On Borel summation and Stokes phenomena for rank one nonlinear systems of ODE’s

Ovidiu Costin ^{∗}

1 Introduction

In this paper we study analytic (linear or) nonlinear systems of ordinary differential equations, at an irregular singularity of rank one, under nonresonance conditions. It is shown that the formal asymptotic exponential series solutions (transseries solutions: countable linear combinations of formal power series multiplied by small exponentials) are Borel summable in a generalized sense along any direction in which the exponentials decay. Conversely, any solution that decreases along some direction is the Borel sum of a transseries.

The summation procedure introduced is an extension of Borel summation which is linear, multiplicative, commutes with differentiation and complex conjugation. The summation algorithm uses the formal solutions alone (and not the differential equation that they solve). Along singular (Stokes) directions, the functions reconstructed by summation are shown to be given by Laplace integrals along special paths, a subset of Écalle’s median paths.

The one-to-one correspondence established between actual solutions and generalized Borel sums of transseries is constant between Stokes lines and changes if a Stokes line is crossed (local Stokes phenomenon). We analyze the connection between local and classical Stokes phenomena.

We study the analytic properties of the Borel (formal inverse Laplace) transform of the series contained in the transseries of the transseries and give a systematic description of their singularities. These Borel transforms satisfy a hierarchy of convolution equations, for which we give the general solution in a space of hyperfunctions. In addition, we show that they are resurgent functions in the sense of Écalle.

The summation procedure is not unique; we classify all proper extensions of Borel summation to transseries solutions of nonresonant systems.

We find formulas connecting the different series contained in the transseries among themselves (resurgence equations). Resurgence turns out to be closely linked to the local Stokes phenomenon.

∗Mathematics Department, University of Chicago, 5734 University Avenue, Chicago IL 60637; e-mail: costin@math.uchicago.edu

The connection to Berry’s hyperasymptotics and applications to the classification of differential equations are briefly discussed.

1.1 General setting

We consider the differential system

y′ = f(x, y)  y ∈ ℂ^{n}  (1.1)

under the following assumptions:

(a1) The function f is analytic at (∞, 0).

(a2) Nonresonnance: the eigenvalues λ_{i} of the linearization

Λ̂ := − (\frac{∂f_{i}}{∂y_{j}} (∞, 0))_{i,j=1,2,...n}  (1.2)

are linearly independent over ℤ (in particular nonzero) and such that the Stokes lines are distinct (a somewhat less restrictive condition is actually used, cf. §1.1.2). Normalization. It is convenient to prepare (1.1) in the following way. Pulling out the inhomogeneous and the linear terms (relevant to leading order asymptotics) we get

y′ = f_{0}(x) − Λ̂y − \frac{1}{x}B̂y + g(x, y)  (1.3)

Under the assumptions (a1) and (a2), by means of normal form calculations it is possible to arrange (1.3) so that ([11], [19])

(n1) Λ̂ = diag(λ_{i}) and

(n2) B̂ = diag(β_{i}) For convenience, we rescale x and reorder the components of y so that

(n3) λ_{1} = 1, and, with ϕ_{i} = arg(λ_{i}), we have ϕ_{i} < ϕ_{j} if i < j. To simplify notations, we formulate some of our results relative to λ_{1}; they can be easily adapted to any other eigenvalue.

To unify the treatment we make, by taking y = y_{1}x^{−N} for some N > 0, (n4) ℜ(β_{j}) < 0, j = 1, 2, …, n.

(there is an asymmetry at this point: the opposite inequality cannot be achieved, in general, as simply and without violating analyticity at infinity). Finally, through a transformation of the form y ↔ y − ∑_{k=1}^{M} a_{k}x^{−k} we arrange that

(n5) f_{0} = O(x^{−M−1}) and g(x, y) = O(y^{2}, x^{−M−1}y). We choose M > 1 + max_{i} ℜ(−β_{i}).

Formal solutions. In prepared form, given (a1) and (a2), (1.3) admits an n–parameter family of formal exponential series solutions (transseries)

ỹ = ỹ_{0} + ∑_{k≥0;|k|>0} C_{1}^{k_{1}} ⋯ C_{n}^{k_{n}}e^{−(k·λ)x}x^{k·m}ỹ_{k}  (1.4)

(see [11], [17],[21], and also § 2.4 below) where m_{i} = 1−⌊β_{i}⌋, (⌊·⌋ = integer part), C ∈ ℂ^{n} is an arbitrary vector of constants, and ỹ_{k} = x^{−k(β+m)} ∑_{l=0}^{∞} a_{k;l}x^{−l} are formal power series.

When x is large in some direction d in ℂ, an important role is played by the subset of transseries which are at the same time asymptotic expressions^{1}: When there are infinitely many exponentials in (1.4) we ask that for all i with C_{i} ≠ 0 we have |e^{−λ_{i}x}| ≪ 1 for large x in the given direction d in ℂ. Formally, agreeing to omit the terms with C_{i} = 0, with x in d, any ascending chain ℜ(−k_{1} · λx) ≤ ℜ(−k_{2} · λx) ≤ …, k_{i} ≠ k_{j}, in (1.4) must be finite (the terms of an asymptotic transseries are well-ordered with respect to ′′ ≪′′). Thus for x in some direction d we only consider those transseries that satisfy the condition:

(c1) ξ + ϕ_{i} := arg(x) + ϕ_{i} ∈ (−π/2, π/2) for all i such that C_{i} ≠ 0. In other words, C_{i} ≠ 0 implies that λ_{i} lies in a half-plane centered on \overline{d}, the complex conjugate direction to d.

From now on, λ = (λ_{i_{1}}, …, λ_{i_{n_{1}}} ), β = (β_{i_{1}}, …, β_{i_{n_{1}}} ), m = (m_{i_{1}}, …, m_{i_{n_{1}}} ) and β′ = β + m where the indices i_{1}, …, i_{n_{1}} satisfy (c1).

We will henceforth consider that (1.3) is presented in prepared form, and use the designation transseries only for those formal solutions satisfying (c1).

The series ỹ_{0} is a formal solution of (1.3) while, for k ≠ 0, ỹ_{k} satisfy a hierarchy of linear differential equations [11] (see also § 2.4 for a brief exposition and notations). Generically all the series ỹ_{k} are factorially divergent and there is no immediate way to uniquely associate actual functions to them. Neither can ỹ be viewed as a classical asymptotic expansion since the ỹ_{k} are beyond all orders of each other (e.g., for k ≠ 0 and all l ∈ ℕ, e^{−λ·kx}x^{k·m}ỹ_{k} = o(x^{−l})).

