, ..., κu: S i, ,m :"

J i,i ˚, ÿ j"0 ! j! Γp1 `ps `1qi ˚mq ` ÿ j"J i,i ˚, `1 ! j! Γp1 `ps `1qpi ˚m `i˚´i `jqq,
with J i,i ˚, " minpi ´i˚´1 , q. Of course, the second sum is zero as soon as J i,i ˚, " , that is ď i ´i˚´1 .

Applying then Lemma 5.11 below, we get

ˇˇB t w m`1 pt, xq ˇˇď CpIi ˚`1qBB 1m K i ˚pm`1q` Γp1 `ps `1qpi ˚pm `1q ` qq ˆÿ iPKYt0u ÿ qPQi pmp i ˚q! pmp i ˚`p i q! pB q´p i x P m qp|x|q
and inequalities (5.5) follow by observing that the double-sum of the right-hand side satisfies

ÿ iPKYt0u ÿ qPQi pmp i ˚q! pmp i ˚`p i q! pB q´p i x P m qp|x|q ď pκ `1qpB ´pi x P m qp|x|q `ÿ iPK 1 ÿ qPQ 1 i pmp i ˚q! pmp i ˚`p i q! pB ´q x P m qp|x|q; hence, ÿ iPKYt0u ÿ qPQi pmp i ˚q! pmp i ˚`p i q! pB q´p i x P m qp|x|q ď pκ `1qP m`1 p|x|q.
Indeed, K Ă t1, ..., κu, the coefficients of the polynomial P m are positive and the quotients pmp i ˚q!{pmp i ˚`p i q! are ď 1 for all i P K. This ends the proof of Proposition 5.7.

Lemma 5.11. Let i P K Y t0u, ě 0 and m ě 0 be. Then,

(5.6) S i, ,m ď pIi ˚`1q pmp i ˚q! pmp i ˚`p i q!
Γp1 `ps `1qpi ˚pm `1q ` qq.

This technical lemma will be proved later in Section 5.4. For the moment, let us end the proof of Theorem 5.4.

The following proposition, already proved in [START_REF] Remy | Gevrey order and summability of formal series solutions of certain classes of inhomogeneous linear integro-differential equations with variable coefficients[END_REF], allows to bound the P m p|x|q's.

ˆ p i ˚`mp i mp i ˚ď 2 p i ˚`mp i ˚Γp1 `ps `1q q.
We are now able to complete the proof of Theorem 5.4.

Ÿ Fourth step: conclusion.

Let us choose for Σ a sector containing a proper subsector Σ 1 bisected by the direction θ and opening larger than πs (such a choice is already possible by definition of a proper subsector, see Footnote 5).

Let us also choose r, ε ą 0 so that 0 ă r ă r `ε ă minpr 2 , 2 ´pi ˚{B 2 q and let us set C 1 :" C ÿ mě0 p2 p i ˚B 2 rq m P R `and K 1 :" 2 p i ˚K .

Thanks to inequalities (5.7), the series ÿ mě0 B t w m pt, xq are normally convergent on Σ ˆDr`ε for all ě 0 and satisfy the inequalities ÿ mě0 ˇˇB t w m pt, xq ˇˇď C 1 K 1 Γp1 `ps `1q q for all pt, xq P Σ ˆDr`ε . In particular, the sum wpt, xq of the series ÿ mě0 w m pt, xq is well-defined, holomorphic on Σ ˆDr`ε and satisfies the inequalities ˇˇB t wpt, xq ˇˇď C 1 K 1 Γp1 `ps `1q q for all ě 0 and pt, xq P Σ ˆDr`ε . Hence, Conditions 1 and 3 of Definition 5.1.

To prove the second condition of Definition 5.1, we proceed as follows. The removable singularities theorem implies the existence of lim tÑ0 tPΣ 1 B t wpt, xq for all x P D r and, thereby, the existence of the Taylor series of w at 0 on Σ 1 for all x P D r (see for instance [START_REF] Malgrange | Sommation des séries divergentes[END_REF]Cor. 1.1.3.3]; see also [START_REF] Loday-Richaud | Divergent Series, Summability and Resurgence II. Simple and Multiple Summability[END_REF]Prop. 1.1.11]). On the other hand, Proof. Lemma 5.13 stems from the increase of the Gamma function on r2, `8r and from the fact that Γpcq ď 1 " Γp2q for all c P r1, 2s. Indeed, we have the inequalities 2 ď 1 `a ď b for all a ě 1 and the inequalities 1 ď 1 `a ď 2 ď b for all a P r0, 1s.

