On Boundary Layer Expansions for a Singularly Perturbed Problem with Conﬂuent Fuchsian Singularities

: We consider a family of nonlinear singularly perturbed PDEs whose coefﬁcients involve a logarithmic dependence in time with conﬂuent Fuchsian singularities that unfold an irregular singularity at the origin and rely on a single perturbation parameter. We exhibit two distinguished ﬁnite sets of holomorphic solutions, so-called outer and inner solutions, by means of a Laplace transform with special kernel and Fourier integral. We analyze the asymptotic expansions of these solutions relatively to the perturbation parameter and show that they are (at most) of Gevrey order 1 for the ﬁrst set of solutions and of some Gevrey order that hinges on the unfolding of the irregular singularity for the second.

We have shown that these equations possess families of holomorphic solutions y p (t, z, ) that can be expressed through Laplace transforms of order k and Fourier inverse integrals y p (t, z, ) = k (2π) 1/2 +∞ −∞ L γp ω p (u, m, ) exp − u t k e izm du u dm along halflines L γ p = [0, +∞)e √ −1γ p enclosed in a suitable unbounded sector S d p with bisecting direction d p ∈ R, where ω p stands for a holomorphic function near the origin and with exponential growth regarding u on S d p , continuous regarding m on R suffering exponential decay and relying analytically on the perturbation parameter on a punctured disc D(0, 0 ) \ {0}. These functions y p are bounded holomorphic on domains T × H β × E p , for some proper chosen bounded sector T with vertex at 0 and E = {E p } 0≤p≤ι−1 a set of bounded sectors whose union covers a full neighborhood of 0 in C * called a good covering (see Definition 4). Furthermore, they share a common asymptotic expansionŷ(t, z, ) = ∑ k≥0 y k (t, z) k that turns out to be of Gevrey order 1/k with respect to the perturbation parameter , meaning that two constants C p , K p > 0 can be found with sup t∈T ,z∈H β |y p (t, z, ) − n−1 ∑ k=0 y k (t, z) k | ≤ C p (K p ) n Γ(1 + n k )| | n for all n ≥ 1, provided that ∈ E p . In this work, we depart from Equation (2) in the special case k = 1. We unfold the operator t 2 ∂ t into a family of singular operators depending on a parameter α ≥ 3, assumed to be an odd natural number, which are now Fuchsian at the points t = ± α−1 2 . Notice that these singular points merge to 0 as borders the origin. In other words, we focus on equations of the next form Q(∂ z )u(t, z, ) = Q 1 (∂ z )u(t, z, )Q 2 (∂ z )u(t, z, ) + (D ,α (∂ t )) δ D R D (∂ z )u(t, z, ) +P(t, z, , D ,α (∂ t ), ∂ z )y(t, z, ) + f (t, z, ) where P is analytic in t,z, and polynomial in the operators D ,α (∂ t ),∂ z and f is a forcing term as above. The coefficients of P and the forcing term f represent bounded analytic functions in z and and, as a new feature, involve a well-chosen logarithmic function in time variable (described in (21)) that is adapted to our problem. They belong to a class of functions that is more restrictive than the one considered in our previous Equation (2). However, notice that according to Lemma 9 and 10, these functions are good approximations of general bounded analytic functions of the form h( t, z, ) on appropriate domains in time t, for z ∈ H β , provided that is close enough to 0. It is worthwhile noting that PDEs with Fuchsian singularities and logarithmic coefficients appear in several recent works by different authors. Specifically, in papers [2,3], Jose Ernie C. Lope investigates existence and uniqueness properties of solutions to linear Fuchsian equations with the shape (t∂ t ) m u(t, x) = G t, x, ((µ(t)∂ x ) α (t∂ t ) j u(t, x)) (α,j)∈I m where G(t, x, (u α,j ) (α,j)∈I m ) stands for a linear form in u α,j , holomorphic in x on a bounded domain of C n and continuous in t on [0, T], T > 0 and where the so-called weight function µ(t) may involve functions like 1/(− log(t)) k+1 provided that k > 0. In paper [4], Hidetoshi Tahara and Hideshi Yamane solve nonlinear equations of the form of Fuchsian type at t = 0, where H represents a bounded holomorphic function whose coefficients includes powers of log(t). In the study [5], Hideshi Yamane constructs solutions to nonlinear wave equations that blow up along prescribed noncharacteristic hypersurfaces using the so-called Fuchsian reduction method introduced by S. Kichenassamy which transforms the initial problem into a Fuchsian PDE which, in general, contains logarithmic terms, see the excellent textbook [6] for a reference. The idea of considering such special types of confluence (3) stems from a work by M. Klimes, see [7], where nonlinear differential systems with irregular singularity at x = 0 with invertible matrix M of dimension n ≥ 1 and unknown vector function where f stands for an analytic function near the origin in C n+1 , are unfolded into a deformed complex parameter depending differential system with Fuchsian singularities at x = ± 1/2 that coalesce to the origin as tends to 0. This paper [7] can be seen as a continuation of the contribution by B. Sternin and V. Shatalov [8] who focused on linear scalar ODEs with holomorphic coefficients of the form for a small complex parameter . Further important recent works on confluence of singularities that are somehow related to our study need to be mentioned. In paper [9], Reinhard Schäfke expounds the confluence of so-called hypergeometric systems of Okubo type to Birkhoff normal forms. This result has been extended to general linear systems of ODEs in two seminal papers [10,11] by Alexey Glutsyuk that describe the convergence of monodromy data of well-chosen fundamental solutions to the Stokes matrices in the confluence process. More recently in [12], Tsvetana Stoyanova has studied particular cases of [10,11] and obtained explicit