One question is therefore to understand the relation between these (algorithmically obtained) formal solutions and the actual solutions of (1.3). In the present paper we show that a suitable generalization of Borel summation provides a one-to-one correspondence between transseries and actual solutions of (1.3):

y ⇌ ỹ_{0} + ∑_{k≥0;|k|>0} C_{1}^{k_{1}} ⋯ C_{n}^{k_{n}}e^{−(k·λ)x}x^{k·m}ỹ_{k} (x → ∞, arg(x) = ξ) (1.5)

Given y, the value of C_{i} can change only when ξ + arg(λ_{i} − k · λ) = 0, k_{i} ∈ ℕ ∪ {0}, i.e. when crossing one of the (finitely many by (c1)) Stokes lines. The correspondence (1.5) defines a summation method, in the sense that it is an extension of convergent summation which preserves its basic properties: linearity, multiplicativity, commutation with differentiation and with complex conjugation. These properties are essential for obtaining true solutions out of transseries for nonlinear differential equations. Our procedure is similar to the medianization proposed by Écalle, but (due to the structure of (1.3)) requires substantially fewer analytic continuation paths. In addition we classify in the context of (1.3) all admissible summation methods (there is a one-parameter family of them, preserving the properties of usual summation). Summation recovers from transseries actual solutions of (1.3) without resorting to (1.3) in the process. In addition, the analysis reveals a rich analytic structure and formulas linking the various ỹ_{k} among themselves (resurgence relations). In [13] we studied this problem under further restrictions on the transseries (decay of the exponentials in a full half-plane) and on the differential equation. Removing those restrictions creates difficulties that required a new approach. New resurgence relations are found and in addition we provide a complete description, needed in applications, of the singularity structure of the Borel transforms of ỹ_{k}.

1An asymptotic expansion of a function carries immediate information about behavior of the function near the expansion point (in contrast to antiasymptotic expansions, e.g. a convergent doubly infinite Laurent series)

1.1.1 Notes on Borel summation

The following is a very brief description; for more details on classical Borel summation see [9], [8] and for recent developments see [7] and especially [2].

If f̃ = ∑_{k=0}^{∞} a_{k}x^{−k−r} is a formal series with ℜ(r) > 0, its Borel transform is defined as the (still formal) series Bf̃ = ∑_{k=0}^{∞} p^{k−1+r}/Γ(k + r + 1), the termby-term inverse Laplace transform of f̃. If r ∈ ℕ^{+} and f̃ converges (to f), then Bf̃ converges in ℂ to an analytic function which is Laplace (L) transformable and LBf̃ = f. A similar property holds more generally when ℜ(r) > 0, with now f and Bf̃ ramified analytic functions. Even when f̃ is divergent (not faster than factorially), Bf̃ may have a nonzero radius of convergence and define a germ of an analytic function F (p). If F (p) can be analytically continued along a ray arg(p) = ϕ, and its growth is at most exponential, then f = L_{ϕ}F = ∫_{e^{iϕ}ℝ^{+}} F (p)e^{−xp}dp defines a function with the property f ∼ f̃ as x → ∞ with ℜ(xe^{iϕ}) > 0. In general now F (p) is singular (not only for p = 0), and L_{ϕ}F (when it exists) will depend on ϕ; the usual convention is to choose ϕ so that

xp ∈ ℝ^{+}  (1.6)

Thus, the Borel sum of f̃ in the direction x, if it exists, is defined as L_{ϕ(x)}Bf with −ϕ(x) = ξ := arg(x). However, when f̃ is a series with real coefficients, it is a common occurrence that F (p) is singular for p ∈ ℝ^{+} (because of conjugation symmetry), and then the classical Borel sum of f̃ along the real axis (the interesting direction in many cases) is undefined.

The difficulty is more serious than it may seem. Summation along paths that avoid the singularities from above or from below give different results and thus would lead to an ambiguous (or unnatural) procedure. More importantly, a “summation” procedure using such paths would not commute with complexconjugation since the “sum” will be, in general, complex for real f̃ and would thus fail to be a (proper) summation method. Symmetry considerations suggest a first step towards averaging: summation along the half-sum of the two paths does commutes with complex conjugation. But this solves a problem only to create another one. The half-sum process fails to commute with multiplication (of series) and is thus not a summation method, either.

It turns out that there exist more sophisticated averages which have all the required properties to define a summation procedure. The technique of averaging, as well as the fundamental concepts of analyzable functions and transseries, were discovered and studied by Ecalle in his constructive approach to the Dulac conjecture (see [1], [2] and [3]). The concept of analyzable function (also discovered by Kruskal in the context of surreal analysis) is regarded as a very comprehensive generalization of analyticity/ quasianalyticity. The widely held belief is that all functions of “natural origin” must be analyzable. In particular, analyzable functions have uniquely associated transseries which are generalized-Borel summable, after a finite number of transformations [3]. We show that, in the particular case of (1.3), decreasing solutions are analyzable.

There is a wide class of admissible, all-purpose averaging methods ([4]). As yet there is no unique, natural average and the problem in its full generality is highly nontrivial. We obtain the balanced average directly from the study of the general solution of the inverse Laplace transform of (1.3). Its potential nonuniqueness is lifted, in our context, by imposing compatibility with hyperasymptotics an important improvement in asymptotic calculations proposed by M. Berry ([22], [23], [24], [25]).

1.1.2 Nonresonance

(1) λ_{i}, i = 1, ..., n_{1} are assumed ℤ-linearly independent for any d. (2) Let θ ∈ [0, 2π) and λ̃ = (λ_{i_{1}}, ..., λ_{i_{p}}) where |arg λ_{i_{j}} − θ| ∈ (−π/2, π/2) (those eigenvalues contained in the open half-plane H_{θ} centered along e^{iθ}). We require that for any θ the complex numbers in the set {λ̃_{i} − k · λ̃ ∈ H_{θ} : k ∈ ℕ^{p}, i = 1, ..., p} (note: the set is finite) have distinct directions. These are the Stokes lines d_{i;k}.