Let us also recall that, according to our assumptions pA 1 q and pA 2 q, Lemma 4.10 implies the following inequalities (5.8) ps `1qi ě p i for all i P K.

Observe that these latter still hold when i " 0 since p 0 " 0. As we shall see below, inequalities (5.8) will play a crucial role in our proof.

Let us now prove Lemma 5.11.

Ÿ First case. When i P t0, ..., i ˚u, inequalities (5.6) are a consequence of the three following lemmas.

Lemma 5.14. Let ě 0, j P t0, ..., u and m ě 0 be. Then, ! j! Γp1 `ps `1qpi ˚m `i˚´i `jqq ď Γp1 `ps `1qpi ˚m `i˚´i `jq ` ´jq.

Proof. Lemma 5.14 is clear when j " and stems obvious from the inequality ! j! " Γp1 `ps `1qpi ˚m `i˚´i `jq ` ´jq Γp1 `ps `1qpi ˚m `i˚´i ` qq ď Ii ˚`1.

Lemma 5.16. Let ě 0 and m ě 0 be. Then, Γp1 `ps `1qpi ˚m `i˚´i ` qq ď pmp i ˚q! pmp i ˚`p i q! Γp1 `ps `1qpi ˚pm `1q ` qq.

Proof. Thanks to the relations (5.8), we have the inequalities 1 `ps `1qpi ˚pm `1q ` q loooooooooooooooomoooooooooooooooon ě2 " 1 `ps `1qpi ˚m `i˚´i ` q `ps `1qi ě 1 `ps `1qpi ˚m `i˚´i ` q `pi looooooooooooooooooomooooooooooooooooooon ě0 .

Hence, the inequalities Γp1 `ps `1qpi ˚pm `1q ` qq ě Γp1 `ps `1qpi ˚m `i˚´i ` q `pi q by applying Lemma 5.13. Lemma 5.16 is then proved when p i " 0 and follows from the identity Γp1 `ps `1qpi ˚m `i˚´i ` q `pi q "

Γp1 `ps `1qpi ˚m `i˚´i ` qq pi ź n"1 pps `1qpi ˚m `i˚´i ` q `nq and from the relations

pi ź n"1 pps `1qpi ˚m `i˚´i ` q `nq ě pi ź n"1
pps `1qi ˚m `nq (since i ˚´i ě 0)

" pi ź n"1 pmp i ˚`nq (since ps `1qi ˚" p i ˚)
" pmp i ˚`p i q! pmp i ˚q! when p i ě 1. Ÿ Second case. When i P ti ˚`1, ..., κu6 , Lemma 5.11 is proved in a similar way as the previous case. However, the calculations are much more complicated because of the term J i,i ˚, " minpi ´i˚´1 , q and of the fact that i ˚´i is negative. Lemma 5.17. Let ě 0, j P t0, ..., u and m ě 0 be. Then, (5.11) ! j! Γp1 `ps `1qi ˚mq ď Γp1 `ps `1qpi ˚m `jq ` ´jq.

Moreover, if i ´i˚ď j ď , then (5.12) ! j! Γp1 `ps `1qpi ˚m `i˚´i `jqq ď Γp1 `ps `1qpi ˚m `i˚´i `jq ` ´j `i ´i˚q .

Proof. Lemma 5.17 is proved in a similar way as Lemma 5.14 by respectively using the relation Γp1 `ps `1qi ˚mq ď Γp1 `ps `1qpi ˚m `jqq for inequality (5.11) and the relations

! j! " ´j ź n"1 pj `nq " ´j ź n"1 pi ˚´i `j loooomoooon ě0 `n `i ´il o omo on ą0 q " ´j`i´i ź n"1`i´i ˚pi ˚´i `j `nq ď ´j`i´i ź n"1 pi ˚´i `j `nq ď ´j`i´i ź n"1
pps `1qpi ˚m `i˚´i `jq `nq for inequality (5.12). Observe that the two conditions j ě i ´i˚a nd i ´i˚ą 0 play a key role in these various calculations.

Lemma 5.18. Let ě 0 and m ě 0 be. Then,

J i,i ˚, ÿ j"0 (5.13)
Γp1 `ps `1qpi ˚m `jq ` ´jq Γp1 `ps `1qpi ˚m ` qq ď i ´i˚.

Moreover, if ě i ´i˚( hence, J i,i ˚, " i ´i˚´1 ), then

(5.14) ÿ j"i´i ˚Γp1 `ps `1qpi ˚m `i˚´i `jq ` ´j `i ´i˚q Γp1 `ps `1qpi ˚m ` qq ď Ii ˚`i ˚´i `1.