formulas for the solutions of an unfolding of a third order linear scalar ODE with irregular singularity at the origin. Confluence under the additional constraint of isomonodromic deformation has been investigated for linear systems of ODEs for Fuchsian singularities by Andrey Bolibrukh, ref. [13] and later extended to the case of irregular singularities by Yulia Bibilo in [14]. Confluence of Fuchsian singularities for linear scalar ODEs has also been undertaken from the point of view of the so-called index of rigidity by Toshio Oshima in [15] and extensions to Pfaffian systems of PDEs are investigated using the middle convolution procedure in [16]. In the series of papers [17][18][19][20], the authors classify and provide normal forms for generic unfolding of nonresonant linear systems of ODEs with irregular singularity at the origin. In [21], confluent non-autonomous Hamiltonian systems are studied and applications to the degeneration processes for the famous Painlevé sixth equation are discussed. We mention also the novel work [22] by Jorge Mozo-Fernández and Reinhard Schäfke where confluence of Fuchsian and more general singularities can be unified in a general theory through the concept of summability regarding an analytic germ.
From a wider standpoint, our study falls in the asymptotic theory of singularly perturbed problems. For a far-reaching introduction and foundation of this active domain of research we may refer to the monograph [23] by S.A. Lomov. For other recent textbooks providing useful examples we can suggest [24][25][26][27].
In this paper, our aim is the study of the effect of the small perturbation α in the singular operators D ,α (∂ t ) on the asymptotic properties of (well-chosen) holomorphic solutions to our problem (4) in comparison to the ones of the departing problem (2).
One first major change in our new setting is the fact that the holomorphic solutions we build up for (4) (see Theorems 1 and 2) are no longer defined on a full sector T centered at 0 in time t. Instead, we manufacture two specific families of analytic functions. The elements of the first one, we call the outer solutions, u out p (t, z, ), 0 ≤ p ≤ ι − 1, are defined in time t on a fixed sectorial annulus A (see (65)) that does not rely on and is confined apart of the origin. The second family is comprised by so-called inner solutions, u in p (t, z, ), 0 ≤ p ≤ η − 1 that are built up on a domain in time T (see (81)) that hinges on , stays close to the Fuchsian singularities ± α−1 2 and borders the origin as comes close to 0. The second substantial difference dwells on the parametric asymptotic expansions of these solutions. Specifically, provided that t is kept (at least) at some fixed distance of the origin in A, the outer solutions u out p (t, z, ) have similar asymptotic expansions in as the one observed for the analytic solutions y p (t, z, ) of (2) (in the case k = 1), indeed they are (at most) of Gevrey order 1. On the other hand, when t is next to the Fuchsian singularities ± α−1 2 on T , the Gevrey order of the asymptotic expansions relatively to of the inner solutions u in p (t, z, ) turns out to be sensitive on the small perturbation α in (3) and becomes (at most) equal to 2 α+1 , see Theorem 3. We undertake the main problem (4) by following a closely related approach to the one used [7] and which actually originates from [8]. Specifically, we exhibit a first family of bounded holomorphic solutions to (4), we name outer solutions, in the form of special Laplace transforms and Fourier integrals where E out p are bounded sectors that belong to a well-chosen good covering E out , see Theorem 1. We distinguish a second set of so-called inner solutions also expressed in a similar manner as the outer functions, namely on domains T × H β , provided that ∈ E in p for a collection of bounded sectors E in p which constitutes a convenient good covering E in , see Theorem 2. In both constructions, the function a(T, ) isproperly selected in a way that the differential operator D ,α (∂ t ) acts on u out/in p as a multiplication by τ on the Borel map W out/in p (τ, m, ), see (20). Notice that in [7], bilateral Laplace transforms are introduced which operate on Borel maps that are holomorphic on strips. The restricted class of admissible coefficients for our main Equation (4) allows us to obtain Borel maps that are defined on unbounded sectors in the Borel plane. We can merely use the classical single side Laplace transforms to construct our solutions (5), (6) and apply similar constructions as in our previous study [1].
Our paper is organized as follows: In Section 2, we set forth the main problem of the work (11) and we depict the strategy which leads to its resolution. Specifically, we search for a solution expressed through special Laplace and Fourier transform (19). The main achievement of this section is the statement of a convolution Equation (29) fulfilled by the Borel map W of the solution. In Section 3, we solve the convolution problem (29) within Banach spaces of holomorphic and continuous functions with exponential growth on unbounded sectors in the Borel plane and exponential decay in real phase. In Section 4.1, we build up a family of outer solutions defined on a covering set of sectors in the perturbation parameter and on a fixed sectorial annulus in time (Theorem 1). In Section 4.2, a collection of inner solutions is exhibited which are defined relatively to on a good covering in C * and regarding time on a domain that remains close to the moving Fuchsian singularities of the main problem. In the last Section 4.3, we analyze the asymptotic behavior of the latter two distinguished families of outer/inner solutions, by means of the classical Ramis-Sibuya approach (Theorem 3).