That the set of λ which satisfy (1) and (2) has full measure follows from the fact that (1) and (2) follow from the condition:

(m, m′ ∈ ℤ^{n}, α ∈ ℝ and (m − αm′) · λ = 0) ⇒ (m = αm′)  (1.7)

Indeed, if (1.7) fails, one of ℜλ_{j}, ℑλ_{j} is a rational function with rational coefficients of the other ℜλ_{j} and ℑλ_{j}, corresponding to a zero measure set in ℝ^{2n}.

1.2 Further notations and conventions

If y_{1} and y_{2} are inverse Laplace transformable functions, then in a neighborhood of the origin L^{−1}(y_{1}y_{2}) = (L^{−1}y_{1}) ∗ (L^{−1}y_{2}), where for f, g ∈ L^{1} convolution is given by

f ∗ g := p ↦ ∫_{0}^{p} f(s)g(p − s)ds  (1.8)

We use the convention ℕ ∋ 0. Let

W = {p ∈ ℂ : p ≠ kλ_{i} , ∀k ∈ ℕ, i = 1, 2, …, n}  (1.9)

The directions d_{j} = {p : arg(p) = ϕ_{j}}, j = 1, 2, …, n (cf. (a2)) are the Stokes lines of ỹ_{0} (note: sometimes known as anti-Stokes lines!). We construct over W a surface R, consisting of homotopy classes of smooth curves in W starting at the origin, moving away from it, and crossing at most one Stokes line, at most once (see Fig. 1):

R := \{γ : (0, 1) ↦ W : γ(0_{+}) = 0; \frac{d}{dt}|γ(t)| > 0; arg(γ(t)) monotonic\}

modulo homotopies(1.10)

Define R_{1} ⊂ R by (1.10) with the supplementary restriction arg(γ) ∈ (ψ_{n} − 2π, ψ_{2}) where ψ_{n} = max{−π/2, ϕ_{n} − 2π} and ψ_{2} = min{π/2, ϕ_{2}}. R_{1} may be viewed as the part of the covering R, above a sector containing the real axis. Similarly we let R′_{1} ⊂ R_{1} with the restriction that the curves γ do not cross the Stokes directions d_{i,k} (cf. §1.1.2), other than ℝ^{+}, and we let ψ_{±} = ± max(± arg γ) with γ ∈ R′_{1}.

Fig 1. The paths near λ_{2} belong to R. The paths near λ_{1} relate to the balanced average

By AC_{γ}(f) we denote the analytic continuation of f along a curve γ. For the analytic continuations near a Stokes line d_{i;k} we use symbols similar to Écalle’s: f^{−} is the branch of f along a path γ with arg(γ) < ϕ_{i}, while f^{−j+} denotes the branch along a path that crosses the Stokes line between jλ_{i} and (j + 1)λ_{i} (see also [13]).

We use the notations Pf for ∫_{0}^{p} f(s)ds and P_{γ}f if integration is along the curve γ.

We write k ⪰ k′ if k_{i} ≥ k′_{i} for all i and k ≻ k′ if k ⪰ k′ and k ≠ k′. The relation ≻ is a well ordering on ℕ^{n_{1}}. We let e_{j} be the unit vector in the j^{th} direction in ℕ^{n_{1}}.

Formal expansions are denoted with a tilde, and capital letters Y, V … will usually denote Borel transforms or other functions naturally associated to Borel space. For notational convenience, we will not however distinguish between the series Ỹ_{k} = Bỹ_{k}, which in our case turn out to be convergent, and the sums Y_{k} of these series as germs of ramified analytic functions.

By symmetry (renumbering the directions) it suffices to analyze the singularity structure of Y_{0} in R_{1} only. However, (c1) breaks this symmetry for k ≠ 0 and the properties of these Y_{k} will be analyzed along some other directions as well.

χ_{A} will denote the characteristic function of the set A. We write |f| := max_{i}{|f_{i}|}. We have

g(x, y) = ∑_{|l|≥1} g_{l}(x)y^{l} = ∑_{s≥0;|l|≥1} g_{s,l}x^{−s}y^{l} (|x| > x_{0}, |y| < y_{0})  (1.11)

where y^{l} = y_{1}^{l_{1}} ⋯ y_{n}^{l_{n}} and |l| = l_{1} + ⋯ + l_{n}. By construction g_{s,l} = 0 if |l| = 1 and s ≤ M.

The formal inverse Laplace transform of g(x, y(x)) (formal since y is still unrestricted) is given by:

L^{−1} ( ∑_{|l|≥1} y(x)^{l} ∑_{s≥0} g_{s,l}x^{−s}) = ∑_{|l|≥1} G_{l} ∗ Y^{∗l} + ∑_{|l|≥2} g_{0,l}Y^{∗l} =: N (Y)

(1.12)

with G_{l}(p) = ∑_{s=1}^{∞} g_{s,l}p^{s−1}/s! and (G_{l} ∗ Y^{∗l})_{j} := (G_{l})_{j} ∗ Y_{1}^{∗l_{1}} ∗ .. ∗ Y_{n}^{∗l_{n}}. By

(n5), G_{1,l}^{(l)}(0) = 0 if |l| = 1 and l ≤ M. The inverse Laplace transform of (1.3) is the convolution equation:

−pY = F_{0} − Λ̂Y − B̂PY + N (Y)  (1.13)

Let d_{j}(x) := ∑_{l≥j} \binom{l}{j}g_{l}(x)ỹ_{0}^{l−j}. Straightforward calculation (see Appendix § 2.4; cf. also [13]) shows that the components ỹ_{k} of the transseries satisfy the hierarchy of differential equations

y′_{k} + (Λ̂ + \frac{1}{x} (B̂ + k · m) − k · λ) y_{k} + ∑_{|j|=1} d_{j}(x)(y_{k})^{j} = t_{k}

(1.14)

where t_{k} = t_{k} (y_{0}, {y_{k′}}_{0≺k′≺k}) is a polynomial in {y_{k′}}_{0≺k′≺k} and in {d_{j}}_{j≤k} (see (2.158)), with t(y_{0}, ∅) = 0; t_{k} satisfies the homogeneity relation

t_{k} (y_{0}, {C^{k′}y_{k′}}_{0≺k′≺k}) = C^{k}t_{k} (y_{0}, {y_{k′}}_{0≺k′≺k})  (1.15)

Taking L^{−1} in (1.14) we get, with D_{j} = ∑_{l≥j} \binom{l}{j}[G_{l} ∗ Y_{0}^{∗(l−j)} + g_{0,l} ∗ Y_{0}^{∗(l−j)}],

(−p + Λ̂ − k · λ) Y_{k} + (B̂ + k · m) PY_{k} + ∑_{|j|=1} D_{j} ∗ Y_{k}^{∗j} = T_{k}

(1.16)

where T_{k} is now a convolution polynomial, cf. (2.162).

