On parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems

Transcription

1 On parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems A. Lastra, S. Malek University of Alcalá, Departamento de Física y Matemáticas, Ap. de Correos 2, E-2887 Alcalá de Henares (Madrid), Spain, University of Lille, Laboratoire Paul Painlevé, Villeneuve d Ascq cedex, France, alberto.lastra@uah.es Stephane.Malek@math.univ-lille.fr January, 7 24 Abstract We study a nonlinear initial value Cauchy problem depending upon a complex perturbation parameter ɛ with vanishing initial data at complex time t = and whose coefficients depend analytically on (ɛ, t) near the origin in C 2 and are bounded holomorphic on some horizontal strip in C w.r.t the space variable. This problem is assumed to be non-kowalevskian in time t, therefore analytic solutions at t = cannot be expected in general. Nevertheless, we are able to construct a family of actual holomorphic solutions defined on a common bounded open sector with vertex at in time and on the given strip above in space, when the complex parameter ɛ belongs to a suitably chosen set of open bounded sectors whose union form a covering of some neighborhood Ω of in C. These solutions are achieved by means of Laplace and Fourier inverse transforms of some common ɛ depending function on C R, analytic near the origin and with exponential growth on some unbounded sectors with appropriate bisecting directions in the first variable and exponential decay in the second, when the perturbation parameter belongs to Ω. Moreover, these solutions satisfy the remarkable property that the difference between any two of them is exponentially flat for some integer order w.r.t ɛ. With the help of the classical Ramis-Sibuya theorem, we obtain the existence of a formal series (generally divergent) in ɛ which is the common Gevrey asymptotic expansion of the built up actual solutions considered above. Key words: asymptotic expansion, Borel-Laplace transform, Fourier transform, Cauchy problem, formal power series, nonlinear integro-differential equation, nonlinear partial differential equation, singular perturbation. 2 MSC: 35C, 35C2. The author is partially supported by the project MTM of Ministerio de Ciencia e Innovacion, Spain The author is partially supported by the french ANR--JCJC 5 project and the PHC Polonium 23 project No. 2827SG.

2 2 Introduction In this paper, we consider a family of parameter depending nonlinear initial value Cauchy problems of the form () Q( z )( t u(t, z, ɛ)) = (Q ( z )u(t, z, ɛ))(q 2 ( z )u(t, z, ɛ)) + D l= ɛ l t d l δ l t R l( z )u(t, z, ɛ) + c (t, z, ɛ)r ( z )u(t, z, ɛ) + f(t, z, ɛ) for given vanishing initial data u(, z, ɛ), where D 2, l, d l, δ l, l D are integers which satisfy the inequalities = δ, δ l < δ l+, d D = (δ D )(k + ), D = d D δ D +, d l > (δ l )(k + ), δ D δ l + 2 k, l + k( δ D ) + for all l D and for some integer k. Besides, Q(X), Q (X), Q 2 (X), R l (X), l D are polynomials submitted to the constraints deg(q) deg(r D ) deg(r l ), deg(r D ) deg(q ), deg(r D ) deg(q 2 ), Q(im), R D (im) for all m R, all l D. The coefficient c (t, z, ɛ) and the forcing term f(t, z, ɛ) are bounded holomorphic functions on a product D(, r) H β D(, ɛ ), where D(, r) (resp. D(, ɛ )) is a disc centered at with small radius r > (resp. ɛ > ) and H β = {z C/ Im(z) < β} is some strip of width β >. In order to avoid cumbersome statements and to improve the readability of the computations, we have restricted our study to a quadratic nonlinearity and monomial coefficients in t in front of the derivatives with respect to t and z but the method described here can also be extended to higher order nonlinearities, with polynomial coefficients w.r.t t in the linear part on the right handside of the equation (). This work can be seen as a continuation of the study described in [23] where the second author has studied nonlinear integro-differential initial values problems with the shape (2) R( z )P ( t, z )Y (t, z) = t b(t s, z) s z Y (s, z)ds + t s z Y (t s, z) s 2 z Y (s, z)ds where R(X) C[X], P (T, X) C[T, X] and s, s, s 2 are non negative integers. The coefficient b(t, z) = k I b k(z)t k is a polynomial in t and its coefficients b k (z) are Fourier inverse transform of some function b k (m) belonging to a Banach space E (β,µ) of continuous functions h : R C endowed with the norm h(m) (β,µ) = sup m R ( + m ) µ exp(β m ) h(m) and define bounded holomorphic functions on any strip H β, < β < β. The initial conditions are defined by Y (, z) = Y (z), ( j t Y )(, z), for all j deg T P (T, X), where Y is also assumed to be the Fourier inverse transform of some Y (m) belonging to E (β,µ). We focused on the case when the degree of R(X)P (T, X) with respect to T is smaller than its degree in X. In that case the classical Cauchy-Kowalevski theorem (see [2]) cannot be applied and the unique formal power series solution Ŷ (t, z) = l Y l(z)t l, with coefficients belonging to the Banach space of bounded holomorphic functions on H β equipped with the sup norm, is in general divergent. Nevertheless, under suitable constraints on the roots of the polynomial T P (T 2, im) and for sufficiently small data b k (β,µ), Y (β,µ), one can construct by means of classical Borel-Laplace

3 3 procedure and Fourier inverse transform an actual holomorphic solution Y (t, z) on C + H β of (2) for the given initial data (C + denotes the set of complex numbers t such that Re(t) > ), which possess the formal series Ŷ as Gevrey asymptotic expansion of order as t tends to, meaning that for any compact subsector W C + centered at, there exist constants C, M > with n sup Y (t, z) Y l (z)t l CM n n! t n z H β l= for all n, all t W. Compared to the work [23], the problem () now involves an additional complex parameter ɛ. Provided that δ D + deg(r D ) > deg(q) + holds, the problem () is singularly perturbed in the parameter ɛ and belongs to a class of so-called PDEs with irregular singularity at t = in the sense of [25]. In the paper [22], the second author has already considered a similar problem of the form (3) ɛt 2 t S z X p (t, z, ɛ) = F (t, z, ɛ, t, z )X p (t, z, ɛ) + P (t, z, ɛ, X p (t, z, ɛ)) for given initial data (4) ( j zx p )(t,, ɛ) = φ j,p (t, ɛ), p ς, j S, where S, ς 2 are some positive integers, F is some differential operator with polynomial coefficients and P a polynomial. The initial data φ j,p (t, ɛ) were assumed to be holomorphic on products T E p C 2 for some sector T centered at and where E = {E p } pς denotes a family of open bounded sectors with aperture larger than π which form a so-called good covering in C, meaning that E p E p+ for all p ς (with the convention that E ς = E ) with the property that the intersection of any three different elements in {E p } pς is empty and that ς p= E p = U \ {}, where U is some neighborhood of in C. Under convenient assumptions on the shape of the equation (3) and on the initial data (4), the existence of a formal series ˆX(t, z, ɛ) = k h k(t, z)ɛ k /k! solution of (3) is established with coefficients h k (t, z) belonging to the Banach space F of bounded holomorphic functions on T D(, δ) (for some δ > small enough) equipped with the sup norm. This formal series ˆX(t, z, ɛ) is the Gevrey asymptotic expansion of order of actual holomorphic solutions X p (t, z, ɛ) of (3), (4) on E p as F valued functions, for all p ς, in other words for any closed subsector W E p centered at, there exist constants C, M > such that n sup X p (t, z, ɛ) h k (t, z)ɛ k /k! CM n n! ɛ n t T,z D(,δ) k= for all n, all ɛ W. In this work we address the same queries as in [22], [23], namely our main purpose is the construction of actual holomorphic solutions u p (t, z, ɛ) to the problem () on domains T H β E p using some Borel-Laplace procedure and Fourier inverse transform and the analysis of their asymptotic expansions as ɛ tends to. More specifically, we can present our main statements as follows. Main results Assume the existence of an unbounded sector S Q,RD = {z C/ z r Q,RD, arg(z) d Q,RD η Q,RD } with direction d Q,RD R, aperture η Q,RD > and radius r Q,RD > such that the quotient Q(im)/R D (im) belongs to S Q,RD for all m R. This sector S Q,RD is prescribed in such a