Short Review of Fourier transforms
We restate the definition of some family of Banach spaces mentioned in [1].
We remind the reader the definition of the inverse Fourier transform acting on the latter Banach spaces and some of its handy formulas relative to derivation and convolution product as stated in [1].
The inverse Fourier transform of f is given by for all x ∈ R. The function F −1 ( f ) extends to an analytic bounded function on the strips for all given 0 < β < β.
(a) Define the function m → φ(m) = im f (m) which belongs to the space E (β,µ−1) . Then, the next identity as the convolution product of f and g. Then, ψ belongs to E (β,µ) and moreover, for all z ∈ H β , for 0 < β < β.

Statement of the Main Problem
Let D ≥ 1, δ D ≥ 2 be natural numbers, α ≥ 3 be an odd natural number, c 1,2 ∈ C * a non-vanishing complex number and ∈ C * a complex parameter. Let δ l ≥ 0 be non-negative integers with for all 0 ≤ l ≤ D − 1. We define the differential operator and for any integer l ≥ 1 we denote (D ,α (∂ t )) l the iteration of order l of the operator D ,α (∂ t ).
We set Q(X), for all m ∈ R. We state the main initial value problem of our study for given vanishing initial data u(0, z, ) ≡ 0. The precise shape of the function a( t, ) is displayed later in the work at (21) and will be justified by the approach we undertake in this work. The coefficients c l and the forcing term f are described as follows. For each 0 ≤ l ≤ D − 1, we consider a sequence of functions m → C l,n (m, ), for n ≥ 0, i.e., part of the Banach space E (β,µ) for some parameter β, µ > 0 (see Definition 1) and depends in an analytic way on on a disc D(0, 0 ) and for which two constants C l,1 > 0, C l,2 > 0 can be found with for any n ≥ 0. We set the sequence of functions c l,n (z, ) defined as the Fourier transform of C l,n (m, ), whenever 0 ≤ l ≤ D − 1 and n ≥ 0. We define the coefficient c l as the function We consider a family of functions m → ψ n (m, ), for n ≥ 1 that belong to the Banach space E (β,µ) and rely analytically on in the disc D(0, 0 ) and for which constants D 1 > 0, D 2 > 0 can be singled out with sup for all n ≥ 1. We set the functions d n (z, ) as the Fourier transform of ψ n (m, ), provided that n ≥ 1. We define the forcing term f (a( t, ), z, ) as the sum The domains where these functions (14) and (17) are well defined and holomorphic will be specified later in the paper in Theorems 1 and 2.
Throughout this work, we are searching for time rescaled solutions of (11) with the shape In fact, if we define the next differential operator the expression U(T, z, ), by means of the change of variable T = t, is required to solve the next singular problem We need to set forth the definition of Banach spaces that have already been introduced in [1].
Definition 3. Let S d be an unbounded sector centered at 0 with bisecting direction d ∈ R and D(0, ρ) the open disc centered at 0 with radius ρ > 0. We fix ν, β > 0 and µ > 1 some positive real numbers. We set E d (ν,β,µ) as the vector space of continuous At a first stage, we undertake a similar strategy as in [7,8] and seek for solutions to (18) in the form of special Laplace and Fourier transforms where represents a sector as described in Definition 3 and a(T, ) stands for a well selected analytic function. Within this section, we assume that the map (τ, m) → W(τ, m, ) belongs to the Banach space E d (ν,β,µ) given in Definition 3 provided that ∈ D(0, 0 ).