1.3 Main results

(a) Analytic structure.

Theorem 1 (i) Y_{0} = Bỹ_{0} is analytic in R ∪ {0}.

The singularities of Y_{0} (which are contained in the set {lλ_{j} : l ∈ ℕ^{+}, j = 1, 2, …, n}) are described as follows. For l ∈ ℕ^{+} and small z, using the notations explained in §1.2,

Y_{0}^{±}(z + lλ_{j}) = ±[(±S_{j})^{l}(ln z)^{0,1}Y_{le_{j}}(z)]^{(lm_{j})} + B_{lj}(z) =

[z^{lβ′_{j}−1}(ln z)^{0,1} A_{lj}(z)]^{(lm_{j})} + B_{lj}(z) (l = 1, 2, …) (1.17)

where the power of ln z is one iff lβ_{j} ∈ ℤ, and A_{lj}, B_{lj} are analytic for small z. The functions Y_{k} are, exceptionally, analytic at p = lλ_{j}, l ∈ ℕ^{+}, iff,

S_{j} = r_{j}Γ(β′_{j}) (A_{1,j})_{j} (0) = 0  (1.18)

where r_{j} = 1−e^{2πi(β′_{j}−1)} if lβ_{j} ∉ ℤ and r_{j} = −2πi otherwise. The S_{j} are Stokes constants, see Theorem 5.

(ii) Y_{k} = Bỹ_{k}, |k| > 1, are analytic in R\{−k′ · λ + λ_{i} : k′ ≤ k, 1 ≤ i ≤ n}. For l ∈ ℕ and p near lλ_{j}, j = 1, 2, …, n there exist A = A_{kjl} and B = B_{kjl} analytic at zero so that (z is as above)

Y_{k}^{±}(z + lλ_{j}) = ±[(±S_{j})^{l}\binom{k_{j} + l}{l}(ln z)^{0,1}Y_{k+le_{j}}(z)]^{(lm_{j})} + B_{klj}(z) =

[z^{k·β′+lβ′_{j}−1}(ln z)^{0,1} A_{klj}(z)]^{(lm_{j})} + B_{klj}(z) (l = 0, 1, 2, …) (1.19)

where the power of ln z is 0 iff l = 0 or k · β + lβ_{j} − 1 ∉ ℤ and A_{k0j} = e_{j}/Γ(β′_{j}). Near p ∈ {λ_{i} − k′ · λ : 0 ≺ k′ ≤ k}, (where Y_{0} is analytic) Y_{k}, k ≠ 0 have convergent Puiseux series.

Remark: The fact that the singular part of Y_{k}(p+lλ_{j}) in (1.17) and (1.19) is a multiple of Y_{k+le_{j}}(p) is the effect of resurgence and provides a way of determining the Y_{k} given Y_{0} provided the S_{j} are nonzero. Since, generically, the S_{j} are nonzero this is a surprising upshot: given one formal solution, (generically) an n parameter family of solutions can be constructed out of it, without using (1.3) in the process; the differential equation itself is then recoverable ([16]).

By Theorem 1 the Borel transforms Y_{k} = Bỹ_{k} define germs of ramified analytic functions and are continuable on the surface R. In order to be able to take Laplace transforms we need to define Bỹ_{k} along any direction d in S. If d ≠ d_{j,k} then Y_{k} are analytically continuable along d and the continuations turn out to have all the properties that we need. But along Stokes lines d_{j,k} analytic continuation is impossible: in general the functions Y_{k} have an infinite array of branch points (1.19). In addition, while both Y_{k}^{+} and Y_{k}^{−} turn out to be Laplace transformable (in distributions) along d_{j,k}, LY_{k}^{+} and LY_{k}^{−} are generically different. Neither the upper nor the lower continuation would give rise to a definition of Borel summation which commutes with complex-conjugation, as discussed in the introduction. Other analytic continuations along paths γ that cross d_{j,k} have even worse problems, namely that AC_{γ}(Y_{k} ∗ Y_{k}) ≠ AC_{γ}(Y_{k}) ∗_{γ} AC_{γ}(Y_{k}), (see [13]; for notations only, see also §2.1.5). As B transforms differential equations into convolution equations, the implication is that with such a γ, LAC_{γ}(Y_{k}) would not be, in general, solutions of their differential equations. Individual analytic continuations are thus inadequate for solving (1.3), but some averages of analytic continuations do satisfy all the requirements. Let α = 1/2 + iσ with σ ∈ ℝ and Bỹ_{k} be extended along d_{j,k} by the weighted average of analytic continuations

B_{α}ỹ_{k} = Y_{k}^{α} = Y_{k}^{+} + ∑_{j=1}^{∞} α^{j} (Y_{k}^{−} − Y_{k}^{−(j−1)+})  (1.20)

Remark 2 Relation (1.20) gives the most general reality preserving, linear operator mapping formal power series solutions of (1.3) to solutions of (1.13) in distributions (more precisely in D′_{m,ν}; see §2.1.2).

This remark follows easily from Proposition 23 and Theorem 4 below.

The choice α = 1/2 has special properties; we call B_{\frac{1}{2}} ỹ_{k} = Y_{k}^{ba} the balanced average of Y_{k}. For this choice the expression (1.20) coincides with the one in which + and − are interchanged (Proposition 34), accounting for the realitypreserving property. Clearly, if Y_{k} is analytic along d_{j,k}, then the terms in the infinite sum vanish and Y_{k}^{α} = Y_{k}; we also let Y_{k}^{α} = Y_{k} if d ≠ d_{j,k}, where again Y_{k} is analytic. It follows from (1.20) and Theorem 3 below that the Laplace integral of Y_{k}^{α} along ℝ^{+} can be deformed into contours as those depicted in Fig. 1, with weight −(−α)^{k} for a contour turning around kλ_{1}.