4 4 way that there exists a set of adequate directions d p R, p ς, with the feature that the distinct complex roots q l (m), l (δ D )k, of the polynomial P m (τ) = Q(im)k R D (im)k δ Dτ (δd )k fulfill estimates of the form : there exist constants M, M 2 > such that τ q l (m) M ( + τ ), τ q l (m) M 2 q l (m) for all l (δ D )k, some integer l {,..., (δ D )k }, for all m R, all τ S dp D(, ρ), for some well chosen unbounded sectors S dp centered at with direction d p and for some radius ρ >. Then, we choose a family E = {E p } pς of sectors with aperture slightly larger than π/k which defines a good covering of C and we take an open bounded sector T centered at such that for every p ς, the product ɛt belongs to a sector with direction d p and aperture slightly larger than π/k, for all ɛ E p, all t T. We make the assumption that the coefficient c (t, z, ɛ) and the forcing term f(t, z, ɛ) can be written as convergent series of the special form c (t, z, ɛ) = c,n (z, ɛ)(ɛt) n, f(t, z, ɛ) = f n (z, ɛ)(ɛt) n, n n on a domain D(, r) H β D(, ɛ ) (where H β is a strip of width β ) such that T D(, r), pς E p D(, ɛ ) and < β < β are given positive real numbers. The coefficients c, (z, ɛ), c,n (z, ɛ) and f n (z, ɛ), n, are supposed to be inverse Fourier transform of functions m C, (m, ɛ), m C,n (m, ɛ) and m F n (m, ɛ) that belong to the Banach space E (β,µ) for some µ > max(deg(q ) +, deg(q 2 ) + ) and that depend holomorphically on ɛ in D(, ɛ ). Our first result stated in Theorem claims that if the norm C, (m, ɛ) (β,µ) and the radius ɛ are chosen small enough and if the radius r Q,RD is taken sufficiently large then we can construct a family of holomorphic bounded functions u p (t, z, ɛ), p ς, defined on the products T H β E p, which solves the problem () with vanishing initial data u p (, z, ɛ) and which can be written as Laplace-Fourier transform k + u p (t, z, ɛ) = (2π) /2 ω dp k (u, m, ɛ)e ( u ɛt )k e izm du L γp u dm. where the inner integration is made along some halfline L γp S dp where ω dp k (u, m, ɛ) denotes a function with at most exponential growth of order k in u/ɛ and exponential decay in m R which satisfies more precisely estimates of the form ω dp k (u, m, ɛ) C( + m ) µ e β m u ɛ + u ɛ 2k exp(ν u ɛ k ) for some constants C, ν >, for all m R, all u S dp D(, ρ), all ɛ D(, ɛ ) \ {}. Our second main result, described in Theorem 2, asserts that the functions u p, p ς, turn out to be the k sums on E p of a common formal power series û(t, z, ɛ) = m h m (t, z) ɛm m! F[[ɛ]] where F is the Banach space of bounded holomorphic functions on T H β equipped with the sup norm. Namely, for any closed subsector W E p centered at, there exist constants C, M > such that sup t T,z H β u p (t, z, ɛ) n m= h m (t, z) ɛm m! CM n Γ( + n k ) ɛ n

5 5 for all n, all ɛ W. It is worth remarking that when deg(q)+ > δ D +deg(r D ), the equation () is not singularly perturbed in ɛ and possess no irregular singularity at t =. However, the asymptotic expansion û of u p as ɛ tends to on E p remains divergent in general. The reason for this phenomenon to appear relies on the way one constructs the actual solutions u p as Laplace transforms of order k in the new variable ɛt and from the fact that for any fixed ɛ D(, ɛ ) \ {}, the problem () is not Kowalevskian with respect to t at (meaning that formal series solutions ˆv(t, z, ɛ) = n v n(z, ɛ)t n, with coefficients z v n (z, ɛ) bounded holomorphic on H β, are in general divergent, as a consequence of Propositions 8 and 9) as it was already the case in our previous paper [23]. The Cauchy problem () we consider here comes within the new trend of research concerning Borel-Laplace summability procedures applied to partial differential differential equation going back to the seminal work of D. Lutz, M. Miyake and R. Schäfke on the linear complex heat equation, see [9]. We quote below some important results in this field not pretending to be exhaustive. This construction of Borel-Laplace k summable or even multi-summable formal series solutions has been extended to general linear PDEs in two complex variables with constant coefficients by W. Balser in [3] and [4] provided that their initial data are analytic functions near the origin that can be analytically continued with exponential growth on some unbounded sectors. A similar result has been obtain for the so-called fractional linear PDEs with noninteger derivatives by S. Michalik, see [24]. Latter on, linear complex heat like equations with variable coefficients have been explored by several authors, see [5], [7], [2]. Recently, general linear PDEs with time dependent coefficients taking for granted that their initial data are entire functions in C N, N, have been investigated by H. Tahara and H. Yamazawa in [28]. In the context of nonlinear PDEs, we mention the work [2] of G. Lysik who constructed summable formal solutions of the one dimensional Burgers equations with the help of the so-called Cole- Hopf transform. We also point out that O. Costin and S. Tanveer have constructed summable formal series in time variable to the celebrated 3D Navier Stokes equations in [9]. We also refer to the work of S. Ouchi who constructed multisummable formal solutions to nonlinear PDEs which come from perturbations of ordinary differential equations, see [26]. We also mention the fact that, these last years, a lot of attention has been payed to singularly perturbed PDEs in the complex domain partly motived by a conjecture of B. Dubrovin which concerns the question of universal behaviour of generic solutions near gradient catastrophe of singularly Hamiltonian perturbations of first order hyperbolic equations, see []. In this active direction, we refer namely to the works of B. Dubrovin and M. Elaeva who investigated the case of generalized Burgers equations in [] and of T. Claeys and T. Grava in [6] who solved the problem for KdV equations. We indicate the recent important studies of T. Koike on Garnier systems, [5], [6] and of S. Hirose on the reduction of general singularly perturbed holonomic systems in two complex variables to Pearcy systems normal forms, [3]. In the sequel, we explain our principal intermediate key results and the arguments needed in their proofs. In a first part, we depart from an auxiliary parameter depending initial value differential and convolution equation which is singular in its perturbation parameter ɛ at, see (72). This equation is formally constructed by making the change of variable T = ɛt in the equation () (as done in our previous works [22], [7]) and by taking the Fourier transform with respect to the variable z. Under the constraint (7) and the assumption that d l δ l, l D (which follows from the hypothesis (69)) we can construct a formal power series solution Û(T, m, ɛ) = n U n(m, ɛ)t n of (72) whose coefficients m U n (m, ɛ) depend holomorphically on ɛ C near the origin and belong to a Banach space E (β,µ) of continuous

