On a $q-$analog of a singularly perturbed problem of irregular type with two complex time variables

Analytic solutions and their formal asymptotic expansions for a family of the singularly perturbed $q-$difference-differential equations in the complex domain are constructed. They stand for a $q-$analog of the singularly perturbed partial differential equations considered in our recent work [A. Lastra, S. Malek, Boundary layer expansions for initial value problems with two complex time variables, submitted 2019]. In the present work, we construct outer and inner analytic solutions of the main equation, each of them showing asymptotic expansions of essentially different nature with respect to the perturbation parameter. The appearance of the $-1$-branch of Lambert $W$ function will be crucial in this respect.

parameter on a small disc centered at the origin, say D(0, 0 ) for some 0 > 0. Throughout the present work, σ q;t stands for the dilation operator on t variable for some fixed q > 1, i.e. σ q;t (f (t)) := f (qt). We adopt the following notation σ δ q;t (f (t)) = f (q δ t), for any δ ∈ Q. The forcing term f (t, z, ) is constructed under certain growth conditions, and turns out to be a holomorphic function in C × C × H β × (D(0, 0 ) \ {0}), for 0 < β < β.
The precise assumptions on the elements involved in the main equation under study are detailed in Section 3.
The problem under study (1) turns out to be a q−analog of the main problem studied in [15], under null initial data u(t 1 , 0, z, ) ≡ u(0, t 2 , z, ) ≡ 0, and where Q(X) ∈ C[X], and P is a polynomial with respect to its first three variables, with holomorphic coefficients on H β ×D(0, 0 ), and the forcing term is holomorphic on C × C × H β × (D(0, 0 ) \ {0}). The analytic solutions and their asymptotic expansions of those singularly perturbed partial differential equations are obtained in [15]. More precisely, the so-called inner solutions are holomorphic solutions of (2), holomorphic on domains in time which depend on the perturbation parameter and approach infinity, admits Gevrey asymptotic expansion of certain positive order, with respect to , whereas the so-called outer solutions are holomorphic solutions of (2), holomorphic on a product of finite sectors with vertex at the origin with respect to the time variables, admit Gevrey asymptotic expansion of a different positive order, with respect to . The previous phenomena of existence of different asymptotic expansions regarding different domains of the actual solution of the main problem is enhanced in the present work, in the sense that different nature on the asymptotic expansions are observed: Gevrey and q−Gevrey asymptotic expansions.
In this work, we search for the analytic solutions of the main problem as the inverse Fourier transform and q−Laplace transform of a positive order in the form (see (21) and (23)), for some appropriate γ ∈ R and k 1 , k 2 , λ 1 , λ 2 > 0 (see Section 3 for their definition). The function ω(τ, m, ) is obtained from a fixed point argument (see Proposition 6), and belongs to qExp d (k 1 ,β,µ,α) , a Banach space of holomorphic functions with q−exponential growth and exponential decay with respect to τ and m, respectively (see Definition 4).
The form (3) of the analytic solutions is motivated on the shape of those of the main problem in [15], mixing both time variables in a common Laplace operator.
A first family of analytic solutions of (1) is constructed on domains of the form T 1 × T 2, × H β × E ∞ h 1 , and a second on domains of the form T 1 × (T 2 ∩ D(0, ρ 2 )) × H β × E 0 h 2 , where T 1 is a finite sector, T 2 is an unbounded sector and where T 2, ⊆ T 2 is a bounded sector which depends on ∈ E ∞ h 1 , and tends to infinity with approaching the origin. The sets E 0 = (E 0 h 2 ) 0≤h 2 ≤ι 2 −1 and E ∞ = (E ∞ h 1 ) 0≤h 1 ≤ι 1 −1 represent good coverings (see Definition 5). Different path deformations performed on the analytic solutions give rise to Theorem 2 and Theorem 3, where upper bounds on the difference of two consecutive solutions are attained (consecutive solutions in the sense that they are related to consecutive sectors in a good covering).
Such bounds are related to null Gevrey and q−Gevrey asymptotic expansions of some positive order. As a matter of fact, the previous differences allow to apply a novel ((q, k); s)−version of the cohomological criteria known as Ramis-Sibuya theorem. Such result is related to functions admitting q−Gevrey asymptotic expansions of order k and a Gevrey sub-level of order s, see Theorem 4. We also apply a q−analog of Ramis-Sibuya Theorem, see Theorem 5.