In addition to symmetry (the balanced average equals the half sum of the upper and lower continuations on (0, 2λ_{j}), [16]), an asymptotic property uniquely picks C = 1/2. Namely, for C = 1/2 alone are the LBỹ_{k} always summable to the least term cf. § 2.1.6.

(b) Connection with (1.3) and (1.15). Generalized Borel summation coincides with the usual Borel summation when the transseries consists of only one term, the first series, when that series is classically Borel summable. This is clear from theorem 3 (ii) below. Furthermore, generalized summation is a map from a class of formal series to functions which is linear, multiplicative, commutes with differentiation and complex conjugation (cf. § 2.1.6, § 2.1.2) so it is a summation procedure, which furthermore, establishes along every direction a one to one correspondence between transseries and decaying actual solutions of (1.3) cf. § 2.1.6, Proposition 34 and Theorem 3 below.

For clarity we state the results for x ∈ S_{x}, a sector in the right half plane containing λ_{1} = 1 in which (c1) holds and for p in the associated domain R′_{1}, but λ_{1} plays no special role as discussed in the introduction.

Theorem 3 (i) The branches of (Y_{k})_{γ} in R′_{1} (R_{1} if k = 0) have limits in a C^{∗}-algebra of distributions, D′_{m,ν}(ℝ^{+}) ⊂ D′ (cf. § 2.1.2). Their Laplace transforms in D′_{m,ν}(ℝ^{+}) L(Y_{k})_{γ} exist simultaneously and with x ∈ S_{x} and for any δ > 0 there is a constant K and an x_{1} large enough, so that for ℜ(x) > x_{1} we have |L(Y_{k})_{γ} (x)| ≤ Kδ^{|k|}.

In addition, Y_{k}(pe^{iϕ}) are continuous in ϕ with respect to the D′_{m,ν} topology, (separately) on (ψ_{−}, 0] and [0, ψ_{+}) .

If m > max_{i}(m_{i}) and l < min_{i} |λ_{i}| then Y_{0}(pe^{iϕ}) is continuous in ϕ ∈ [0, 2π]\{ϕ_{i} : i ≤ n} in the D′_{m,ν}(ℝ^{+}, l) topology and has (at most) jump discontinuities for ϕ = ϕ_{i}. For each k, |k| ≥ 1 and any K there is an l > 0 and an m such that Y_{k}(pe^{iϕ}) are continuous in ϕ ∈ [0, 2π]\{ϕ_{i}; −k′ ·λ+λ_{i} : i ≤ n, k′ ≤ k} in the D′_{m,ν}((0, K), l) topology and have (at most) jump discontinuities on the boundary.

(ii) The sum (1.20) converges in D′_{m,ν} (and coincides with the analytic continuation of Y_{k} when Y_{k} is analytic along ℝ^{+}). For any δ there is a large enough x_{1} independent of k so that Y_{k}^{ba}(p) with p ∈ R′_{1} are Laplace transformable in D′_{m,ν} for ℜ(xp) > x_{1} and furthermore |(LY_{k}^{ba})(x)| ≤ δ^{|k|}. In addition, if d ≠ ℝ^{+}, then for large ν, Y_{k} ∈ L_{ν}^{1}(d).

The functions LY_{k}^{ba} are analytic for ℜ(xp) > x_{1}. For any C ∈ ℂ^{n_{1}} there is an x_{1}(C) large enough so that the sum

y = LY_{0}^{ba} + ∑_{|k|>0} C^{k}e^{−k·λx}x^{−k·β}LY_{k}^{ba}  (1.21)

converges uniformly for ℜ(xp) > x_{1}(C), and y is a solution of (1.3). When the direction of p is not the real axis then, by definition, Y_{k}^{ba} = Y_{k}, L is the usual Laplace transform and (1.21) becomes

y = LY_{0} + ∑_{|k|>0} C^{k}e^{−k·λx}x^{−k·β}LY_{k}  (1.22)

In addition, LY_{k}^{ba} ∼ ỹ_{k} for large x in the half plane ℜ(xp) > x_{1}, for all k, uniformly.

iii) More generally, for any α and any solution y of (1.3) such that y ∼ ỹ_{0} for large x along a ray in S_{x} there exists a constant vector C = C_{α;y} so that

y = LB_{α}ỹ_{0} + ∑_{|k|>0} C^{k}e^{−k·λx}x^{−k·β}LB_{α}ỹ_{k}  (1.23)

Given α the representation (1.23) of y is unique (see also § 1.1.1 above for the convention on the direction of Laplace integration).

Of special interest are the cases α = 1/2, discussed above, and also α = 0, 1 which give:

y = LY_{0}^{±} + ∑_{|k|>0} C^{k}e^{−k·λx}x^{−k·β}LY_{k}^{±}  (1.24)

(c) Resurgence properties; local Stokes phenomenon.

It turns out that the formal series ỹ_{k} are connected among each-other via their Borel transforms. Resurgence formulas link Y_{k} to analytic continuations of Y_{k′} with k′ ≺ k, in a way that, generically, Y_{0} contains enough information to compute all Y_{k}.

Various resurgence properties have been observed in different contexts, and the term resurgence has been used with slightly different interpretations. In the hyperasymptotic theory of M. Berry, it was discovered that the first asymptotic series reappears in various shapes in the process of computing higher terms of the expansions. J. Écalle, in his comprehensive theory of analyzable functions, has obtained a general resurgence principle, the bridge equation [2]. The common denominator of resurgence is the reappearance of “earlier” terms in the formulas of “later” ones. It turns out that, for our problem, resurgence is fundamentally linked to the Stokes phenomenon. In the following formulas we make the convention Y_{k}(p − j) = 0 for p < j as an element of D′_{m,ν}(ℝ^{+}). We again state the results is stated for p ∈ R′_{1} and x ∈ S_{x} but hold in any sector where (c1) is valid.