6 6 function with exponential decay on R introduced in the paper [9] by O. Costin and S. Tanveer. This series turns out to be in general divergent as we will see below. In the next step, we follow the strategy developped recently by H. Tahara and H. Yamazawa in [28], namely we multiply each hand side of (72) by the power T k which transforms it into an equation (76) which involves only differential operators in T of irregular type at T = of the form T β T with β k + due to our assumption (69) on the shape of the equation (72). Then, we apply a formal Borel transform of order k (defined as a slightly modified version of the classical Borel transform of order k from the reference book []), that we call m k Borel transform in Definition 3, to the formal series Û with respect to T, denoted ω k (τ, m, ɛ) = n U n (m, ɛ) τ k Γ( n k ). From the commutation rules of the m k Borel transform with respect to the weighted convolution product of formal series (introduced in Proposition 5) and the differential operators T β T for β k + described in Proposition 6, we get that ω k (τ, m, ɛ) formally solves a convolution equation in both variables τ and m, see (8). Under some size constraint on the E (β,µ) norm of the constant term C, of one coefficient of the equation (8) and for all ɛ C close enough to, we show that ω k (τ, m, ɛ) is actually convergent for τ on some fixed neighborhood of and can be extended to a holomorphic functions ω d k (τ, m, ɛ) on unbounded sectors S d centered at zero with bisecting direction d and tiny aperture provided that S d stays away from the roots of some polynomial P m (τ), for all m R. Besides, the function ω d k (τ, m, ɛ) satisfies estimates of the form : there exist constants ν > and ϖ d > with ω d k (τ, m, ɛ) ϖ d( + m ) µ e β m τ ɛ + τ ɛ 2k exp(ν τ ɛ k ) for all τ S d, m R, all ɛ C near the origin (see Proposition 9). The technical constraints (69) and (87) together with (8), (84) and (85) allow, by means of lower bound estimates (86) for the polynomial P m (τ), the transformation of equation (8) into a fixed point equation H ɛ (ω k ) = ω k where the map H ɛ is given by (89) for which we can find a solution ωk d in some Banach space of holomorphic functions F(ν,β,µ,k,ɛ) d studied in Section 2. It is worth noting that the formal series Û(T, m, ɛ) diverges since the function ω k(τ, m, ɛ) cannot in general be extended everywhere on C w.r.t τ. But, as a result, we get that these series Û are m k summable w.r.t T (see Definition 3) in all the directions d chosen as above. In other words, some Laplace transform of order k of ωk d denoted U d (T, m, ɛ) can be constructed for all T belonging to a sector S d,k,h ɛ with bisecting direction d, aperture slightly larger than π/k and radius h ɛ (for some h > ). This function T U d (T, m, ɛ) is the unique E (β,µ) valued map which admits Û(T, m, ɛ) as Gevrey asymptotic expansion of order /k on S d,k,h ɛ. Moreover, U d (T, m, ɛ) solves the auxiliary problem (72) with vanishing initial data U d (, m, ɛ), see Proposition. In Theorem, we construct a family of actual bounded holomorphic solutions u p (t, z, ɛ), p ς of our original problem () on domains of the form T H β E p. The sectors E p, p ς constitute a so-called good covering in C (Definition 4). The strip H β has width < β < β and T is a fixed bounded sector centered at which fulfills the constraint ɛt S dp,k for all ɛ E p, t T, and S dp,k is a sector of bisecting direction d p and aperture slightly larger than π/k where d p are suitable directions for which the unbounded sectors S dp with small aperture and bisecting direction d p satisfy the restrictions described above. Namely, the functions u p are set as Fourier inverse transforms of U dp, u p (t, z, ɛ) = F (m U dp (ɛt, m, ɛ))(z)

7 7 where the definition of F is pointed out in Proposition 7. In addition to that, one can prove that the difference of any two neighboring functions u p+ (t, z, ɛ) u p (t, z, ɛ) tends to zero as ɛ on E p E p+ faster than a function with exponential decay of order k, uniformly w.r.t. t T and z H β, see (9). The last section of the paper is devoted to deal with this latter growth information in order to show the existence of a common asymptotic expansion û(t, z, ɛ) = m h m(t, z)ɛ m /m! of Gevrey order /k for all the functions u p (t, z, ɛ) as ɛ tends to on E p, uniformly w.r.t. t T and z H β, see Theorem 2. The key tool in proving the result is the classical Ramis-Sibuya theorem (Theorem (RS)). The layout of this work reads as follows. In Section 2, we define some weighted parameter depending Banach spaces of continuous functions on C R with exponential growth on sectors w.r.t the first variable and exponential decay on R w.r.t the second one. We study the continuity properties of several kind of linear and nonlinear integral operators acting on these spaces that will be useful in Section 4. In Section 3, we give a definition of k summability (that we call m k summability) which is a minor modification of the classical one given in the textbook [] and which is appropriate for the problem we have to deal with. We also give conditions for the set of m k sums of formal series to be a differential algebra. This fact will be important in the next section where we construct actual solutions of the auxiliary equation (72). We provide explicit commutation formulas for the m k Borel transform w.r.t products and differential operators of irregular type. In Section 4, we introduce an auxiliary differential and convolution problem (72) for which we construct a formal solution. We show that the m k Borel transform of this formal solution satisfies a convolution problem (8). Under suitable assumptions, we can solve uniquely this latter problem in the Banach spaces described in Section 2 using some fixed point theorem argument. Then, applying Laplace transform, we can give a uniquely determined actual solution to (72) having the formal solution mentioned above as Gevrey asymptotic expansion. In Section 5, with the help of Section 4, we build a family of actual holomorphic solutions to our initial Cauchy problem () on a full neighborhood of the origin in C w.r.t the perturbation parameter ɛ. We show that the difference of any two neighboring solutions is exponentially flat for some integer order in ɛ (Theorem ). In Section 6, we show that the actual solutions constructed in Section 5 share a common formal series as Gevrey asymptotic expansion as ɛ tends to on sectors (Theorem 2). The result relies on the classical so-called Ramis-Sibuya theorem. 2 Banach spaces functions with exponential growth and decay We denote by D(, r) the open disc centered at with radius r > in C and by D(, r) its closure. Let S d be an open unbounded sector in direction d R and E be an open sector with finite radius r E, both centered at in C. By convention, these sectors do not contain the origin in C. Definition Let ν, β, µ > and ρ > be positive real numbers. Let k be an integer and let ɛ E. We denote F(ν,β,µ,k,ɛ) d the vector space of continuous functions (τ, m) h(τ, m) on ( D(, ρ) S d ) R, which are holomorphic with respect to τ on D(, ρ) S d and such that h(τ, m) (ν,β,µ,k,ɛ) = sup ( + m ) µ + τ ɛ 2k τ D(,ρ) S d,m R τ ɛ exp(β m ν τ ɛ k ) h(τ, m)

8 8 is finite. One can check that the normed space (F d (ν,β,µ,k,ɛ),. (ν,β,µ,k,ɛ)) is a Banach space. Remark: These norms are appropriate modifications of the norms defined by O. Costin and S. Tanveer in [9] and by the second the author in [22] and [23]. Throughout the whole section, we assume ɛ E, µ, β, ν > are fixed. In the next lemma, we check the continuity property by multiplication operation with bounded functions. Lemma Let (τ, m) a(τ, m) be a bounded continuous function on ( D(, ρ) S d ) R, which is holomorphic with respect to τ on D(, ρ) S d. Then, we have ( ) (5) a(τ, m)h(τ, m) (ν,β,µ,k,ɛ) for all h(τ, m) F d (ν,β,µ,k,ɛ). sup a(τ, m) τ D(,ρ) S d,m R h(τ, m) (ν,β,µ,k,ɛ) In the next proposition, we study the continuity property of some convolution operators acting on the latter Banach spaces. Proposition Let γ 2 > be a real number. Let k be an integer such that /k γ 2. Then, there exists a constant C > (depending on ν, k, γ 2 ) with (6) for all f(τ, m) F d (ν,β,µ,k,ɛ). (τ k s) γ 2 f(s /k, m) ds s (ν,β,µ,k,ɛ) C ɛ kγ 2 f(τ, m) (ν,β,µ,k,ɛ) Proof Let f(τ, m) F d (ν,β,µ,k,ɛ). For any τ D(, ρ) S d, the segment [, τ k ] is such that the map s [, τ k ] f(s /k, m) is well defined, provided that m R. By definition, we have that (7) (τ k s) γ 2 f(s /k, m) ds s (ν,β,µ,k,ɛ) = sup ( + m ) µ + τ ɛ 2k τ D(,ρ) S d,m R τ ɛ exp(β m ν τ ɛ k ) {( + m ) µ e β m exp( ν s / ɛ k ) + s 2 ɛ 2k s /k ɛ f(s /k, m)} A(τ, s, m, ɛ)ds where Therefore, A(τ, s, m, ɛ) = ( + m ) µ e β m exp(ν s / ɛ k ) + s 2 ɛ 2k s /k ɛ (τ k s) γ 2 s (8) (τ k s) γ 2 f(s /k, m) ds s (ν,β,µ,k,ɛ) C (ɛ) f(τ, m) (ν,β,µ,k,ɛ)