The main two results of the present work are Theorem 6 and Theorem 7 relating the analytic solutions of (1) to their formal power series expansions obtaining asymptotic results of different nature. Such solutions are known as inner and outer solutions (see Definition 8 and Definition 9, resp.). Such asymptotic solutions have also been observed in the previous study [15], in the framework of singularly perturbed PDEs. However, the different nature of the asymptotic expansions regarding the outer and inner solutions is a novel phenomena which has firstly been observed in the present study.
The inner and outer expansions appear in the study of matched asymptotic expansions (see [22,28], among others for the classical theory). In the work [6] by A. Fruchard and R. Schäfke, the method of matching is developed, studying the nature of such asymptotic expansions, under Gevrey settings.
The proof of this result leans on the application of accurate estimates related to the −1branch of Lambert W function (see Lemma 3).
Concerning the outer solutions of the main problem under study, we consider a good covering (E 0 h 2 ) 0≤h 2 ≤ι 2 −1 , and the holomorphic solutions of (1), u h 2 (t, z, ) defined on E 0 h 2 w.r.t. for all 0 ≤ h 2 ≤ ι 2 − 1. In Theorem 7, we prove that is an outer solution of (1) with values in the Banach space F 2 of holomorphic and bounded functions on T 1 ×(T 2 ∩D(0, ρ 2 ))×H β . Moreover, there exists a formal power seriesû 0 ( ) which is the common q−Gevrey asymptotic expansion of some positive order of each solution In recent years, an increasing interest on the study of the asymptotic behavior of solutions to q−difference-differential equations in the complex domain has been observed. New theories giving rise to q−analogs of the classical theory of Borel-Laplace summability have been discussed and studied, as in the case of the work [29], by H. Tahara, where the author also provides information about q−analogs of Borel and Laplace transforms and related properties on convolution or Watson-type results. The use of procedures based on the Newton polygon is also exploited in recent studies, such as the work [30] by H. Tahara and H. Yamazawa. Also, it is worth mentioning the study of q−analogs of Briot-Bouquet type partial differential equations by H. Yamazawa, in [32]. Integral transforms involving special functions have also been considered in the study of q−difference-differential equations in [7,23]. Other references in this context by the authors and collaborators are listed in the references. Different kinds of Advanced/delayed partial differential equations are the cornerstone of mathematical models which have been recently studied. Examples of such studies have been applied to tsunamis and rogue waves which can be found in [26]. We also refer to other studies such as [24,25], and the references therein.
The outline of the work is as follows. In Section 2, we recall some known facts about formal q−Borel transform, analytic q−Laplace transform and inverse Fourier transform together with some properties which are applied to transform the main equation under study into auxiliary problems. Afterwards, we provide the definition and related properties of some Banach spaces involved in the construction of the solution. In Section 3 we state the main problem under study (17) and two auxiliary equations. The elements involved in them in addition to the domains of existence and upper bounds of the solutions of such equations are detailed. Section 4 is devoted to the existence and description of the domain of existence for the auxiliary equation (22), and associated estimates. In the following section, Section 5, we provide analytic solutions of (17) (see Theorem 1) and estimates on the difference of two of them (see Theorem 2 and Theorem 3). After a brief summary on q−asymptotic expansions in the first part of Section 6, we provide formal power series expansions in the perturbation parameter of the analytic solutions and relate them asymptotically in adequate domains. These results are attained in Theorem 6 and Theorem 7. The work concludes with two technical sections, Section 7 and Section 8, left to the end of the work in order not to interfere with our reasonings.