Theorem 4 i) For all k and ℜ(p) > j, ℑ(p) > 0 as well as in D′_{m,ν} we have

Y_{k}^{±j∓}(p) − Y_{k}^{±(j−1)∓}(p) = (±S_{1})^{j}\binom{k_{1} + j}{j} (Y_{k+je_{1}}^{∓}(p − j))^{(mj)} (1.25)

and also,

Y_{k}^{±} = Y_{k}^{∓} + ∑_{j≥1} \binom{j + k_{1}}{k_{1}}(±S_{1})^{j}(Y_{k+je_{1}}^{∓}(p − j))^{(mj)}  (1.26)

ii) Local Stokes transition. Consider the expression of a fixed solution y of (1.3) as a Borel summed transseries (1.21). As arg(x) varies, (1.21) changes only through C, and that change occurs when Stokes lines are crossed (cf. §1.1.2; the Stokes lines of Y_{0} are the directions of λ_{i}). We have, in the neighborhood of ℝ^{+}, with S_{1} defined in (1.18):

C(ξ) = \{ \begin{matrix}C^{−} = C(−0)  for ξ < 0 \\ C^{0} = C(−0) + \frac{1}{2}S_{1}e_{1}  for ξ = 0 \\ C^{+} = C(−0) + S_{1}e_{1}  for ξ > 0 \end{matrix}  (1.27)

(d) Classical Stokes phenomena and local Stokes transitions. Again we formulate the result below for λ_{1} but with straightforward adjustments it holds relative to any other eigenvalue. Let C be of the form C_{1}e_{1}. Along the imaginary axis, condition (c1) fails. The positive and negative imaginary are the antistokes lines corresponding to λ_{1} = 1 (note: sometimes called Stokes lines!). If we choose paths in the right half plane approaching the positive/negative imaginary axis in such a way that |x^{−β_{1}−l}e^{−x}| → K ≠ 0 along them, where l + β ∈ (0, M), then y ∼ C^{±}x^{−l−β_{1}}e^{−x} + ỹ_{0} for large x and the term multiplied by K is now the leading behavior of y. The particular choice of K and l within this range is rather arbitrary, the main point being that along such special curves, the constant C is definable in terms of classical asymptotics. Within the right half plane, it is only near the imaginary axis that this happens, since otherwise the exponential term is smaller than all terms of ỹ_{0}. On the other hand Borel summation makes possible the definition of C throughout the right half plane, and we now address the issue of the relation between classical asymptotics and exponential asymptotics.

Theorem 5 Let γ^{±} be two paths in the right half plane, near the positive/ negative imaginary axis such that |x^{−β_{1}+1}e^{−xλ_{1}}| → 1 as x → ∞ along γ^{±}.

Consider the solution y of (1.3) given in (1.21) with C = Ce_{1} and where the path of integration is p ∈ ℝ^{+}. Then

y = (C ± \frac{1}{2}S_{1})e_{1}x^{−β_{1}+1}e^{−xλ_{1}}(1 + o(1))  (1.28)

for large x along γ^{±}, where S_{1} is the same as in (1.18), (1.27).

Classical asymptotics loses track of the value of C along any ray other than the imaginary directions, as the terms multiplied by C will be hidden “beyond all orders” of the classically divergent series ỹ_{0}. In contrast to the classical picture, we see that through generalized Borel summation the constant C is precisely defined throughout the positive half-plane and the question of where the change in C occurs is well defined.

Formula (1.27) is the exponential asymptotic expression of the Stokes phenomenon. It shows that the constant jumps as the Stokes line is crossed, (1.27), as originally predicted by Stokes himself [10]. Subsequently, the original ideas of Stokes, based on optimal truncation of series were greatly refined by M. Berry, leading to his theory of hyperasymptotic expansions and a description of Stokes transitions for saddle integrals [22].

If more than one component of C is nonzero, then in general there is no direction along which C can be defined through classical asymptotics. Part of the difficulty of studying nonlinear Stokes phenomena using classical tools stems from this fact.

Relation (1.27) expresses the evolution of C and the presence of a Stokes phenomenon beyond all orders of Poincaré asymptotics.

2 Proofs and further results

The layout of the proofs is as follows. We study the formal inverse Laplace transforms of (1.3) and (1.14) in a C^{∗}-algebra of distributions that we introduce. Using a fixed point principle we find the general solution of these convolution equations, and then study their properties. We then show that generalized Laplace transforms of these distributions exist and have all the required properties. The resurgence formulas are obtained by comparing different expressions for the same solution of (1.3) near a Stokes line.

Since the proofs rely to a large extent on the detailed study of the convolution equations (1.13), (1.16), we start with a few heuristic remarks. In a convolution equation such as (1.13), the term (−p + Λ̂)Y plays a role similar to that of the highest derivative term in a differential equation. To illustrate this, assume a solution Y is already given on an interval, say (0, a), and we wish to extend it to (0, a + ϵ). We look for such a solution in the form Y + δ, where we take Y = 0 on (a, a + ϵ) and δ = 0 on (0, a). If ϵ is small, then δ ∗ δ = 0 and the equation in δ is linear inhomogeneous. The terms that involve integrals of δ are of order O(ϵ)∥δ∥ as ϵ → 0, so that the dominant terms are the forcing term, together with (−p+Λ̂)δ provided the coefficient is invertible. If in addition the forcing is non-singular then δ can be found, e.g., by a convergent ϵ expansion; this is the analog of an ordinary point of a differential equation. δ can be singular if either Δ_{p} = det (p − Λ̂) = 0 or the forcing is singular. To understand the qualitative behavior near a zero of Δ_{p} one has to keep (at least) one more term, the leading term among those previously discarded (i.e. the second term in the notation (1.16)). In this approximation, δ satisfies a differential equation.

In our problem, there are n roots of Δ_{p} but because of the nonlinearity and nonlocality of the equations, a singularity generates (when convolved with itself) a whole array of singularities affecting δ through the forcing term. Through convolution, a nonintegrable singularity produces further singularities of even lower regularity. We introduce a distribution space, D′_{m,ν,l}, whose degree of regularity decreases with the distance from the origin, at a “convolution-like” rate; these distributions form a convolution Banach algebra (cf. § 2.1.2).

Technically, the proofs rely on suitable fixed-point theorems in spaces having some of the properties we want to prove, notably in terms of regularity and behavior at infinity. This is combined with a local analysis near noninvertibility points of the dominant term, which is done by treating the convolution equations as a perturbation of the approximating differential equation mentioned above, which splits the singularity thus making again possible the use of fixed point theorems. This analysis is used in order to find the resurgence properties, which in turn are used to prove (using among others Lemma 11 below) the sharper results on global analyticity and structure of singularities.

We start by introducing some useful functional spaces and derive specific fixed point theorems.