Proof. ‹ Inequality (5.13) is clear when m " " 0 (we have indeed J i,i ˚, " 0; hence, j " 0 too) and stems from the relations 1 `ps `1qpi ˚m `jq ` ´j looooooooooooooomooooooooooooooon ě0 " 1 `ps `1qi ˚m ` `sj ď 1 `ps `1qpi ˚m ` q loooooooooooomoooooooooooon ě2 and Lemma 5.13 otherwise. Indeed, we have in this case

J i,i ˚, ÿ j"0 Γp1 `ps `1qpi ˚m `jq ` ´jq Γp1 `ps `1qpi ˚m ` qq ď J i,i ˚, ÿ j"0 1 " J i,i ˚, `1 ď i ´i˚.
‹ Let us now prove inequality (5.14) and let us suppose for the moment that P ti ´i˚, ..., Ii ˚u7 . Then, the condition ě i ´i˚ą 0 implies 1 `ps `1qpi ˚m `i˚´i `jq ` ´j `i ´il ooooooooooooooooooooooooooomooooooooooooooooooooooooooon ě0 " 1 `ps `1qi ˚m ` `spi ˚´i `j loooomoooon ďjď q ď 1 `ps `1qpi ˚m ` q loooooooooooomoooooooooooon ě2 7 This set makes sense since, thanks to Remark 9, we have Ii ˚ą κ ´i˚ě i ´i˚.

Proof. Using the relation ps `1qi ˚" p i ˚(see (5.1)) and the fact that p i ˚is the maximum of the p i (see Assumption pA 3 q), we successively have 1 `ps `1qpi ˚pm `1q ` q loooooooooooooooomoooooooooooooooon ě2 " 1 `ps `1qpi ˚m ` q `pi ě 1 `ps `1qpi ˚m ` q `pi loooooooooooomoooooooooooon ě0 and the inequalities Γp1 `ps `1qpi ˚pm `1q ` qq ě Γp1 `ps `1qpi ˚m ` q `pi q by applying Lemma 5.13. Then, we conclude as in Lemma 5.16.

Let r 2 ą 0 be as in Definition A.1 and let us choose Σ 1 Ť Σ a proper subsector of Σ. For any J ě 1, we derive from condition (A. pps `1qJ ` q ď AB J Γp1 `ps `1qJq with convenient constants A, B ą 0 independent of J. Consequently, there exist C 1 , K 1 ą 0 such that the following inequalities

sup |x|ďr2 |u J,˚p xq| ď C 1 K 1J Γp1 `ps `1qJq
hold for all J ě 1. Condition (A.2) follows then by choosing

r 2 2 " r 2 , C 2 " max ˜C1 , sup |x|ďr2 |u 0,˚p xq| ¸and K 2 " K 1 .
This ends the proof.

Following Proposition A.4 gives us a characterization of the s-Gevrey asymptotic in terms of conditions on the successive derivatives B J t u of the function u with respect to t. To prove Ccondition 2, we consider 0 ă r 2 ă minpρ, ρ 2 q as in Definition A.1 and a proper subsector Σ 1 Ť Σ and we choose a radius 0 ă r 1 2 ă r 2 , a sector Σ 2 such that Σ 1 Ť Σ 2 Ť Σ and a positive constant δ ą 0 small enough so that, for all t P Σ 1 , the closed disc centered at t with radius |t| δ be contained in Σ 2 . Then, the Cauchy integral formula implies

B J t upt, xq " J! p2iπq 2 ż
|t 1 ´t|"|t|δ |x 1 ´x|"r2´r 1 2 upt 1 , x 1 q pt 1 ´tq J`1 px 1 ´xq dt 1 dx 1 " J! p2iπq 2 ż |t 1 ´t|"|t|δ |x 1 ´x|"r2´r 1 2 ˜upt 1 , x 1 q ´J´1 ÿ j"0 u j,˚p xq t j j! ¸dt 1 dx 1 pt 1 ´tq J`1 px 1 ´xq for all J ě 0, all t P Σ 1 and all |x| ď r 1 2 . Indeed, the sum is 0 when J " 0 and the J-th derivative of a polynomial of degree J ´1 is 0 too when J ě 1. Hence, This proves the sufficient condition; hence, Proposition A.4.