9 9 where τ + ɛ C (ɛ) = sup 2k τ D(,ρ) S d τ ɛ exp( ν τ ɛ k ) τ k exp(νh/ ɛ k ) + h2 ɛ 2k h k ɛ ( τ k h) γ 2 dh Making the change of variable h = ɛ k h in the integral inside C (ɛ) yields (9) C (ɛ) = ɛ kγ τ + 2 ɛ sup 2k τ D(,ρ) S d τ ɛ exp( ν τ ɛ k ) where A(x) = + x2 x /k τ ɛ k For any x >, we have A(x) Ã(x), where Using L Hospital rule, we know that lim Ã(x) = x + lim x + exp(νh ) + h 2 (h ) k ( τ ɛ k h ) γ 2 dh ɛ kγ 2 sup x x exp( νx) exp(νh) + h 2 h k (x h) γ 2 dh Ã(x) = ( + x 2 )x γ 2 k exp( νx) x exp(νx)x k /( + x 2 ) x ( exp(νx) (+x 2 )x γ 2 k = lim x + ) exp(νh) + h 2 h k dh. ( + x 2 )x 2(γ 2 k ) x k A(x) ν( + x 2 )x γ 2 k (2x γ 2 k + + (γ 2 k )xγ 2 k ( + x 2 )) and this latter limit is finite if γ 2 holds. Hence, we deduce that there exists a constant à > such that () sup Ã(x) à x Gathering the estimates (8), (9), (), we see that (6) holds. Proposition 2 Let γ and χ 2 > be real numbers. Let ν 2 be an integer. We consider a holomorphic function a γ,k(τ) on D(, ρ) S d, continuous on D(, ρ) S d, such that a γ,k(τ) ( + τ k ) γ for all τ D(, ρ) S d. i) Assume that χ 2. If ν 2 + χ 2 γ, then there exists a constant C 2. > (depending on ν, ν 2, χ 2, γ ) such that () a γ,k(τ) (τ k s) χ 2 s ν 2 f(s /k, m)ds (ν,β,µ,k,ɛ) C 2. ɛ k(+ν 2+χ 2 γ ) f(τ, m) (ν,β,µ,k,ɛ)

10 for all f(τ, m) F(ν,β,µ,k,ɛ) d. ii) Assume that χ 2 = χ k for some real number χ. If ν 2 + k γ, then there exists a constant C 2.2 > (depending χ, k, ν, γ, ν 2 ) on such that (2) a γ,k(τ) for all f(τ, m) F d (ν,β,µ,k,ɛ). (τ k s) χ 2 s ν 2 f(s /k, m)ds (ν,β,µ,k,ɛ) C 2.2 ɛ k(+ν 2+χ 2 γ ) f(τ, m) (ν,β,µ,k,ɛ) Proof In the first part of the proof, let us assume that i) holds. Let f(τ, m) F d (ν,β,µ,k,ɛ). By definition, we have (3) a γ,k(τ) (τ k s) χ 2 s ν 2 f(s /k, m)ds (ν,β,µ,k,ɛ) a γ,k(τ) = sup ( + m ) µ + τ ɛ 2k τ D(,ρ) S d,m R τ ɛ exp(β m ν τ ɛ k ) {( + m ) µ e β m exp( ν s / ɛ k ) + s 2 ɛ 2k s /k ɛ f(s /k, m)} B(τ, s, m, ɛ)ds where Therefore, B(τ, s, m, ɛ) = ( + m ) µ e β m exp(ν s / ɛ k ) + s 2 ɛ 2k s /k ɛ (τ k s) χ 2 s ν 2. (4) a γ,k(τ) where (τ k s) χ 2 s ν 2 f(s /k, m)ds (ν,β,µ,k,ɛ) C 2 (ɛ) f(τ, m) (ν,β,µ,k,ɛ) τ + ɛ C 2 (ɛ) = sup 2k τ D(,ρ) S d τ ɛ exp( ν τ ɛ k ) ( + τ k ) γ τ k exp(νh/ ɛ k ) + h2 ɛ 2k h k ɛ ( τ k h) χ 2 h ν 2 dh Making the change of variable h = ɛ k h in the integral inside C 2 (ɛ) yields (5) C 2 (ɛ) = ɛ k(+ν τ 2+χ 2 ) + ɛ sup 2k τ D(,ρ) S d τ ɛ exp( ν τ ɛ k ) ( + ɛ k τ ɛ k ) γ τ ɛ k exp(νh ) + h 2 (h ) k ( τ ɛ k h ) χ 2 h ν 2 dh ɛ k(+ν 2+χ 2 ) sup B(x, ɛ) x

11 where B(x, ɛ) = + x2 x /k exp( νx) x exp(νh) ( + ɛ k x) γ + h 2 h k +ν 2 (x h) χ 2 dh. For any x >, we get that B(x, ɛ) B(x, ɛ), where B(x, ɛ) = ( + x2 )x χ 2 ( + ɛ k x) γ exp( νx) x exp(νh) + h 2 hν 2 dh Let x >. From the inequality + ɛ k x, for all x [, x ] and ɛ E, there exists a constant B > such that (6) sup x [,x ],ɛ E B(x, ɛ) B. On the other hand, since + ɛ k x ɛ k x holds for all x and ɛ E, we get that B(x, ɛ) B 2 (x)/ ɛ kγ where (7) B2 (x) = ( + x 2 )x χ 2 γ exp( νx) for all x x. By L Hospital rule we get that lim B 2 (x) = x + lim x + x exp(νh) + h 2 hν 2 dh ( + x 2 )x 2(χ 2 γ ) x ν 2 ν( + x 2 )x χ 2 γ (2x χ 2 γ + + (χ 2 γ )x χ 2 γ ( + x 2 )) which is finite if we assume that ( + ν 2 + χ 2 γ ). We deduce that there exists a constant B 2 > such that (8) sup x x B(x, ɛ) ɛ kγ sup B2 (x) B 2 x x ɛ kγ Bearing in mind the estimates (4), (5), (6) and (8), we obtain (). In the second part of the proof, assume now that the condition ii) holds. Let f(τ, m). By definition, we have F d (ν,β,µ,k,ɛ) (9) a γ,k(τ) where B(τ, s, m, ɛ) = (τ k s) χ k s ν 2 f(s /k, m)ds (ν,β,µ,k,ɛ) a γ,k(τ) = sup ( + m ) µ + τ ɛ 2k τ D(,ρ) S d,m R τ ɛ exp(β m ν τ ɛ k ) e β m s exp(ν ( + m ) µ {( + m ) µ e β m exp( ν s / ɛ k ) + s 2 ɛ 2k {exp( ν τ k s ɛ k ) + ɛ k ) exp(ν τ k s ɛ k τ k s 2 ɛ 2k τ k s /k ɛ ) s /k ɛ s /k ɛ τ k s /k ɛ + s 2 ɛ 2k f(s /k, m)} (τ k s) χ k } B(τ, s, m, ɛ)ds (τ k s) s ν2. + τ k s 2 ɛ 2k

12 2 Hence, (2) a γ,k(τ) where (2) C 2.2 (ɛ) = sup x (τ k s) χ k s ν 2 f(s /k, m)ds (ν,β,µ,k,ɛ) C 2.2 (ɛ)c 2.3 (ɛ) f(τ, m) (ν,β,µ,k,ɛ) exp( ν x ɛ By using the classical estimates ɛ 2k k ) + x2 x /k ɛ x χ k, τ + ɛ C 2.3 (ɛ) = sup 2k τ D(,ρ) S d τ ɛ ( + τ k ) γ τ k h /k ( τ k h) /k ( τ k h) h ν2 dh. ɛ ɛ + h2 + ( τ k h) 2 ɛ 2k (22) sup x m exp( m 2 x) = ( m ) m e m x m 2 for any real numbers m and m 2 >, we get that ( (23) C 2.2 (ɛ) ɛ χ ( χ kν ) χ k e ( χ k ) + ( 2 + χ k ) ν χ 2+ k ɛ 2k ) e (2+ χ k ). Making the change of variable h = ɛ k h in the integral involved in the definition of C 2.3 (ɛ) yields τ + ɛ (24) C 2.3 (ɛ) = sup 2k τ D(,ρ) S d τ ɛ ( + ɛ k τ ɛ k ) γ τ ɛ k (h ) /k ( τ ɛ k h ) /k + (h ) 2 + ( τ ɛ k h ) 2 ( τ ɛ k h ) ɛ kν 2 (h ) ν 2 dh ɛ kν 2 sup B 2.3 (x, ɛ) x where B 2.3 (x, ɛ) = + x2 x x /k ( + ɛ k x) γ ( + h 2 )( + (x h) 2 h ) (x h) k +ν 2 dh. k For any x >, we have that B 2.3 (x, ɛ) B 2.3 (x, ɛ), where B 2.3 (x, ɛ) = + x 2 ( + ɛ k x) γ x ( + h 2 )( + (x h) 2 h ν 2 dh. ) (x h) k Let x >. From the inequality + ɛ k x, for all x [, x ], ɛ E, there exists a constant B 2.3 > such that (25) sup x [,x ],ɛ E B 2.3 (x, ɛ) B 2.3.