Statement of a Convolution Problem Satisfied by the Expression W(τ, m, )
Our principal intention is to find a related tractable problem fulfilled by the function W. In this respect, the function a(T, ) is properly singled out in a way that the next identity formally holds Specifically, the function a(T, ) can be chosen as any primitive of the rational function 1/(s 2 − α+1 ). Among them, we pick up the simplest one For later need, the next lemma is also easily checked. The proof follows from the application of Definition 2 (b) and the use of Fubini theorem.
Then, the next identity holds (2) Let m → H 2 (m, ) be an element of E (β,µ) for any given ∈ D(0, 0 ). We put The following identity Moreover, we observe that for any natural number h ≥ 1, the building block (a(T, )) −h can be easily expressed as special Laplace transforms that involve the function a(T, ) described in (21). Indeed, a direct computation shows that for any integer h ≥ 1 provided that the integral makes sense for a properly chosen path L γ . As a result, we can express the coefficients c l , for 0 ≤ l ≤ D − 1 and forcing term f as special Laplace and Fourier transforms. Specifically, according to the integral representations (13) and (16) and the expansions (14) and (17), we obtain where and We are now ready to state the main convolution problem satisfied by the map (τ, m) → W(τ, m, ) provided that U γ given by (19) solves (18). Specifically,

Bounds for Convolution Operators Acting on Banach Spaces with Exponential Growth and Decay
We keep the notations of the previous sections and subsections. Throughout this subsection, we establish a list of technical lemmas that are essential in the resolution of the problem (29) within the Banach spaces presented in Definition 3 from Section 2.2. Although the proofs of the statements are close to the ones given in [1], they will be given in full details for the sake of clarity and readability of the paper.
In a first lemma, we observe that the functions C l and ψ defined in (26) and (28) belong to the Banach space E d (ν,β,µ) .
Proof. According to the assumption (12), for any givenC l,2 > C l,2 , one can find a constant C l,3 > 0 such that for all τ ∈ S d ∪ D(0, ρ), all m ∈ R and all ∈ D(0, 0 ). This yields the first bounds (30). Similar estimates can be obtained for ψ(τ, m, ) owing to the conditions (15) which give rise to the second bounds (31).
In the next lemma, we check the continuity property for the operator of multiplication by a bounded function. Its proof is straightforward.
In the sequel, we need to fix some holomorphic function a δ D (τ) on S d ∪ D(0, ρ), continuous on its closureS d ∪D(0, ρ), with the bounds for all τ ∈S d ∪D(0, ρ).
In the next three lemmas, we analyze the continuity of convolutions operators acting on E d (ν,β,µ) .

Lemma 6.
Under the condition (41), we can select a constant K 3 > 0 (hinging on µ,δ l , R l for 0 ≤ l ≤ D) such that Proof. From Lemma 2, we know that C l (τ, m, ) belongs to E d (ν,β,µ) and fulfills the bounds for all τ ∈ S d ∪ D(0, ρ), all m ∈ R. Likewise, h(τ, m) is subjected to the bounds (36). Taking into account the definition of the constant K in (37), the next estimates follow where B(m) is given in the proof of Lemma 5 and satisfies the bounds (44) under the constraints (10) and (41). Lemma 6 follows.