ˇˇB
In the sequel, we denote by ' A s pΣ, D ρ2 q the set of all the functions which are s-Gevrey asymptotic on Σ to a formal series of OpD ρ2 qrrtss; ' T s;Σ,Dρ 2 : A s pΣ, D ρ2 q ÝÑ OpD ρ2 qrrtss s the map which assigns to each upt, xq P A s pΣ, D ρ2 q its s-Gevrey asymptotic series. Observe that T s;Σ,Dρ 2 is well-defined due to Remark A.2 and Proposition A.4. Following Proposition A.5 specifies the algebraic properties of A s pΣ, D ρ2 q and T s;Σ,Dρ 2 .

Proposition A.5. Let s ě 0 and Σ be an open sector with vertex 0 P C.

(1) pA s pΣ, D ρ2 q, B t , B x qis a C-differential algebra stable under the anti-derivations B ´1 t and B ´1

x .

(2) The map T s;Σ,Dρ 2 : A s pΣ, D ρ2 q ÝÑ OpD ρ2 qrrtss s is a homomorphism of Cdifferential algebras for the derivations B t and B x . Moreover, it commutes with the anti-derivations B ´1 t and B ´1 x . Proof. The proof is the same that the one given in [START_REF] Remy | Gevrey order and summability of formal series solutions of certain classes of inhomogeneous linear integro-differential equations with variable coefficients[END_REF]Prop. 2].

A.2. The s-Gevrey Borel-Ritt Theorem.

Theorem A.6. Supposons that Σ has opening ď πs. Then, the map T s;Σ,Dρ 2 is onto.

Proof. It is sufficient to consider a sector Σ with opening πs. Moreover, by means of a rotation, we can besides assume that Σ is bisected by the direction θ " 0. We denote by R its radius. ‹ Let r upt, xq " ÿ jě0 u j,˚p xq t j j! P OpD ρ2 qrrtss s a s-Gevrey formal series. By assumption, the coefficients u j,˚p xq satisfy the following two conditions:

' u j,˚p xq P OpD ρ2 q for all j ě 0, ' there exist 0 ă r 2 ă ρ 2 , C ą 0 and K ą 0 such that |u j,˚p xq| ď CK j Γp1 ps `1qjq for all j ě 0 and |x| ď r We shall prove below that upt, xq is s-Gevrey asymptotic to r upt, xq on Σ.

‹ Let 0 ă r

t

  vi,q a pi,qq pt, xqB κ´i t B q x

Theorem 2 . 1 .

 21 D is a linear automorphism of OpD ρ2 qrrtss. Proof. Let r f pt, xq P OpD ρ2 qrrtss. Then, a series r upt, xq " ÿ jě0 u j,˚p xq t j j! is solution of Dr u " r f pt, xq if and only if its coefficients u j,˚p xq satisfy, for all j ě 0, the identities (2.1) u j,˚p xq " f j,˚p xqÿ iPK u j´vi,q´i´m,˚p xq

Definition 4 . 1

 41 (s-Gevrey formal series). Let s ě 0 be. A formal series r upt, xq " ÿ jě0 u j,˚p xq t j j! P OpD ρ2 qrrtss is said to be Gevrey of order s (in short, s-Gevrey) if there exist three positive constants 0 ă r 2 ă ρ 2 , C ą 0 and K ą 0 such that the inequalities sup |x|ďr2 |u j,˚p xq| ď CK j Γp1 `ps `1qjq

Definition 4 . 7 (

 47 Nagumo norms). Let f P OpD ρ q, n ě 0 and 0 ă r ă ρ be. Let d r pxq " r ´|x| denote the Euclidian distance of x P D r to the boundary of the disc D r . Then, the Nagumo norm }f } n,r of f is defined by }f } n,r :" sup |x|ăr |f pxqd r pxq n | . Following Proposition 4.8 gives us some properties of the Nagumo norms.

  n`p i ˚ptq " r u ˚,n ptq ´r f ˚,n ptq ´n ÿ m"1 ˆn m ˙api ˚,p i ˚q ptqB ´i t r u ˚,n´m`q ptq

Ÿ

  Second step: the fixed point procedure. Let us set r wpt, xq " ÿ mě0 r w m pt, xq and let us consider the solution of equation (5.2), where the r w m pt, xq's belong to OpD ρ qrrtss for a suitable common ρ ą 0 and are recursively determined, for all m ě 0, by the relations(5.3)

5 . 4 .