Review on certain integral operators. Study of Banach spaces involved in the problem
We briefly describe the foundations of the analytic and formal operators that will allow the transformation of the main problem under study in terms of auxiliary equations. The solutions of such equations belong to certain Banach spaces which are constructed subsequently.

Review of some formal and analytic operators
This section is devoted to recall some of the basic facts on formal and analytic transformations corresponding to the q−analogs of those appearing in the classical Borel-Laplace summability theory. These tools were developed in [4,27]. Through the whole section, q > 1 stands for a real number, and k ≥ 1 is a positive integer. E stands for a complex Banach space.
The next result, whose proof can be found in Proposition 5 [19], is crucial in order to transform the main equation into an auxiliary one lying in the q−Borel plane. The q−analog of Laplace transform used in this work was introduced in [31], and makes use of a kernel given by the Jacobi Theta function of order k, given by We recall that Jacobi Theta function satisfies From Lemma 4.1 [11], given anyδ > 0, there exists C k,q > 0 (independent ofδ) such that for all x ∈ C such that |1 + xq m k | >δ, for every m ∈ Z. This last property allows to define a q−analog of Laplace transform with appropriate properties.
Definition 2 Let ρ > 0 and S d be an unbounded sector with vertex at 0, and bisecting direction d ∈ R. Let f : S d ∪ D(0, ρ) → E be a holomorphic function, continuous up to the boundary, such that there exist K > 0, and α ∈ R with We choose an argument γ ∈ R within the set of arguments in S d and define the q−Laplace transform of order k of f in direction γ by where L γ = {re γ √ −1 : r ∈ (0, ∞)}, and π q 1/k := log(q) The proof of the next results can be found in detail in Lemma 4 and Proposition 6 [19].
We conclude with the definition and properties regarding the inverse Fourier transform.
Proposition 3 Let µ > 1, β > 0 and f ∈ E (β,µ) . The inverse Fourier transform of f is defined by This function can be extended to an analytic function on the horizontal strip H β = {z ∈ C : |Im(z)| < β}. Moreover, it holds that the function φ(m) = imf (m) belongs to E (β,µ−1) and In addition to this, it holds that the convolution product of f ∈ E (β,µ) and g ∈ E (β,µ) , defined by

Banach spaces of functions of q−exponential growth and exponential decay
In this section, we state the definition of the complex Banach space qExp d (k,β,µ,α) . The analytic solution of the main equation under study is built departing from one element in such Banach space via q−Laplace transform and inverse Fourier transform. Similar versions of this Banach space have already appeared in previous works by the authors such as [11,12], and the contribution [19] of the second author.
In the whole section, we fix real numbers β, µ, k > 0, q, δ > 1 and α. We also set d ∈ R and choose an infinite sector S d of bisecting direction d, with vertex at the origin; and the closed disc D(0, ρ), for some ρ > 0.
In the forthcoming results, we describe some properties on the elements of the previous Banach space when combined with certain operators. The first result is a direct consequence of the previous definition.
, and it holds that In order to give upper bounds for C 21 (|τ |), we have The mean value theorem guarantees that for some positive constant C 22 which does not depend on τ ∈ S d ∪ D(0, ρ). This entails that for every τ ∈ S d ∪ D(0, ρ). We conclude the result by taking into account (7). 2 The next result is stated without proof, heavily rests on Proposition 2 [16], and it is based on bounds stated in Lemma 2.2 [3].