13 3 On the other hand, since + ɛ k x ɛ k x holds for all x and ɛ E, we get that (26) B2.3 (x, ɛ) B 2.4 (x) ɛ kγ where B 2.4 (x) = ( + x 2 )x γ x ( + h 2 )( + (x h) 2 h ν 2 dh ) (x h) k for all x x. Now, we make the change of variable h = xu in the integral inside B 2.4 (x). We can write B 2.4 (x) = ( + x 2 )x ν 2+ k γ F k (x) where F k (x) = u ν 2 du. ( + x 2 u 2 )( + x 2 ( u) 2 )( u) k Using a partial fraction decomposition, we can split F k = F,k (x) + F 2,k (x), where F,k (x) = 4 + x 2 (2u + )u ν 2 du, ( + x 2 u 2 )( u) k F 2,k (x) = 4 + x 2 (3 2u)u ν 2 du. ( + x 2 ( u) 2 )( u) k In particular, we observe that there exist two constants F,k, F 2,k > such that (27) F,k (x) F,k 4 + x 2, F 2,k (x) F 2,k 4 + x 2 for all x x. Hence, if one assumes that ν 2 + k γ, then we get a constant B 2.4. > such that (28) sup x x B2.3 (x, ɛ) ɛ kγ sup B2.4 (x) B 2.4. x x ɛ kγ Finally, gathering all the estimates (2), (23), (24), (25), (28), we get (2). Proposition 3 Let k be an integer. Let Q (X), Q 2 (X), R(X) C[X] such that (29) deg(r) deg(q ), deg(r) deg(q 2 ), R(im) for all m R. Assume that µ > max(deg(q ) +, deg(q 2 ) + ). Let m b(m) be a continuous function on R such that b(m) R(im) for all m R. Then, there exists a constant C 3 > (depending on Q, Q 2, R, µ, k, ν) such that (3) b(m) (τ k s) k ( s for all f(τ, m), g(τ, m) F d (ν,β,µ,k,ɛ). Q (i(m m ))f((s x) /k, m m ) Q 2 (im )g(x /k, m ) (s x)x dxdm )ds (ν,β,µ,k,ɛ) C 3 ɛ f(τ, m) (ν,β,µ,k,ɛ) g(τ, m) (ν,β,µ,k,ɛ)

14 4 Proof Let f(τ, m), g(τ, m) F d (ν,β,µ,k,ɛ). For any τ D(, ρ) S d, the segment [, τ k ] is such that for any s [, τ k ], any x [, s], the expressions f((s x) /k, m m ) and g(x /k, m ) are well defined, provided that m, m R. By definition, we can write b(m) where (τ k s) k ( s (τ k s) /k ( Q (i(m m ))f((s x) /k, m m ) Q 2 (im )g(x /k, m ) (s x)x dxdm )ds (ν,β,µ,k,ɛ) = sup ( + m ) µ + τ ɛ 2k τ D(,ρ) S d,m R τ ɛ exp(β m ν τ ɛ k ) s {( + m m ) µ e + s x 2 β m m ɛ 2k exp( ν s x / ɛ κ ) f((s x) /k, m m )} {( + m ) µ e β m + x 2 ɛ 2k x /k ɛ s x /k ɛ exp( ν x / ɛ k )g(x /k, m )} C(s, x, m, m, ɛ)dxdm )ds C(s, x, m, m, ɛ) = exp( β m ) exp( β m m ) ( + m m ) µ ( + m ) µ b(m)q (i(m m ))Q 2 (im ) s x /k x /k ɛ 2 Now, we know that there exist Q, Q 2, R > with ( + s x 2 )( + x 2 ) exp(ν s x / ɛ k ) exp(ν x / ɛ k ) ɛ 2k ɛ 2k (s x)x (3) Q (i(m m )) Q ( + m m ) deg(q ), Q 2 (im ) Q 2 ( + m ) deg(q 2), R(im) R( + m ) deg(r) for all m, m R. Therefore, (32) b(m) (τ k s) k ( s Q (i(m m ))f((s x) /k, m m ) Q 2 (im )g(x /k, m ) (s x)x dxdm )ds (ν,β,µ,k,ɛ) C 3 (ɛ) f(τ, m) (ν,β,µ,k,ɛ) g(τ, m) (ν,β,µ,k,ɛ)

15 5 where (33) C 3 (ɛ) = sup ( + m ) µ + τ ɛ 2k τ D(,ρ) S d,m R τ ɛ exp(β m ν τ ɛ k ) R( + m ) deg(r) τ k ( τ k h) /k ( h Q Q 2 ( + m m ) deg(q ) ( + m ) deg(q 2) exp( β m ) exp( β m m ) ( + m m ) µ ( + m ) µ (h x) /k x /k ɛ 2 ( + (h x)2 )( + x2 ) ɛ 2k ɛ 2k exp(ν(h x)/ ɛ k ) exp(νx/ ɛ k ) (h x)x dxdm )dh Using the triangular inequality m m + m m, for all m, m R, we get that C 3 (ɛ) C 3. C 3.2 (ɛ) where (34) C 3. = Q Q 2 R sup ( + m ) µ deg(r) m R ( + m m ) µ deg(q ) ( + m ) µ deg(q 2) dm which is finite whenever µ > max(deg(q ) +, deg(q 2 ) + ) under the assumption (29) using the same estimates as in Lemma 4 of [23] (see also the Lemma 2.2 from [9]), and where τ + ɛ (35) C 3.2 (ɛ) = sup 2k τ D(,ρ) S d τ ɛ exp( ν τ ɛ k ) τ k h ( τ k h) /k exp(νh/ ɛ κ ) (h x) /k x /k ɛ 2 ( + (h x)2 )( + x2 ɛ 2k Making the changes of variables h = ɛ k h and x = ɛ k x, we get that (36) τ k = ɛ h ( τ k h) /k exp(νh/ ɛ k ) τ ɛ k ( τ h ɛ k h ) /k exp(νh ) (h x) /k x /k ɛ 2 ( + (h x)2 )( + x2 ɛ 2k From (35) and (36), we get that C 3.2 (ɛ) ɛ C 3.3, where + x 2 (37) C 3.3 = sup x x /k exp( νx) x ɛ 2k ) ɛ 2k ) (h x)x dxdh (h x)x dxdh. ( + (h x ) 2 )( + x 2 dx dh ) (h x ) k x k (x h ) /k exp(νh ) ( h Again by the change of variable x = h u, for u [, ], we can write ( + (h x ) 2 )( + x 2 dx )dh ) (h x ) k x k (38) h ( + (h x ) 2 )( + x 2 dx ) (h x ) k x k = h 2 k du = J ( + (h ) 2 ( u) 2 )( + h 2 u 2 )( u) k u k (h ) k

16 6 Using a partial fraction decomposition, we can split J k (h ) = J,k (h ) + J 2,k (h ), where (39) J,k (h ) = h 2 k (h 2 + 4) 3 2u du ( + h 2 ( u) 2 )( u) k u k J 2,k (h ) = h 2 k (h 2 + 4) 2u + du ( + h 2 u 2 )( u) k u k From now on, we assume that k 2. By construction of J,k (h ) and J 2,k (h ), we see that there exists a constant j k > such that (4) J k (h ) j k h 2 k (h 2 + 4) for all h. From (37) and (4), we deduce that C 3.3 sup x C3.3 (x), where (4) C3.3 (x) = ( + x 2 ) exp( νx) From L Hospital rule, we know that lim C 3.3 (x) = x + lim x + x j k x 2 k j k exp(νh ) h 2 k (h 2 + 4) dh. (+x 2 ) 2 x 2 +4 ν( + x 2 ) 2x is finite when k 2. Therefore, we get a constant C 3.3 > such that (42) sup x C 3.3 (x) C 3.3. Taking into account the estimates for (33), (34), (35), (37), (4) and (42), we obtain the result (3) when k 2. In the remaining case k =, from Corollary 4.9 of [8] one can check the existence of a constant j > such that (43) J (h ) j h 2 + for all h. From (37) and (43), we deduce that C 3.3 sup x C3.3. (x), where (44) C3.3. (x) = ( + x 2 ) exp( νx) From L Hospital rule, we know that lim C 3.3. (x) = x + lim x + x is finite. Therefore, we get a constant C 3.3. > such that (45) sup x C 3.3. (x) C j exp(νh ) h 2 + dh. j ( + x 2 ) ν( + x 2 ) 2x Taking into account the estimates for (33), (34), (35), (37), (44) and (45), we obtain the result (3) for k =.