 54 considering recurrence relations (5.3) with w m and the k-sum gpt, xq instead of r w m and r gpt, xq, it is clear that wpt, xq satisfies equation (5.2) with right-hand side gpt, xq in place of r gpt, xq and, consequently, so does its Taylor series. Then, since equation (5.2) has a unique formal series solution r wpt, xq (proceed similarly as Theorem 2.1 by exchanging the roles of x and t), we then conclude that the Taylor expansion of wpt, xq is r wpt, xq. Hence, Condition 2 of Definition 5.1. This achieves the proof of the k-summability of r wpt, xq and, thereby, the sufficient condition of point 1 of Theorem 5.4. Proof of Lemma 5.11. We are left to prove the technical Lemma 5.11. Before starting the calculations, let us first recall a classical result on the Gamma function which will be us useful in the sequel. Lemma 5.13. Let a and b be two nonnegative numbers satisfying b ě maxp2, 1`aq. Then, Γp1 `aq ď Γpbq.

Lemma 5 . 15 .

 515 ˚m `i˚´i `jq `nq and from the relation Γp1 `ps `1qpi ˚m `i˚´i `jq ` ´jq " Γp1 `ps `1qpi ˚m `i˚´i `jqq ´j ź n"1 pps `1qpi ˚m `i˚´i `jq `nq when j ă . Let ě 0 and m ě 0 be. Then,

Proposition A. 4 .( 2 )

 42 Let s ě 0 and Σ be an open sector with vertex 0 P C. Then, a function upt, xq holomorphic on a domain Σ ˆDρ for some ρ ą 0 is s-Gevrey asymptotic to a formal series r upt, xq " ÿ jě0 u j,˚p xq t j j! P OpD ρ2 qrrtss on Σ if and onlyif there exists 0 ă r 2 ă minpρ, ρ 2 q such that (1) for any |x| ď r 2 , the map t Þ Ñ upt, xq has r upt, xq as Taylor series at 0 on Σ, for any proper subsector Σ 1 Ť Σ, there exist two positive cConstants C ą 0 and K ą 0 such that, for all J ě 0 and all t P Σ 1 , sup |x|ďr2 ˇˇB J t upt, xq ˇˇď CK J Γp1 `ps `1qJq. Proof. Ÿ Necessary condition. Let us suppose that upt, xq is s-Gevrey asymptotic to r upt, xq on Σ and let us prove Conditions 1 and 2 of Proposition A.4. Due to Definition A.1, Condition 1 is straightforward.

2 .

 2 Therefore, the series p upτ, xq " ÿ jě0 u j,˚p xqτ j Γp1 `sjqj! converges for all pτ, xq P D ρ ˆDr2 , where ρ is the radius of convergence of ÿ jě0 Γp1 `ps `1qjq Γp1 `sjqj! pKτ q j . ‹ Let us now fix b P D ρ , b ą 0, and let us consider the holomorphic function upt, xq P OpΣ ˆDr2 q defined by upt, xq " t ´k ż b k 0 p upξ s , xqe ´ξ{t k dξ, where s " 1 k and ξ " τ k .

  Proposition 4.2 specifies the algebraic structure of the OpD ρ2 qrrtss s 's. Let s ě 0 be. Then, pOpD ρ2 qrrtss s , B t , B x q is a C-differential algebra stable under the anti-derivations B ´1

	Proposition 4.2. t	and B ´1 x .
	Proof. See for instance [42, Prop. 1] or [2, p. 64].
	4.2. Main result. Let us first begin by observing that Proposition 4.2 implies the
	following.

  by using a technique of majorant series. Remark 4.19. Like in relation (2.1), some terms }u j,˚p xq} σj,r2

  1 2 , we have |u j,˚p xq| " ˇˇˇu j,˚p xqd r2 pxq σj 1 d r2 pxq σj ˇˇˇď ˇˇu j,˚p xqd r2 pxq σj ˇpr

	2 ´r1 2 q σj	ď	}u j,˚p xq} σj,r2

  or [15, I.2]) times t0u (in the plane C of the variable x). We denote by O I θ,πs ˆt0u the space of such sections. OpD ρ2 qttu k;θ ÝÑ O I θ,πs ˆt0u r upt, xq Þ ÝÑ upt, xq is a homomorphism of C-differential algebras for the derivations B t and B x . Moreover, it commutes with the anti-derivations B ´1 Before stating the main result of this section, let us first begin with a preliminary remark on the series r upt, xq. According to Notation 1.1, let us write the coefficients a pi,qq pt, xq on the form a pi,qq pt, xq "

	Corollary 5.3. The operator of k-summation
	S k;θ : t	and B ´1 x .
	Let us now turn to the study of our formal series solution r upt, xq.
	5.2. Main result.
		ÿ	a pi,qq ˚,n ptq
		ně0

5.2.1.

A preliminary remark.