Statement of the main problem and auxiliary equations
Let β > 0 and k 1 , k 1 , k 2 ≥ 1 with k 1 > k 1 , and D 1 , D 2 ≥ 2 be integer numbers. We also consider a real number q > 1, and choose real numbers In addition to that, there exist natural numbers λ 1 , λ 2 with Let us also fix polynomials Q, R D 1 D 2 , and R 1 2 for all 1 ≤ 1 ≤ D 1 − 1 and 1 ≤ 2 ≤ D 2 − 1, with complex coefficients such that . These polynomials are chosen in such a way that R D 1 D 2 (im) = 0 for m ∈ R, and Let µ > 1 such that The main problem under study in this work is It is constructed as follows. Let ψ(τ, m, ) be a continuous function, continuous in C × R × D(0, 0 ), for some 0 > 0, entire with respect to the first variable, and holomorphic with respect to the third one in D(0, 0 ). We moreover assume there exists C ψ > 0 such that and where µ satisfies (16).
In view of the definition of q−Laplace transform and the results described in Section 2.1, one can define for some fixed γ ∈ R. Here, T := (T 1 , T 2 ). The function turns out to be holomorphic on the set Observe that in the previous construction, given T 1 , T 2 ∈ C , one can choose γ ∈ R with cos(γ − k 2 arg(T 2 )) > 0 and |1 + (uq m/k )/T 1 | > 0 for all m ∈ Z and with u ∈ L γ . For every 0 ≤ 1 ≤ D 1 − 1 and 0 ≤ 2 ≤ D 2 − 1, the function c 1 2 (z, ) is constructed in the following way: . In addition to that, we assume that uniform bounds with respect to the perturbation parameter are satisfied, i.e. there exist C 1 2 > 0 with (20) sup

Study of auxiliary equations
In this section we preserve the statements and constructions concerning the main problem under study (17), and the geometric and algebraic conditions held on the elements involved in the main equation.
We search for solutions of (17) of the form Assuming the solution is of the form (21), the expression U (T 1 , T 2 , m, ) solves We reduce the study of solutions of (17) to those of (22), which are linked through (21). In order to solve (22), we adapt a recent approach developed in [15] to a new situation involving both partial differential and q−difference operators. We seek for solutions of (22) of the special form for some appropriate function ω(τ, m, ) and γ ∈ R. We refer to Section 2.1 for the definitions of the elements involved in the previous expression. Let us consider a second auxiliary equation: We define the polynomial P m (τ ) by whose factorization is given by .
Let d ∈ R be such that the infinite sector S d of bisecting direction d satisfies the following geometric construction: there exists D(0, ρ). The previous is a feasible condition for an appropriate choice of small enough η Q,R D 1 D 2 > 0 and large enough r Q,R D 1 D 2 > 0, in view of (14) and the definition of q (m). More precisely, one chooses S d such that q (m)/τ has positive distance to 1, for all 0 ≤ ≤ d D 1 +δ D 2 − 1, m ∈ R, and τ ∈ S d ∪ D(0, ρ). The previous choice of d yields for some C P > 0, valid for all τ ∈ S d ∪ D(0, ρ) and all m ∈ R.
4 Domains of existence for the solutions of (22), and associated estimates In this section, we describe appropriate domains on the time variables in which the solution of the main problem under consideration is well defined, within appropriate geometric conditions. LetT 1 be a bounded sector with vertex at the origin such that there exists δ 1 > 0 with for all r ≥ 0 and all T 1 ∈T 1 , and γ being an argument in S d . We also fix an unbounded sector T 2 , with vertex at the origin, such that for some δ 2 > 0 and well chosen γ ∈ arg(S d ). Observe that, in particular, there exists δ 3 > 0 with cos(γ − k 2 arg(T 2 )) > δ 3 . The next technical result describes accurate bounds for the solutions of (22) in different domains. The proof is left to Section 7.

Analytic solutions of the main problem: inner and outer solutions
In this section, we preserve the values of the elements involved in the main problem (17) stated in Section 3. More precisely, we assume (11)- (16), and also the hypotheses on the forcing term (19) in (18) and the coefficients in (20). Let d ∈ R and S d an infinite sector with vertex at 0 ∈ C under the geometric condition imposed in Proposition 6. Our main aim is to construct analytic solutions of (17) and their asymptotic behavior in different domains. For this purpose, we consider the analytic solutions as stated in Section 3.1. Such solutions are defined in families of sectors with respect to the perturbation parameter, conforming good coverings of C (see Definition 5). We also provide information about the difference of two solutions in consecutive sectors of the good covering, which will be crucial to determine the asymptotic behavior of the analytic solutions. We refer to consecutive solutions to solutions which are associated to consecutive elements in a fixed good covering of C . Let us first recall the notion of good covering in C .
, for some neighborhood of the origin U.
A family of sectors (E h ) 0≤h≤ι−1 under these assumptions is known as a good covering in C .
LetT 1 andT 2 be sectors following the construction in Section 4.