2.1 Technical constructions and results

2.1.1 Focusing spaces and algebras

We say that a family of norms ∥∥_{ν} depending on a parameter ν ∈ ℝ^{+} is focusing if for any f with ∥f∥_{ν_{0}} < ∞

∥f∥_{ν} ↓ 0 as ν ↑ ∞  (2.1)

Let E be a linear space and {∥∥_{ν}} a family of norms satisfying (2.1). For each ν we define a Banach space B_{ν} as the completion of {f ∈ E : ∥f∥_{ν} < ∞}. Enlarging E if needed, we may assume that B_{ν} ⊂ E. For α < β, (2.1) shows that the identity is an embedding of B_{α} in B_{β}. Let F ⊂ E be the projective limit of the B_{ν}. That is to say

F := ⋃_{ν>0} B_{ν}  (2.2)

is endowed with the topology in which a sequence is convergent if it converges in some B_{ν}. We call F a focusing space.

Consider now the case when (B_{ν}, +, ∗, ∥∥_{ν}) are commutative Banach algebras. Then F inherits a structure of a commutative algebra, in which ∗ (“convolution”) is continuous. We say that (F, ∗, ∥∥_{ν}) is a focusing algebra.

2.1.2 Examples

Let K ∈ ℝ^{+} and S = S_{K,α_{1},α_{2}} = {p : arg(p) ∈ [α_{1}, α_{2}] ⊂ (−π/2, π/2), |p| ≤ K} (or a finite union of such sectors) and V be a small neighborhood of the origin. \overline{V} will be the closure of V, cut along the negative axis, and together with these upper and lower cuts.

(1). L^{1}_{ν}(K). Let K = S_{K,ϕ,ϕ}. The space L^{1}_{ν}(K) with convolution (1.8) is a commutative Banach algebra under each of the (equivalent) norms

∥f∥_{ν} = ∫_{0}^{K} e^{−νt}|f(t exp(iϕ))|dt  (2.3)

Indeed, with F (s) := f(se^{iϕ}) and G(s) := g(se^{iϕ}) we have:

∫_{0}^{K} dte^{−νt} |∫_{0}^{t} dsF (s)G(t − s)| ≤ ∫_{0}^{K} dte^{−νt} ∫_{0}^{t} ds|F (s)G(t − s)|

= ∫_{0}^{K} ∫_{0}^{K−v} e^{−ν(u+v)}|F (v)||G(u)|dudv

≤ ∫_{0}^{K} ∫_{0}^{K} e^{−ν(u+v)}|F (v)||G(u)|dudv = ∥f∥_{ν}∥g∥_{ν} (2.4)

By dominated convergence ∥f∥_{ν} ↓ 0 as ν ↑ ∞ and thus L^{1}(K) is a focusing algebra.

(2) If K = ∞ in example (1), then the norms (2.3) are not equivalent anymore for different ν, but convolution is still continuous in (2.3) and the projective limit of the L_{ν}^{1}(ℝ^{+}e^{iϕ}), F(ℝ^{+}e^{iϕ}) ⊂ L_{loc}^{1}(ℝ^{+}e^{iϕ}), is a focusing algebra.

(3a) T_{β}(S ∪ \overline{V}). For ℜ(β) > 0 and ϕ_{1} ≠ ϕ_{2}, this space is given by \{f : f(p) = p^{β}F (p)}, where F is analytic in the interior of S ∪ V and continuous in its closure. We take the family of (equivalent) norms

∥f∥_{ν,β} = K sup_{s∈S∪\overline{V}} |e^{−νp}f(p)|  (2.5)

It is clear that convergence of f in ∥∥_{ν,β} implies uniform convergence of F on compact sets in S ∪ V (for p near zero, this follows from Cauchy’s formula). T_{β} are thus Banach spaces and focusing spaces by (2.5). The spaces {T_{β}}_{β} are isomorphic to each-other. Taking s = pt in (1.8) we find that

p^{−β_{1}−β_{2}−1}(f_{1} ∗ f_{2})(p) = ∫_{0}^{1} t^{β_{1}}F_{1}(pt)(1 − t)^{β_{2}}F_{2}(p(1 − t))dt = F (p) (2.6)

where F is manifestly analytic, and that the application

(· ∗ ·) : T_{β_{1}} × T_{β_{2}} ↦ T_{β_{1}+β_{2}+1}  (2.7)

is continuous:

∥f_{1} ∗ f_{2}∥_{ν,β_{1}+β_{2}+1} = K sup_{p} |e^{−νp} ∫_{0}^{p} s^{β_{1}}F_{1}(s)(p − s)^{β_{2}}F_{2}(p − s)ds|

≤ K^{−1} sup_{p} ∫_{0}^{p} |Ke^{−νs}s^{β_{1}}F_{1}(s)Ke^{−ν(p−s)}(p − s)^{β_{2}}F_{2}(p − s)| d|s|

≤ ∥f_{1}∥_{ν,β_{1}}∥f_{2}∥_{ν,β_{2}}  (2.8)

A natural generalization of T_{β} is obtained taking β_{1}, …, β_{N} ∈ ℂ with positive real parts, no two of them differing by an integer. If f_{β} = ∑_{i=1}^{k} p^{β_{i}}A_{i}(p) with A_{i} analytic, then f_{β} ≡ 0 iff A_{i} ≡ 0 for all i (e.g., by a Puiseux series argument). It is then natural to identify the space T_{{β_{1},... ,β_{k}}} of functions of the form f_{β} with ⊕_{i=1}^{k}T_{β_{i}}. Convolution with analytic functions is defined on T_{{β_{1},... ,β_{k}}} while convolution of two functions in T_{{β_{1},... ,β_{k}}} takes values in T_{{β_{i}+β_{j} mod 1}}. We write T_{{·}} when the concrete values of β_{1}, …, β_{k} do not matter.

(3b) A particular case of the preceding example is A_{z,l}(S ∪ V) consisting of analytic functions in the interior of S∪V, continuous on its closure, and vanishing at the origin together with the first l derivatives. A_{z,l} can be identified with T_{l}.