17 7 Definition 2 Let β, µ R. We denote by E (β,µ) the vector space of continuous functions h : R C such that h(m) (β,µ) = sup( + m ) µ exp(β m ) h(m) m R is finite. The space E (β,µ) equipped with the norm. (β,µ) is a Banach space. Proposition 4 Let k be an integer. Let Q(X), R(X) C[X] be polynomials such that (46) deg(r) deg(q), R(im) for all m R. Assume that µ > deg(q) +. Let m b(m) be a continuous function such that b(m) R(im) for all m R. Then, there exists a constant C 4 > (depending on Q, R, µ, k, ν) such that (47) b(m) (τ k s) k for all f(m) E (β,µ), all g(τ, m) F d (ν,β,µ,k,ɛ). f(m m )Q(im )g(s /k, m )dm ds s (ν,β,µ,k,ɛ) C 4 ɛ f(m) (β,µ) g(τ, m) (ν,β,µ,k,ɛ) Proof The proof follows the same lines of arguments as those of Propositions and 3. Let f(m) E (β,µ), g(τ, m) F(ν,β,µ,k,ɛ) d. We can write (48) N 2 := b(m) where b(m) (τ k s) k f(m m )Q(im )g(s /k, m )dm ds s (ν,β,µ,k,ɛ) = sup ( + m ) µ + τ ɛ 2k τ D(,ρ) S d,m R τ ɛ exp(β m ν τ ɛ k ) {( + m ) µ exp(β m ) exp( ν s ɛ k ) + {( + m m ) µ exp(β m m )f(m m )} s 2 ɛ 2k s /k ɛ D(τ, s, m, m, ɛ) = Q(im )e β m e β m m ( + m m ) µ ( + m ) µ exp( ν s Again, we know that there exist constants Q, R > such that g(s /k, m )} D(τ, s, m, m, ɛ)dm ds ) ɛ k s /k + s 2 ɛ 2k Q(im ) Q( + m ) deg(q), R(im) R( + m ) deg(r) ɛ (τ k s) /k s for all m, m R. By means of the triangular inequality m m + m m, we get that (49) N 2 C 4. (ɛ)c 4.2 f(m) (β,µ) g(τ, m) (ν,β,µ,k,ɛ)

18 8 where τ + ɛ C 4. (ɛ) = sup 2k τ D(,ρ) S d τ ɛ exp( ν τ ɛ k ) τ k exp(νh/ ɛ k ) + h2 ɛ 2k h k ɛ ( τ k h) /k dh and C 4.2 = Q R sup ( + m ) µ deg(r) m R ( + m m ) µ ( + m ) µ deg(q) dm. From the estimates (9) and (), we know that there exists a constant C 4. > such that (5) C 4. (ɛ) C 4. ɛ and from the estimates for (34), we know that C 4.2 is finite under the assumption (46) provided that µ > deg(q) +. Finally, gathering this latter bound estimates together with (49) and (5) yields the result (47). In the next proposition, we show that (E (β,µ),. (β,µ) ) is a Banach algebra for some noncommutative product introduced below. Proposition 5 Let Q (X), Q 2 (X), R(X) C[X] be polynomials such that (5) deg(r) deg(q ), deg(r) deg(q 2 ), R(im), for all m R. Assume that µ > max(deg(q ) +, deg(q 2 ) + ). Then, there exists a constant C 5 > (depending on Q, Q 2, R, µ) such that (52) R(im) Q (i(m m ))f(m m )Q 2 (im )g(m )dm (β,µ) C 5 f(m) (β,µ) g(m) (β,µ) for all f(m), g(m) E (β,µ). Therefore, (E (β,µ),. (β,µ) ) becomes a Banach algebra for the product defined by f g(m) = R(im) Q (i(m m ))f(m m )Q 2 (im )g(m )dm. As a particular case, when f, g E (β,µ) with β > and µ >, the classical convolution product belongs to E (β,µ). f g(m) = f(m m )g(m )dm Proof The proof is similar to the one of Proposition 3. Let f(m), g(m) E (β,µ). We write (53) R(im) Q (i(m m ))f(m m )Q 2 (im )g(m )dm (β,µ) = sup( + m ) µ e β m m R R(im) {( + m m ) µ e β m m f(m m )} {( + m ) µ e β m g(m )} E(m, m )dm

19 9 where E(m, m ) = e β m m e β m ( + m m ) µ ( + m ) µ Q (i(m m ))Q 2 (im ). Using the triangular inequality m m + m m and the estimates in (3), we get that (54) R(im) where C 5 = Q Q 2 R Q (i(m m ))f(m m )Q 2 (im )g(m )dm (β,µ) sup ( + m ) µ deg(r) m R C 5 f(m) (β,µ) g(m) (β,µ) ( + m m ) µ deg(q ) ( + m ) µ deg(q 2) dm which is finite whenever µ > max(deg(q )+, deg(q 2 )+) provided that (5) holds as explained in Proposition 3 (see (34)). 3 Laplace transform, asymptotic expansions and Fourier transform We give a definition of k Borel summability of formal series with coefficients in a Banach space which is a slightly modified version of the one given in [], Section 3.2, in order to fit our necessities. Definition 3 Let k be an integer. Let m k (n) be the sequence defined by for all n. A formal series m k (n) = Γ( n k ) = t n k e t dt ˆX(T ) = a n T n T E[[T ]] n= with coefficients in a Banach space (E,. E ) is said to be m k summable with respect to t in the direction d [, 2π) if i) there exists ρ R + such that the following formal series, called a formal m k Borel transform of ˆX B mk ( ˆX)(τ) a n = Γ( n τe[[τ]], k )τn is absolutely convergent for τ < ρ. n= ii) there exists δ > such that the series B mk ( ˆX)(τ) can be analytically continued with respect to τ in a sector S d,δ = {τ C : d arg(τ) < δ}. Moreover, there exist C > and K > such that B mk ( ˆX)(τ) E Ce K τ k for all τ S d,δ.