Definition 6
Let (E h ) 0≤h≤ι−1 be a good covering in C . We also fix a bounded sector, T 1 , and an unbounded sector T 2 , both with vertex at the origin. For all 0 ≤ h ≤ ι−1, let S d h be an infinite sector of bisecting direction d h ∈ R. We say that the set admissible if the following conditions hold: • For every 0 ≤ h ≤ ι − 1, ∈ E h and t 1 ∈ T 1 we have λ 1 t 1 ∈T 1 .
Theorem 1 Let (E h ) 0≤h≤ι−1 be a good covering in C . For every 0 ≤ h ≤ ι−1 we choose d h ∈ R such that S d h satisfies the geometric conditions of Proposition 6. Let {(S d h ) 0≤h≤ι−1 , T 1 × T 2 } be a family of sectors associated to the good covering (E h ) 0≤h≤ι−1 . If there exist small enough then for every 0 ≤ h ≤ ι − 1 the problem (17) admits a solution u h (t, z, ), which defines a bounded and holomorphic function in T 1 × T 2 × H β × E h , for any fixed 0 < β < β.

Inner solutions of the main problem
Let χ ∞ 2 be a bounded domain, such that the good covering (E ∞ h 1 ) 0≤h 1 ≤ι 1 −1 satisfies the following condition: then we say that u h 1 (t, z, ) represents an inner solution of (17).
Theorem 2 Under the assumptions of Theorem 1 and the constraints on the inner solutions of the main problem of Definition 8, let (E ∞ h 1 ) 0≤h 1 ≤ι 1 be a good covering of C . Then, there exists C inn > 0 such that for every , t 2 ∈ T 2, ,µ 2 , z ∈ H β for any fixed 0 < β < β and t 1 ∈ T 1 , one has for someĈ 1 ,Ĉ 2 ,Ĉ 3 > 0, ∆ 1 ∈ R and a real number k 1 ∈ (0, k 1 ) chosen small enough.

Outer solutions of the main problem
Definition 9 Let ι 2 ≥ 2 and (E 0 h 2 ) 0≤h 2 ≤ι 2 be a good covering of C , and consider an admissible Assume that t 2 ∈ T 2 is such that |t 2 | < ρ 2 for some fixed ρ 2 > 0 which is independent of , then we say that u h 2 (t, z, ) represents an outer solution of (17).
Theorem 3 Under the assumptions of Theorem 1 and the constraints on the outer solutions of the main problem of Definition 9, let (E 0 h 2 ) 0≤h 2 ≤ι 2 be a good covering of C . Then, there exists , t 2 ∈ T 2 with |t 2 | < ρ 2 , z ∈ H β for any fixed 0 < β < β and t 1 ∈ T 1 , one has for someĈ 4 ,Ĉ 5 > 0 and ∆ 2 ∈ R.
Proof Let 0 ≤ h 2 ≤ ι 2 − 1 and consider consecutive solutions u h 2 , u h 2 +1 of (17), constructed in Theorem 1. We proceed to write the difference of two consecutive solutions in the form for all 0 ≤ h 2 ≤ ι 2 − 1, with h 1 substituted by h 2 in the expressions of J 1 , J 2 , J 3 in the proof of Theorem 2. Let ∈ E 0 h 2 ∩ E 0 h 2 +1 and t 2 ∈ T 2 , with |t 2 | < ρ 2 , z ∈ H β for some 0 < β < β and t 1 ∈ T 1 . Owing to analogous bounds as those leading to (50) and (51), we arrive at for someC 10 > 0. Similar estimates hold for |J 2 |. On the other hand, direct computations yield for someC 11 > 0. This concludes the proof. 2

Asymptotic expansions of mixed order
This section is divided in two parts. The first part recalls some facts about q−asymptotic expansions and also describes q−asymptotic expansions which show a sub-Gevrey growth in their estimates, and related results.
In the second part of this section, we provide the existence of a formal solution of the main problem under study, written as a formal power series in the perturbation parameter, and explain the asymptotic relationship between this and the analytic solutions.