Construction of a Unique Solution to the Convolution Equation
The principal objective of this subsection consists of the construction of a unique solution of the equation (29) within the Banach space described in Definition 3.
We first provide further analytic assumptions on the leading polynomials Q(X) and R D (X) that allows us to transform our problem (29) into a fixed-point equation which is stated later on, see (64). Specifically, we impose the existence of an unbounded sectorial annulus A Q,R D = {z ∈ C/|z| ≥ r Q,R D , arg(z) ∈ (α Q,R D , β Q,R D )} for some inner radius r Q,R D > 0 and angles α Q,R D , β Q,R D ∈ R with α Q,R D < β Q,R D such that for all m ∈ R. At this point, we follow a comparable protocol as the one initiated in our erstwhile study [1]. We focus on the polynomial P m (τ) = Q(im) − τ δ D R D (im) for which we ask lower bounds respectively to τ in C and m in R. We factorize this polynomial as follows where the roots can be computed explicitly as We consider an unbounded sector S d centered at 0 with bisecting direction d ∈ R and a small disc D(0, ρ). We adjust the annulus A Q,R D in a way that the next condition hold : a constant q > 0 can be chosen with lower bounds |τ − q l (m)| ≥ q(1 + |τ|) for all 0 ≤ l ≤ δ D − 1, all m ∈ R, whenever τ ∈ S d ∪ D(0, ρ). Indeed, the feature (47) impose all the roots q l (m), 0 ≤ l ≤ δ D − 1 to remain a part of the origin and satisfy |q l (m)| > 2ρ for a well-chosen ρ > 0. Furthermore, provided that the aperture of A Q,R D is taken close enough to 0, all the roots q l (m) stay inside a union U of unbounded sectors centered at 0 that do not cover a full neighborhood of 0 in C * . We select the sector S d such that S d ∩ U = ∅ holds. As a result, the quotient q l (m)/τ is kept at a distance of some disc centered at 1 in C whenever τ ∈ S d , m ∈ R and 0 ≤ l ≤ δ D − 1. Then, Ref.
(49) follows. Lower bounds for P m (τ) are given in the next lemma.

Lemma 7.
When the sector S d and the disc D(0, ρ) are submitted to the constraints stated above, a constant C P > 0 (relying on δ D and q) can be singled out with for all τ ∈ S d ∪ D(0, ρ), all m ∈ R.
Proof. The factorization (48) together with (49) lead to In the next lemma, we discuss properties of a nonlinear map H which acts on a small closed ball centered at the origin in the Banach space E d (ν,β,µ) .

Lemma 8.
Assume that the sector S d and the disc D(0, ρ) fulfill the constraints stated above. Under the additional conditions that for 0 ≤ l ≤ D − 1 and provided that the constants c 1,2 , C l,1 , D 1 for 0 ≤ l ≤ D − 1, defined in Section 2.2 are taken small enough, for all ∈ D(0, 0 ), the map set as is submitted to the next properties: (i) There exists a radius > 0 (which can be chosen independently of in D(0, 0 )) such that the next inclusions hold, whereB(0, ) stands for the closed ball of radius > 0 centered at 0 in the space E d (ν,β,µ) .
In a second part of the proof, we discuss the second property ii). We pick up two elements h 1 (τ, m) and h 2 (τ, m) inside the ballB(0, ) constructed in the first part i).
We first deal with the nonlinear part of H . To provide upper bounds, we need first to rewrite the next difference of convolutions into a sum of convolution terms that involve the difference h 1 − h 2 , namely Then, under the first constraint of (51), Lemma 3, 4 with (50) ensure the existence of a constant K 1 > 0 (depending on Q 1 , Q 2 , R D and µ) such that In this second phase, we sort the constants |c 1,2 |, C l,1 for 0 ≤ l ≤ D − 1 small enough in order that the next inequality is fulfilled Then, gathering the bounds (61) and the corresponding estimates for the linear part of H through (56), (57) overhead where h(τ, m) has to be replaced by the difference h 1 (τ, m) − h 2 (τ, m), we get the expected shrinking character of H given by (54).
Proof. From Lemma 8, we observe that the map H is contractive from the complete metric spacē B(0, ) into itself for the distance d(x, y) = ||x − y|| (ν,β,µ) . As a result, the classical fixed-point theorem guarantees the existence of a unique function (τ, m) → W(τ, m, ) insideB(0, ) with for every ∈ D(0, 0 ). Moreover, this function W relies analytically on in the disc D(0, 0 ). Observe that if one puts the term τ δ D R D (im)W(τ, m, ) from the right to the left-hand side of Equation (29) and divide the resulting equation by the polynomial P m (τ) given by (48), (29) can be exactly recast as the Equation (64) above. In conclusion, the unique fixed point of (64) precisely solves the problem (29) with vanishing initial data W(0, m, ) ≡ 0. The proposition follows.