  The sets Q i are the sets introduced in the preliminary remark of Section 5.2:

							1 2 ą 0.
	Let us now write r upt, xq on the form
	r upt, xq "	p i ˚´1 ÿ n"0	r u ˚,n ptq	x n n!	`B´p i x r vpt, xq
	with r vpt, xq P OpD ρ2 qrrtss and let us set r w :" B	´it	r v. Then, equation (1.2) becomes
	(5.2)					∆ r w " r gpt, xq,
	where ∆ is the linear integro-differential operator
	∆ :" 1 ´bpi ˚,p i ˚q pt, xqB ´pi x B i t	`ÿ iPK	qPQi ÿ	b pi,qq pt, xqB	q´p i x B i ˚´i
				˜pi ˚´1 ÿ n"0	r u ˚,n ptq	n! x n	´r f pt, xq	ÿ
							iPK	ÿ qPQi	p i ˚´1´q ÿ n"0	b pi,qq pt, xqr u ˚,n`q ptq	x n n!	.
	Indeed, we have B i ˚´i t	B	´it	" B ´i t for all i P K.

t and where the inhomogeneity r gpt, xq is defined by r gpt, xq :" b pi ˚,p i ˚q pt, xq

  Let us set B 2 :" B 1 K i ˚κp i ˚2p i ˚p1 `pi ˚qp i ˚´1 . Then, Propositions 5.7 and 5.12 imply, for all ě 0 and pt, xq P Σ ˆDr2 , the inequalities ˇˇB t w m pt, xq ˇˇď CK Γp1 `ps `1qpi ˚m ` qq pB 2 |x|q m ˇˇB t w m pt, xq ˇˇď Cp2 p i ˚K q Γp1 `ps `1q qp2 p i ˚B 2 |x|q m .Indeed, the conditions ps `1qi ˚" p i ˚and s `1 ď p i ˚(see relations (5.1)) imply

				pmp i ˚q!	;
	hence, the inequalities
	(5.7)
	Γp1 `ps `1qpi ˚m ` qq " Γp1 `ps `1q `mp i	˚q
				mp i ź
		" Γp1 `ps `1q q	pps `1q `jq
				j"1
				mp i ź
		ď Γp1 `ps `1q q	p p i ˚`j q
				j"1
		" Γp1 `ps `1q q	p p i ˚`mp i ˚q! p p i ˚q!
	and, consequently,
	Γp1 `ps `1qpi ˚m ` qq pmp i ˚q!	ď Γp1 `ps `1q q
	Proposition 5.12 ([43, Prop. 5]). Let m ě 0 be. Then,
	P m p|x|q ď	`κp i ˚2p i ˚p1 `pi pmp i ˚q! ˚qp i ˚´1 ˘m	|x|

m for all x P D r2 .

  Indeed, we have previously saw in the proof of Proposition A.3 that Γp1 `sJqJ! ď e 1`ps`1qJ Γp1 `ps `1qJq. This proves Condition 2 and, consequently, the necessary condition.Ÿ Sufficient condition. Let us now suppose that Conditions 1 and 2 are satisfied and let us prove condition (A.1) of Definition A.1. To do that, let us consider a proper subsector Σ 1 Ť Σ.For any fixed |x| ď r 2 , the map t Þ Ñ upt, xq admits the Taylor expansion with Bt J pt 1 , xqdt 1 for all J ě 1, all t P Σ 1 and all t 0 P Σ 1 . Due to Condition 1, lim Bt j pt 0 , xq exists for all j ě 0 and is equal to u j,˚p xq. Therefore, the limits of the left-hand and of the right-hand sides of (A.3) both exist when t 0 Ñ 0 and we have Bt J pt 1 , xqdt 1 for all J ě 1, all t P Σ 1 and all |x| ď r 2 . Hence, applying Condition 2:Bt J pt 1 , xq ˇˇˇ| t| for all J ě 1 and all t P Σ 1 . Condition (A.3) follows then from the inequality Γp1 `ps `1qJq J! ď 2 pS`1qJ Γp1 `sJq , S P N, S ě s