Review on q-asymptotic expansions
In the whole subsection, (F, · ) stands for a complex Banach space.
We first recall the notion of q−Gevrey asymptotic expansions, which can be also found in [12] in more detail.
Definition 10 Let V be a bounded open sector with vertex at 0 in C, q ∈ R with q > 1. We also fix a positive integer k. We say that a holomorphic function f : V → F admits the formal power seriesf ( ) = n≥0 f n n ∈ F for every ∈ U and N ≥ 0.
Such pure q−asymptotic expansions have been recently studied when dealing with the asymptotic behavior of the solutions of q−difference-differential equations in the complex domain [5,11,12,19].
We recall the definition of asymptotic expansion of mixed order as introduced in the work [20]. Observe that the set of functions admitting q−Gevrey asymptotic expansions of order 1/k coincide with the functions admitting ((q, k); 0)−Gevrey asymptotic expansion.
We now proceed to state a ((q, k); s)−version of the Ramis-Sibuya Theorem. The classical statement of this result involves Gevrey asymptotic expansions, and guarantee s-summability of some power seires. This cohomological criterion can be found in Proposition 2 [1] and Lemma XI-2-6 [9]. This version has already been stated in the work [20] where a complete proof can be found therein.
Theorem 4 (((q, k); s)−Ramis-Sibuya Theorem) Let (E h ) 0≤h≤ι−1 be a good covering in C . For every 0 ≤ h ≤ ι − 1 we consider a holomorphic function G h : E h → F and define ∆ h := G h+1 − G h , which turns out to be a holomorphic function in Z h := E h ∩ E h+1 (here, E ι and G ι stand for E 0 and G 0 , respectively). Assume that the following statements hold: • G h is a bounded function in a vicinity of 0, for every 0 ≤ h ≤ ι − 1.
Then, there existsĜ ∈ F[[ ]] which is the common ((q, k); s)−Gevrey asymptotic expansion of In [19], a q−Gevrey version of Ramis-Sibuya is obtained. That version is related to q−Gevrey asymptotic expansions of some positive order k, which coincide with (q, k); 0)−Gevrey asymptotic expansions in our framework. Ramis-Sibuya theorem in that framework reads as follows: Theorem 5 ((q-RS)) Let (F, · F ) be a Banach space and (E h ) 0≤h≤ι−1 be a good covering in C . For every 0 ≤ h ≤ ι − 1, let G h ( ) be a holomorphic function from E h into F and let the cocycle ∆ h ( ) = G h+1 ( ) − G h ( ) be a holomorphic function from Z h = E h+1 − E h into F (with the convention that E ι = E 0 and G ι = G 0 ). We make further assumptions: 1. The functions G h ( ) are bounded as tends to 0 on E h , for all 0 ≤ h ≤ ι − 1.

The function
for all ∈ Z h , and all 0 ≤ h ≤ ι − 1.
Then, there exists a formal power seriesĜ( ) ∈ F[[ ]] which is the common q−Gevrey asymptotic expansion of order 1/k of the function G h ( ) on E h , for all 0 ≤ h ≤ ι − 1.