20 2 If this is so, the vector valued Laplace transform of B mk ( ˆX)(τ) in the direction d is defined by L d m k (B( ˆX))(T ) = k B mk ( ˆX)(u)e (u/t du )k L γ u, along a half-line L γ = R + e iγ S d,δ {}, where γ depends on T and is chosen in such a way that cos(k(γ arg(t ))) δ >, for some fixed δ. The function L d m k (B mk ( ˆX))(T ) is well defined, holomorphic and bounded in any sector S d,θ,r /k = {T C : T < R /k, d arg(t ) < θ/2}, where π k < θ < π k + 2δ and < R < δ /K. This function is called the m k sum of the formal series ˆX(T ) in the direction d. We now state some elementary properties concerning the m k sums of formal power series. ) The function L d m k (B mk ( ˆX))(T ) has the formal series ˆX(T ) as Gevrey asymptotic expansion of order /k with respect to t on S d,θ,r /k. This means that for all π k < θ < θ, there exist C, M > such that (55) L d m k (B mk ( ˆX))(T n ) a p T p E CM n Γ( + n ) T n k p= for all n 2, all T S d,θ,r/k. Moreover, from Watson s lemma (see Proposition p. 75 in []), we get that L d m k (B mk ( ˆX))(T ) is the unique holomorphic function that satisfies the estimates (55) on the sectors S d,θ,r /k with large aperture θ > π k. 2) Let us assume that (E,. E ) also has the structure of a Banach algebra for a product. Let ˆX (T ), ˆX 2 (T ) T E[[T ]] be m k summable formal power series in direction d. Let q q 2 be integers. We assume that ˆX (T ) + ˆX 2 (T ), ˆX (T ) ˆX 2 (T ) and T q q 2 ˆX T (T ), which are elements of T E[[T ]], are m k summable in direction d. Then, the following equalities (56) L d m k (B mk ( ˆX ))(T ) + L d m k (B mk ( ˆX 2 ))(T ) = L d m k (B mk ( ˆX + ˆX 2 ))(T ), L d m k (B mk ( ˆX ))(T ) L d m k (B mk ( ˆX 2 ))(T ) = L d m k (B mk ( ˆX ˆX 2 ))(T ) T q q 2 T Ld m k (B mk ( ˆX ))(T ) = L d m k (B mk (T q q 2 T ˆX ))(T ) hold for all T S d,θ,r /k. These equalities are consequence of the unicity of the function having a given Gevrey expansion of order /k in large sectors as stated above in ) and from the fact that the set of holomorphic functions having Gevrey asymptotic expansion of order /k on a sector with values in the Banach algebra E form a differential algebra (meaning that this set is stable with respect to the sum and product of functions and derivation in the variable T ) (see Theorem 8,9 and 2 in []). In the next proposition, we give some identities for the m k Borel transform that will be useful in the sequel. Proposition 6 Let ˆf(t) = n f nt n, ĝ(t) = n g nt n be formal series whose coefficients f n, g n belong to some Banach space (E,. E ). We assume that (E,. E ) is a Banach algebra for some product. Let k, m be integers. The following formal identities hold. (57) B mk (t k+ t ˆf(t))(τ) = kτ k B mk ( ˆf(t))(τ)

21 2 (58) B mk (t m ˆf(t))(τ) = τ k and Γ( m k ) (τ k s) m k B mk ( ˆf(t))(s /k ) ds s (59) B mk ( ˆf(t) ĝ(t))(τ) = τ k B mk ( ˆf(t))((τ k s) /k ) B mk (ĝ(t))(s /k ) (τ k s)s ds Proof First, we show (57). By definition, we have that (6) B mk ( tk+ k t ˆf(t))(τ) = n k f n Γ( n n k + )τn+k By application of the addition formula for the Gamma function which yields Γ( n k + ) = n k Γ( n k ) for any n, we deduce (57) from (6). Now, we prove (58). By definition, we can write (6) B mk (t m ˆf(t))(τ) = Γ( m k ) f n Γ( m Γ( n k )Γ( n k ) n k ) Γ( m+n k ) τ m+n. Using the Beta integral formula (see Appendix B in [2]), we can write (62) Γ( m k )Γ( n k ) Γ( m+n k ) = τ k τ m+n for any m, n. Plugging (62) into (6) yields (58). Finally, we show (59). By definition, we have (63) B mk ( ˆf(t) ĝ(t))(τ) = n 2 ( p+q=n Using again the Beta integral formula, we can write (τ k s) m k s n k ds f p Γ( p k ) g q Γ( q k ) Γ( p k )Γ( q k ) Γ( n k ) )τ n (64) Γ( p k )Γ( q k ) Γ( n k ) = τ k τ n (τ k s) p k s q k ds when p + q = n and p, q. By the substitution of (64) into (63), we deduce (59). In the following proposition, we recall some properties of the inverse Fourier transform Proposition 7 Let f E (β,µ) with β >, µ >. The inverse Fourier transform of f is defined by F + (f)(x) = (2π) /2 f(m) exp(ixm)dm for all x R. The function F (f) extends to an analytic function on the strip (65) H β = {z C/ Im(z) < β}. Let φ(m) = imf(m) E (β,µ ). Then, we have (66) z F (f)(z) = F (φ)(z)

22 22 for all z H β. Let g E (β,µ) and let ψ(m) = (2π) /2 f g(m), the convolution product of f and g, for all m R. From Proposition 5, we know that ψ E (β,µ). Moreover, we have (67) F (f)(z)f (g)(z) = F (ψ)(z) for all z H β. Proof Let f E (β,µ). It is straight to check that F (f) is well defined on the real line. The fact that F (f) extends to an analytic function on the strip H β follows from the next inequality. There exists C > such that f(m) exp(izm) C ( + m ) µ exp((β β) m ) for all m R, z H β, with β < β. The relations (66), (67) are classical and can be found for instance in [27]. 4 Formal and analytic solutions of convolution initial value problems with complex parameters Let k and D 2 be integers. For l D, let d l, δ l, l be nonnegative integers. We assume that (68) = δ, δ l < δ l+, for all l D. We make also the assumption that (69) d D = (δ D )(k + ), d l > (δ l )(k + ), D = d D δ D + for all l D. Let Q(X), Q (X), Q 2 (X), R l (X) C[X], l D, be polynomials such that (7) deg(q) deg(r D ) deg(r l ), deg(r D ) deg(q ), deg(r D ) deg(q 2 ), Q(im), R D (im) for all m R, all l D. We consider sequences of functions m C,n (m, ɛ), for all n and m F n (m, ɛ), for all n, that belong to the Banach space E (β,µ) for some β > and µ > max(deg(q ) +, deg(q 2 ) + ) and which depend holomorphically on ɛ D(, ɛ ). We assume that there exist constants K, T > such that (7) C,n (m, ɛ) (β,µ) K ( T ) n, F n (m, ɛ) (β,µ) K ( T ) n for all n, for all ɛ D(, ɛ ). We define C (T, m, ɛ) = C,n (m, ɛ)t n n, F (T, m, ɛ) = F n (m, ɛ)t n n

23 23 which are convergent series on D(, T /2) with values in E (β,µ). singular initial value problem We consider the following (72) Q(im)( T U(T, m, ɛ)) = ɛ (2π) /2 Q 2 (im )U(T, m, ɛ)dm + + ɛ (2π) /2 + ɛ (2π) /2 for given initial data U(, m, ɛ) =. D l= Proposition 8 There exists a unique formal series Q (i(m m ))U(T, m m, ɛ) R l (im)ɛ l d l +δ l T d l δ l T U(T, m, ɛ) C (T, m m, ɛ)r (im )U(T, m, ɛ)dm C, (m m, ɛ)r (im )U(T, m, ɛ)dm + ɛ F (T, m, ɛ) Û(T, m, ɛ) = n U n (m, ɛ)t n solution of (72) with initial data U(, m, ɛ), where the coefficients m U n (m, ɛ) belong to E (β,µ) for β > and µ > max(deg(q )+, deg(q 2 )+) given above and depend holomorphically on ɛ in D(, ɛ ) \ {}. Proof From Proposition 5 and the conditions in the statement above, we get that the coefficients U n (m, ɛ) of Û(T, m, ɛ) are well defined, belong to E (β,µ) for all ɛ D(, ɛ ) \ {}, all n and satisfy the following recursion relation (73) (n + )U n+ (m, ɛ) = ɛ Q(im) n +n 2 =n,n,n ɛ Q(im) + D l= for all n max ld d l. (2π) /2 Q (i(m m ))U n (m m, ɛ)q 2 (im )U n2 (m, ɛ)dm R l (im) ( ) ɛ l d l +δ l Π δ l j= Q(im) (n + δ l d l j) U n+δl d l (m, ɛ) n +n 2 =n,n,n 2 ɛ (2π) /2 Q(im) (2π) /2 C,n (m m, ɛ)r (im )U n2 (m, ɛ)dm C, (m m, ɛ)r (im )U n (m, ɛ)dm + Using the formula from [28], p. 4, we can expand the operators T δl(k+) δ l T (74) T δ l(k+) δ l T = (T k+ T ) δ l + pδ l A δl,pt k(δ l p) (T k+ T ) p where A δl,p, p =,..., δ l are real numbers. We define integers d l,k to satisfy (75) d l + k + = δ l (k + ) + d l,k ɛ Q(im) F n(m, ɛ) in the form