				upt, xq	´J´1 ÿ j"0	u j,˚p xq	t j j!	"	0 ż t	pJ ´1q! pt ´t1 q J´1	B J u
	sup |x|ďr2	ˇˇˇˇu pt, xq	´J´1 ÿ j"0	u j,˚p xq	t j j!	|x|ďr2 ˇˇˇˇď sup t 1 PΣ 1	ˇˇˇB	J u	J J!	ď CK J Γp1 `ps `1qJq J!	|t|
	which stems from the relations
												J	J
												ź	ź
		Γp1 `ps `1qJq " Γp1 `sjq	psJ `jq ď Γp1 `sJq	pSJ `jq
												j"1	j"1
	and
		J
		ź
		j"1	J t upt, xq ˇˇď pSJ `jq J! " ˆpS `1qJ J! p2πq 2 J	sup |t 1 ´t|"|t|δ ˙ď pS`1qJ ÿ k"0	p|t| δq J ˇˇp2πq 2 ˇˇupt 1 , x 1 q ˆpS `1qJ k ˙" 2 pS`1qJ .
												|x 1 ´x|"r2´r 1 2
									ď CK J Γp1 `sJqJ!	|t| J p1 `δq J |t| J δ J
	integral remainder
	(A.3)	upt, xq	´J´1 ÿ j"0	B j u Bt j pt 0 , xq	pt ´t0 q j j!	"	t0 ż t	pJ ´1q! pt ´t1 q J´1	B J u
												B j u
												t0Ñ0
												t0PΣ 1

ď C 1 K 1J Γp1 `ps `1qJq

with C 1 " eC and K 1 " e s`1 K ˆ1 `1 δ ˙.

J

  1 2 ă r 2 . For any 0 ă δ ă π 2 and 0 ă R 1 ă R, we denote by Σ δ the proper subsector of Σ defined by Since t P Σ δ ñ | argptq| ă π 2 ñ ptq ą 0 ñ ˇˇξ sj e ´ξ{t k ˇˇ" |ξ| sj e sj e ´ξ{t k converges normally on r0, b k s.Therefore, we can permute the sum and the integral. Hence,

			Σ δ "	" t P C; |argptq| ă	π 2k ´δ k	and 0 ă |t| ă R 1
						0	ξ sj Γp1 `sjq	e ´ξ{t k	dξ, j ě 0
	(see [2, pp. 78-79] for instance), we first have
	upt, xq	´J´1 ÿ j"0	u j,˚p xq	t j j!	" t ´k ż b k 0 ˜ÿ jě0	u j,˚p xq Γp1 `sjqj!	ξ sj e ´ξ{t k	¸dξ
								´J´1 ÿ j"0	u j,˚p xq j!	t ´k ż `8 0	ξ sj Γp1 `sjq	e ´ξ{t k	dξ.
									´ξ	pt k q |t| 2k ď b j
	for all ξ P r0, b k s, the series ξ upt, xq ÿ jě0 u j,˚p xq Γp1 `sjqj! ´J´1 ÿ j"0 u j,˚p xq t j j! " ÿ jěJ u j,˚p xq Γp1 `sjqj! t ´k ż b k 0	ξ sj e ´ξ{t k	dξ
									´J´1 ÿ j"0	u j,˚p xq Γp1 `sjqj!	t	´k ż `8 b

*

.

Let J ě 1 and pt, xq P Σ δ ˆDr 1 2 be. From the relation

t j " t ´k ż `8 k

ξ sj e ´ξ{t k dξ.

See Remark

3.4.

See Remark

4.15.

Of course, this case occurs if and only if i ˚ă κ.

Proof. ‹ Let us first suppose ď Ii ˚. Inequality (5.9) is clear when p , m, iq " p0, 0, i ˚q. Otherwise, we have 1 `ps `1qpi ˚m `i˚´i `jq ` ´j looooooooooooooooooooomooooooooooooooooooooon ě0 " 1 `ps `1qpi ˚m `i˚´i q ` `sj ď 1 `ps `1qpi ˚m `i˚´i ` q loooooooooooooooooomoooooooooooooooooon ě2 for all j P t0, ..., u, and inequality (5.9) stems from Lemma 5.13:

Γp1 `ps `1qpi ˚m `i˚´i `jq ` ´jq Γp1 `ps `1qpi ˚m `i˚´i ` qq ď ÿ j"0

1 " `1 ď Ii ˚`1.

‹ Let us now suppose ą Ii ˚and let us write the sum of (5.9) on the form (5.10)

By similarly to the previous case, we get

1 " Ii On the other hand, we have the inequalities

for all j P t0, ..., ´Ii ˚u. Indeed, the relation ps `1qi ˚" p i ˚(see (5.1)) and the definition of i ˚(see Remark 3.4) imply sIi ˚" Ipps `1qi ˚´i ˚q " Ipp i ˚´i ˚q ě I ě 1.

Consequently, by applying Lemma 5.13, the first sum of the right-hand side of (5.10) is bounded as follows:

Γp1 `ps `1qpi ˚m `i˚´i ` q ´sIi ˚q Γp1 `ps `1qpi ˚m `i˚´i ` qq " ´Ii ˚`1 ps `1qpi ˚m `i˚´i ` q ˆΓp1 `ps `1qpi ˚m `i˚´i ` q ´sIi ˚q Γp1 `ps `1qpi ˚m `i˚´i ` q ´1q ď ´Ii ˚`1 ps `1qpi ˚m `i˚´i ` q .