Asymptotic expansions for the analytic solutions of the main problem
In this section, we preserve all the assumptions made on the elements involved in the main problem (17), detailed in Section 3. Moreover, we depart from the geometric construction of the elements used to construct the analytic solutions of (17), described in Sections 4 and 5, and under Assumption (31). The next result shows that the difference of two consecutive solutions of the main problem allow the application of the ((q, k); s)−version of Ramis-Sibuya Theorem, obtained in Theorem 4, in adequate domains.
Lemma 3 Let k 1 , k 1 , k 2 , λ 1 , λ 2 , µ 2 be positive constants with µ 2 > λ 2 . For every n ∈ N, we consider the function Then, it holds that for n ≥ n 0 , where n 0 ≥ 1 is a large enough integer depending on k 1 , k 1 , k 2 , λ 1 , λ 2 , µ 2 with Proof We make the change of variable τ = log(x) and consider the auxiliary function Ψ(τ ) := Ψ(e τ ). We search for the maximum of Ψ for τ ∈ R. It holds that It is known that the solution of the equation log(t) + log(b)t = log(a), for some fixed a, b > 0, is given by t = W (a log(b))/ log(b), where W is the Lambert W function. The maximum of Ψ(τ ) is then attained at where W −1 stands for the −1-branch of Lambert W function. Let Then τ n can be written in the form τ n = −1 which can be applied to estimate τ n : leading to We conclude that Ψ(τ ) ≤ Ψ(τ n ). Taking into account (41) we arrive at for every 0 < ξ < 1, and someĈ 1 ,Â 1 > 0. The definition of A, C in (40) yields which defines a holomorphic and bounded function on E ∞ h 1 , with values in F 1 . Then, there exist a formal seriesû ∞ ( ) ∈ F 1 [[ ]] such that for all 0 ≤ h 1 ≤ ι 1 − 1, the function (43) admitsû ∞ ( ) as its ((q, K); S)−Gevrey asymptotic expansion on E ∞ h 1 , for K and S stated in (39).
Proof Taking into account (36) in Theorem 3, and Theorem 5, we get the existence of a formal power seriesû 0 ( ) ∈ F 2 [[ ]] which is the common q-Gevrey asymptotic expansion of the function (44), as a function on E 0 h 2 with values in F 2 . This holds for all 0 ≤ h 2 ≤ ι 2 − 1. 2

Proof of Proposition 7
In this section, we give proof of the technical Proposition 7. We consider U γ (T , m, ) is constructed in the form (23).
For the first part of the proof, we take into account the property (6) on Jacobi Theta function, and Proposition 6, to arrive at We split I(|T 1 |, |T 2 |) into the sum of I 1 (|T 1 |, |T 2 |) and I 2 (|T 1 |, |T 2 |), where the first element is associated to the integration in (0, ρ) and the second is concern with the integration restricted to (ρ, ∞), for some ρ > 0. We study each part of the splitting: We have for some C 3 > 0, which after the change of variable r = |T 1 |r equals On the other hand, we assume ρ, ρ 1 > 0 are such that ρ ≥ ρ 1 . Then, the positive function φ : [ρ, ∞) → R defined by φ(r) = log 2 (r/|T 1 |) is monotone increasing on [ρ, ∞) for any choice of |T 1 | < ρ 1 . Therefore, (46) Let x = |T 2 | k 2 /δ 3 . We make the change of variable r = xr in the last integral and arrive at log 2 (xr + δ) + α log(xr + δ) exp (−r) xdr.
Taking into account that we get that I 2.1 (x) = I 2.2 (x) + I 2.3 (x), where the splitting is done on the integral by cutting the integration path into (0, 1 − δ/x) and (1 − δ/x, ∞) for I 2.2 (x) and I 2.3 (x), respectively. for some C 7 > 0. The first statement of Proposition 7 holds. We give proof for the second statement. The first arguments in the proof of the first statement can be followed word by word up to the splitting of I(|T 1 |, |T 2 |) into I 1 (|T 1 |, |T 2 |) + I 2 (|T 1 |, |T 2 |). The quantity I 1 (|T 1 |, |T 2 |) is upper bounded by a constant for every |T 1 | > 0 and |T 2 | > 0. We now proceed to give upper estimates on I 2 (|T 1 |, |T 2 |). Let us choose ρ 1 , ρ > 0 such that φ(r) is monotone increasing on [ρ, ∞). It holds that (50) for some C 8 > 0. The conclusion follows from this last upper bound.