Construction of Analytic Solutions to the Main Problem and Their Parametric Asymptotic Expansions
We build up two distinguished families of actual analytic solutions to our main problem (11). The elements of the first family are called outer solutions and those of the second family are named inner solutions in the context of so-called boundary layer expansions (see the textbook [31] for an explanation of this terminology). Indeed, the domain of holomorphy in time t turns out to be a fixed sectorial annulus independent of that is kept apart of the origin for the outer family and a domain that relies on the parameter that comes close to the origin when tends to 0 for the inner family. Both sets of solutions lean on the next definition of so-called good covering in C * .

Definition 4.
Let ι ≥ 2 be an integer. We consider a finite set E = {E p } 0≤p≤ι−1 where E p stand for open sectors with vertex at 0 such that E p ⊂ D(0, 0 ) which fulfills the next three assumptions: The union of all the sectors E p covers a punctured disc centered at 0 in C. Then, the set E is called a good covering in C * .

Outer Solutions
We select a bounded sectorial annulus for given radii ρ 1 , ρ 2 > 0 and angles β 1 < β 2 . The next lemma presents bounds estimates for the function a( t, ) where a(T, ) is displayed in (21) for and t in suitable domains.

Lemma 9.
For any given δ > 0, one can find a small enough radius 0 > 0 such that the next factorization holds, provided that ∈ D(0, 0 ) \ {0}, t ∈ A, where a out ( , t) represents a holomorphic function on D(0, 0 ) × A such that sup Proof. For t ∈ A and ∈ D(0, 0 ) \ {0}, we can write By classical Taylor expansions, we can expand for some holomorphic function ε(z) near the origin such that lim z→0 ε(z) = 0. As a result, we can write that gives rise to the factorization (66) with the bounds (67) whenever 0 is taken close enough to 0.
In the sequel, we define the notion of admissible set of sectors accordingly to a good covering.

Definition 5.
We consider a good covering E out = E out p 0≤p≤ι−1 in C * , a set of unbounded sectors S d out p , 0 ≤ p ≤ ι − 1, with bisecting direction d out p ∈ R and a disc D(0, ρ) for some radius ρ > 0 that suffer the next two conditions: (1) A constant q > 0 can be taken with lower bounds (49) We can find a constant ∆ out > 0 such that for all ∈ E out p , all t ∈ A, a direction γ out p ∈ R (that may rely on , t) with L γ out whenever τ ∈ L γ out p , ∈ E out p and t ∈ A, for all 0 ≤ p ≤ ι − 1. Then, the family of sectors is called admissible accordingly to the good covering E out .
In the next first principal statement of the paper, we construct a family of so-called outer holomorphic solutions to our main Equation (11) defined on the sectors E out p of a good covering E out in C * relatively to the perturbation parameter . The difference between neighboring solutions are also estimated and gives rise to exponential flat bounds of order 1.
As a result, we can recast the difference u out p+1 − u out p as a sum of three integrals ))e izm dτ τ dm are halflines and C ρ/2,γ out p ,γ out p+1 stands for the arc of circle with radius ρ/2 which attaches the two points . We first control the quantity According to the condition (2) of Definition 5, bearing in mind the bounds (72) and the factorization (66), provided that the constant δ > 0 in (67) is taken close enough to 0, we obtain the estimates for some δ 2 > 0 chosen under the condition that holds, for all z ∈ H β , all t ∈ A and ∈ E out p ∩ E out p+1 .
In a similar manner, we can display upper bounds for the second integral piece for some δ 2 > 0 taken in a way that (78) holds, provided that z ∈ H β , t ∈ A and ∈ E out p ∩ E out p+1 . In the last step, we handle the integral part along the arc of circle By construction, under the condition (69) for the directions γ out p , γ out p+1 , we observe that the next inequality Taking into account the bounds (72) and the factorization (66), if the constant δ > 0 in (67) is chosen small enough, we get for a well-chosen δ 2 > 0 submitted to (78), whenever z ∈ H β , t ∈ A and ∈ E out p ∩ E out p+1 . In conclusion, gathering the bounds (77), (79) and (80) above, we obtain the expected estimates (71) from the decomposition (75).