Inequality (5.9) follows then by observing that ´Ii ˚`1 ps `1qpi ˚m `i˚´i ` q ď 1 s `1 ď 1 for all ą Ii ˚. This ends the proof of Lemma 5.15. for all j P ti ´i˚, ..., u and inequality (5.14) follows from Lemma 5.13:

When ą Ii ˚, we proceed similarly as in Lemma 5.15 by writing the sum of (5.14) on the form

and by observing that the two sums of the right-hand side can be respectively bounded as follows:

´Ii ˚`i´i ÿ j"i´i ˚p...q ď 1 and

The first inequality is proved as in Lemma 5.15 by using Lemma 5.13 and the relations

1 `ps `1qi ˚m ` `spi ˚´i `jq ď 1 `ps `1qpi ˚m ` q ´sIi ď 1 `ps `1qpi ˚m ` q ´1 for all j P ti ´i˚, ..., ´Ii ˚`i ´i˚u . As for the second inequality, it stems from Lemma 5.13 and the relation 1 `ps `1qpi ˚m `i˚´i `jq ` ´j `i ´i˚ď 1 `ps `1qpi ˚m ` q proved just above. This ends the proof of Lemma 5.18.

Let us now apply Lemmas 5.17 and 5.18:

Indeed, the second sum of S i, ,m is zero (we have J i,i ˚, " ) and Remark 9 implies Ii ˚ą κ ´i˚ě i ´i˚. ' Case ě i ´i˚. Then, S i, ,m ď pi ´i˚`I i ˚´i `i˚`1 qΓp1 `ps `1qpi ˚m ` qq " pIi ˚`1qΓp1 `ps `1qpi ˚m ` qq.

Thereby, to end the proof of Lemma 5.11, we are left to prove the following.

Lemma 5.19. Let ě 0 and m ě 0 be. Then,

Γp1 `ps `1qpi ˚pm `1q ` qq.

Appendix A. Gevrey asymptotic

In this appendix, we present various results of the general theory of the Gevrey asymptotic expansions in the framework of the formal power series in OpD ρ2 qrrtss.

A.1. s-Gevrey asymptotic. Still considering t as the variable and x as a parameter, one extends, in the similar way as the s-Gevrey formal series (see Definition 4.1), the classical notion of Gevrey asymptotic to a formal series in Crrtss to the one of Gevrey asymptotic to a formal series in OpD ρ2 qrrtss in requiring similar conditions, the estimates being however uniform with respect to x. Definition A.1 (s-Gevrey asymptotic). Let s ě 0 and Σ be an open sector with vertex 0 P C. A function upt, xq holomorphic on a domain Σ ˆDρ for some ρ ą 0 is said to be Gevrey asymptotic of order s (in short, s-Gevrey asymptotic) to a formal series ÿ jě0 u j,˚p xq t j j! P OpD ρ2 qrrtss on Σ if there exists 0 ă r 2 ă minpρ, ρ 2 q such that, for any proper subsector Σ 1 Ť Σ, there exist two positive constants C ą 0 and K ą 0 such that, for all J ě 1 and all t P Σ 1 :

A series which is the s-Gevrey asymptotic expansion of a function is said to be an s-Gevrey asymptotic series on Σ.

Remark A.2. If any exists, the s-Gevrey asymptotic series is unique.

Proposition A.3. Let s ě 0 be. Then, a s-Gevrey asymptotic series on a sector Σ is a s-Gevrey series.

Proof. Let r upt, xq " ÿ jě0 u j,˚p xq t j j! P OpD ρ2 qrrtss be a s-Gevrey asymptotic series of a function upt, xq on Σ. We want to prove that there exist positive constants 0 ă r 2 2 ă ρ 2 , C 2 ą 0 and K 2 ą 0 such that, for all J ě 0, (A.2) sup

Let us now observe that the inequalities pξ{b k q sj ď pξ{b k q Js hold both when ξ ď b k and j ě J and when ξ ě b k and j ă J. This brings then us to the following ˇˇˇˇu pt, xq ´J´1 ÿ ξ sj e ´ξ sinpδq{|t| k q dξ.

Observe that the last inequality stems from the fact that t P Σ δ implies Γp1 `sjqj! pKbq j and K 1 " 1 bpsinpδqq s . The constants C 1 and K 1 depend on Σ δ and on the choice of b, but are independant of t and x. This achieves the proof.