24 24 for all l D. Multiplying the equation (72) by T k+ and using (74), we can rewrite the equation (72) in the form (76) Q(im)(T k+ T U(T, m, ɛ)) = ɛ T k+ (2π) /2 + + l= pδ l + ɛ T k+ (2π) /2 + ɛ T k+ (2π) /2 Q (i(m m ))U(T, m m, ɛ)q 2 (im )U(T, m, ɛ)dm D ( R l (im) ɛ l d l +δ l T d l,k (T k+ T ) δ l U(T, m, ɛ) ) A δl,p ɛ l d l +δ l T k(δ l p)+d l,k (T k+ T ) p U(T, m, ɛ) C (T, m m, ɛ)r (im )U(T, m, ɛ)dm C, (m m, ɛ)r (im )U(T, m, ɛ)dm + ɛ T k+ F (T, m, ɛ) We denote ω k (τ, m, ɛ) the formal m k Borel transform of Û(T, m, ɛ) with respect to T, ϕ k (τ, m, ɛ) the formal m k Borel transform of C (T, m, ɛ) with respect to T and ψ k (τ, m, ɛ) the formal m k Borel transform of F (T, m, ɛ) with respect to T, ω k (τ, m, ɛ) = n U n (m, ɛ) τ n Γ( n k ), ϕ k(τ, m, ɛ) = C,n (m, ɛ) τ n Γ( n n k ) ψ k (τ, m, ɛ) = n F n (m, ɛ) τ n Γ( n k ) Using (7) we get that ϕ k (τ, m, ɛ) F d (ν,β,µ,k,ɛ) and ψ k(τ, m, ɛ) F d (ν,β,µ,k,ɛ), for all ɛ D(, ɛ ) \ {}, any unbounded sector S d centered at and bisecting direction d R, for some ν >. Indeed, we have that (77) ϕ k (τ, m, ɛ) (ν,β,µ,k,ɛ) τ + ɛ C,n (m, ɛ) (β,µ) ( sup 2k n τ D(,ρ) S d τ ɛ exp( ν τ ɛ k ) τ n Γ( n k )), ψ k (τ, m, ɛ) (ν,β,µ,k,ɛ) τ + ɛ F n (m, ɛ) (β,µ) ( sup 2k n τ D(,ρ) S d τ ɛ exp( ν τ ɛ k ) τ n Γ( n k )) By using the classical estimates (22) and Stirling formula Γ(n/k) (2π) /2 (n/k) n k 2 e n/k as n tends to +, we get two constants A, A 2 > depending on ν, k such that τ + ɛ (78) sup 2k τ D(,ρ) S d τ ɛ exp( ν τ ɛ k ) τ n Γ( n k ) = sup ɛ n ( + τ τ D(,ρ) S d ɛ 2k ) τ exp( ν τ ɛ n ɛ k ) Γ( n k ) ɛ n sup( + x 2 )x n e νx ( k x Γ( n k ) ɛn ( n νk ) n k e n k + ( n νk + 2 ) ν ) n k +2 n ( e k +2) /Γ(n/k) A ɛ n (A 2 ) n

25 25 for all n, all ɛ D(, ɛ ) \ {}. Therefore, if ɛ fulfills ɛ A 2 < T, we get the estimates (79) ϕ k (τ, m, ɛ) (ν,β,µ,k,ɛ) A n ψ k (τ, m, ɛ) (ν,β,µ,k,ɛ) A C,n (m, ɛ) (β,µ) (ɛ A 2 ) n A A 2 K T ɛ A 2 T ɛ, n F n (m, ɛ) (β,µ) (ɛ A 2 ) n A A 2 K T ɛ A 2 T ɛ for all ɛ D(, ɛ ) \ {}. Using the computation rules for the formal m k Borel transform in Proposition 6, we deduce the following equation satisfied by ω k (τ, m, ɛ), (8) Q(im)(kτ k ω k (τ, m, ɛ)) = ɛ τ k + + D + pδ l ( (2π) /2 s pδ D ( R l (im) l= ( (2π) /2 s + ɛ τ k Γ( + k ) s Γ( + k ) (τ k s) /k Q (i(m m ))ω k ((s x) /k, m m, ɛ) ) Q 2 (im )ω k (x /k ds, m, ɛ) (s x)x dxdm s ( + R D (im) k δ D τ δdk ω k (τ, m, ɛ) A δd,p A δl,pɛ l d l +δ l s τ k Γ(δ D p) ɛ l d l +δ l τ k Γ( d l,k k ) τ k Γ( d l,k k + δ l p) (τ k s) δ D p (k p s p ω k (s /k, m, ɛ)) ds s + ɛ τ k Γ( + k ) (τ k s) d l,k k (k δ l s δ l ω k (s /k, m, ɛ)) ds s (τ k s) d l,k k (τ k s) /k ) +δ l p (k p s p ω k (s /k, m, ɛ)) ds s ) ϕ k ((s x) /k, m m, ɛ)r (im )ω k (x /k ds, m, ɛ) (s x)x dxdm s (τ k s) /k (2π) /2 ( C, (m m, ɛ)r (im )ω k (s /k, m, ɛ)dm ) ds s + ɛ τ k Γ( + k ) We make the additional assumption that there exists an unbounded sector S Q,RD = {z C/ z r Q,RD, arg(z) d Q,RD η Q,RD } ) (τ k s) /k ψ k (s /k, m, ɛ) ds s with direction d Q,RD R, aperture η Q,RD > for some radius r Q,RD > such that (8) Q(im) R D (im) S Q,R D for all m R. We factorize the polynomial P m (τ) = Q(im)k R D (im)k δ Dτ (δ D )k in the form (82) P m (τ) = R D (im)k δ D Π (δ D )k l= (τ q l (m))

26 26 where Q(im) (83) q l (m) = ( R D (im) k δ D ) (δ D )k exp( Q(im) (arg( R D (im)k δ D ) (δ D )k + 2πl (δ D )k )) for all l (δ D )k, all m R. We choose an unbounded sector S d centered at, a small closed disc D(, ρ) and we prescribe the sector S Q,RD in such a way that the following conditions hold. ) There exists a constant M > such that (84) τ q l (m) M ( + τ ) for all l (δ D )k, all m R, all τ S d D(, ρ). Indeed, from (8) and the explicit expression (83) of q l (m), we first observe that q l (m) > 2ρ for every m R, all l (δ D )k for an appropriate choice of r Q,RD and of ρ >. We also see that for all m R, all l (δ D )k, the roots q l (m) remain in a union U of unbounded sectors centered at that do not cover a full neighborhood of the origin in C provided that η Q,RD is small enough. Therefore, one can choose an adequate sector S d such that S d U = with the property that for all l (δ D )k the quotients q l (m)/τ lay outside some small disc centered at in C for all τ S d, all m R. This yields (84) for some small constant M >. 2) There exists a constant M 2 > such that (85) τ q l (m) M 2 q l (m) for some l {,..., (δ D )k }, all m R, all τ S d D(, ρ). Indeed, for the sector S d and the disc D(, ρ) chosen as above in ), we notice that for any fixed l (δ D )k, the quotient τ/q l (m) stays outside a small disc centered at in C for all τ S d D(, ρ), all m R. Hence (85) must hold for some small constant M 2 >. By construction of the roots (83) in the factorization (82) and using the lower bound estimates (84), (85), we get a constant C P > such that (86) P m (τ) M (δ D )k M 2 R D (im)k δ D Q(im) ( R D (im) k δ D ) (δ D )k ( + τ ) (δ D )k M (δ D )k k δ D M 2 (k δd ) (min x (δ D )k (δ (r Q,RD ) D )k R D (im) ( + x) (δ D )k )( + τ k ) (δ D ) ( + x k ) (δ D ) k k (δ = C P (r Q,RD ) D )k R D (im) ( + τ k ) (δ D ) k for all τ S d D(, ρ), all m R. In the next proposition, we give sufficient conditions under which the equation (8) has a solution ω k (τ, m, ɛ) in the Banach space F(ν,β,µ,k,ɛ) d where β, µ are defined above. Proposition 9 Under the assumption that (87) δ D δ l + 2 k, l + k( δ D ) +,