Inner Solutions
We fix a bounded sectorial annulus for some radii r 1 , r 2 > 0 and angles α 1 < α 2 and we define the next open sectorial domain for all ∈ C * . We start this subsection with a lemma that provides bounds estimates for the function a( t, ) where a(T, ) is given by (21) for and t in appropriate domains.

Lemma 10.
For any fixed δ > 0, we can find a large enough inner radius r 1 > 0 such that we can factorize In the next definition, we introduce the notion of admissible set of sectors relatively to a good covering. Definition 6. We consider a good covering E in = {E in p } 0≤p≤η−1 in C * , a set of unbounded sectors S d in p , 0 ≤ p ≤ η − 1 with bisecting direction d in p ∈ R and a disc D(0, ρ) for some radius ρ > 0 that fulfill the next two of properties: (1) A constant q can be chosen with lower bounds (49) |τ − q l (m)| ≥ q(1 + |τ|) for all 0 ≤ l ≤ δ D − 1, all 0 ≤ p ≤ η − 1, all m ∈ R, whenever τ ∈ S d in p ∪ D(0, ρ).
(2) There exists ∆ in > 0 such that for all ∈ E in p , all t ∈ T , one can choose a direction γ in p ∈ R (that may depend on ) with L γ in p = R + e √ −1γ in p ⊂ S d in p ∪ {0} for which cos(arg(τ) − arg( α+1 2 ) − arg(x)) > ∆ in (84) provided that τ ∈ L γ in p and t = α−1 2 x ∈ T , whenever 0 ≤ p ≤ η − 1. If the conditions above are both satisfied, the collection of sectors S in = {{S d in p } 0≤p≤η−1 , D(0, ρ)} is called admissible relatively to the good covering E in .
In the next second main statement of the work, we exhibit a collection of actual holomorphic solutions, called inner solutions, to our main Equation (11) defined on the sectors E in p of a good covering E in = {E in p } 0≤p≤η−1 in C * with respect to the perturbation parameter . We control also the difference between consecutive solutions on the intersection of sectors E in p ∩ E in p+1 where exponentially flat bounds leaning on α are observed. By the construction above, the function (t, z) → u in p (t, z, ) is bounded and holomorphic and solves our main Equation (11) on the domain T × H β , for all ∈ E in p , provided that the inner radius r 1 of χ is taken large enough and the radius 0 > 0 of the disc D(0, 0 ) containing all the sectors E in p is strained to the bounds where r 2 is the outer radius of χ.
In the second part of the proof, we focus on the bounds (87). For the discussion and the technical content are very similar to the last part of the proof of Theorem 1 dealing with the estimates (71), the main arguments are presented in a more elliptical manner. Take p ∈ {0, . . . , η − 1}. Since the two applications τ → W in j (τ, m, ), for j = p, p + 1 are analytic continuations on the sectors S d in p , S d in p+1 of a joint analytic function we denote τ → W(τ, m, ) on the disc D(0, ρ), provided that m ∈ R, ∈ D(0, 0 ), we can express the difference u in p+1 − u in p into three integral pieces where L ρ/2,γ in j , j = p, p + 1 represent halflines in directions γ in j at a distance ρ/2 from the origin built as in (76) and C ρ/2,γ in p ,γ in p+1 symbolizes an arc of circle with radius ρ/2 joining the above two halflines.
We evaluate the first integral block W in p+1 (τ, m, ) exp(τa( α+1 2 x, ))e izm dτ τ dm According to the condition (2) of Definition 6, bearing in mind the bounds (88) and the factorization (82), provided that the constant δ > 0 in (83) is selected close enough to 0, we reach the next estimates