(4) D′_{m,ν}, the “staircase distributions”. Proofs of the properties stated in this paragraph and more details are given in § 2.3.1. Let D(0, x) be the test functions on (0, x) and D = D(0, ∞). Let D′_{m} ⊂ D′ be the distributions f for which f = F_{k}^{(km)} on D(0, k + 1) with F_{k} ∈ L^{1}[0, k + 1]. There is a uniquely associated “staircase decomposition”, a sequence {Δ_{i}(f)}_{i∈ℕ} = {Δ_{i}}_{i∈ℕ} such that Δ_{i} ∈ L^{1}(ℝ^{+}), Δ_{i} = Δ_{i}χ_{[i,i+1]} and

f = ∑_{i=0}^{∞} Δ_{i}^{(mi)}  (2.9)

Convolution is well defined on D′_{m} by

f ∗ f̃:= (F_{k} ∗ F̃_{k})^{(2km)} in D′(0, k + 1)  (2.10)

and (D′_{m}, +, ∗) is a commutative algebra. We define, for f ∈ D′_{m},

∥f∥_{ν,m} := c_{m} ∑_{i=0}^{∞} ν^{im}∥Δ_{i}∥_{ν}  (2.11)

where c_{m} is defined in Lemma 39, and ∥Δ∥_{ν} is computed from (2.3) with K = ∞. Then (2.11) is a norm on D′_{m} and D′_{m,ν} = (D′_{m}, +, ∗, ∥∥_{m,ν}) is a Banach algebra.

With respect to the family of norms ∥∥_{m,ν}, the projective limit of the D′_{m,ν}, F_{m} is a focusing algebra.

For any f ∈ L_{ν_{0}}^{1}(ℝ^{+}) there is a constant C(ν, ν_{0}) such that f ∈ D′_{m,ν} for all ν > ν_{0} and

∥f∥_{D′_{m,ν}} ≤ C(ν_{0}, ν)∥f∥_{L_{ν_{0}}^{1}}  (2.12)

and formula (2.10) is equivalent to (1.8), in this case.

The operator f(p) ↦ pf(p) is continuous from D′_{m,ν} to D′_{m,ν+δ} for any δ > 0. For a ∈ ℝ^{+}, D′_{m,ν}(a, ∞) = {f ∈ D′_{m,ν} : Δ_{i}(x) = 0 for x < a} is a closed

ideal in D′_{m,ν} (isomorphic to the restriction D′_{m,ν}(a, ∞) of D′_{m,ν} to D(a, ∞)). The restrictions D′_{m,ν}(a, b) of D′_{m,ν} to D(a, b) are for 0 < a < b < ∞ Banach spaces with respect to the norm (2.11) restricted to (a, b).

The functions in D (ℝ^{+}\ℕ) are dense in D′_{m,ν}, with respect to the norm (2.11) (Lemma 41).

If we choose a different interval length l > 0 instead of l = 1 in the partition associated to (2.9), we then write D′_{m,ν}(l). Obviously, dilation gives a natural isomorphism between these structures. If d = {te^{iϕ} : t ∈ ℝ^{+}} is any ray, D′_{m,ν}(d) and F_{m;ϕ} are defined in an analogous way and have the same properties as their real counterpart.

Laplace transforms are naturally defined in D′_{m,ν}.

Lemma 6 Laplace transform extends continuously from D(ℝ^{+}\ℕ) to D′_{m,ν}(ℝ^{+}) by the formula

(Lf)(x) := ∑_{k=0}^{∞} x^{mk} ∫_{0}^{∞} e^{−sx}Δ_{k}(s)ds  (2.13)

In particular, with f, g, h′ ∈ D′_{m,ν} we have

L(f ∗ g) = L(f)L(g)
L(h′) = xL(h) − h(0)
L(pf) = −(L(f))′  (2.14)

For x ∈ S_{ν} = {x : ℜ(x) > ν} the sum (2.13) converges absolutely. Laplace transform is, for fixed x ∈ S_{ν}, a continuous functional (of norm less than one) on D′_{m,ν}.

(Lf)(x) is analytic in S_{ν}.

Furthermore, L is injective in D′_{m,ν}.

The proof is given in § 2.3.4. We conclude this section with a few remarks.

Remark 7 Let U be one of the spaces considered in the examples and ν be large:

i) if g is analytic, (and if g ∈ U in Examples (2) and (4)) then L_{g} := f ↦ f ∗ g is a bounded operator and ∥L_{g}∥ = O(ν^{−1}) (P is such an operator, since Pf = f ∗ 1);

ii) replacing ∗ by ∗_{ϕ} defined as f ∗_{ϕ} g := e^{iϕ}(f ∗ g) leads to an isomorphic structure;

iii) if g ∈ U is a real valued nonnegative function and |f| ≤ g (|P^{km}f| ≤ P^{km}g for all k, on (0, k + 1) if the space is D′_{m,ν}) then ∥f∥ ≤ ∥g∥.

(i) In L_{ν}^{1} and D′_{m,ν} this follows from the continuity of convolution and (2.12). In the examples (3), the natural inclusion T_{β+ℕ} ⊂ T_{β} together with (2.7) and (2.8) makes convolution with an analytic function continuous in T_{β} and the claim follows from the estimate ∥f ∗ g∥ ≤ max |g| ∥f∥ ∥P|e^{νp}|∥.

(ii) The isomorphism is given by f ↦ e^{−iϕ}f.

(iii) Since P is positivity preserving, writing |f| ≤ g as −g ≤ (ℜ, ℑ)f ≤ g the property is obvious when f, g are functions, while for D′_{m,ν} it follows from equation (2.98) below.

2.1.3 Vectorial convolution and focusing spaces

We endow B_{ν}^{n} with a Banach space structure by identifying it with B_{ν} ⊕⋯⊕B_{ν} (n times). The projective limit of the B_{ν}, F^{n} is, clearly, a focusing space. We define a convolution on (· ∗ ·) : F^{n} ↦ F (not F^{n} ↦ F^{n}) by

V ∗ W := ∑_{i,j=1}^{n} V_{i} ∗ W_{j}  (2.15)

We write V^{∗l} := V_{1}^{∗l_{1}} ∗ V_{2}^{∗l_{2}} ∗ ⋯ ∗ V_{n}^{∗l_{n}} with the conventions V^{∗1} = V and that the factors with l_{i} = 0 are omitted.

2.1.4 A fixed point property

Lemma 8 Let F be a focusing space and N be a (linear or nonlinear) operator defined on F. Equivalently, in view of (2.1), let N be defined on ⋃_{ν>ν_{0}} B_{ν}(δ) with B_{ν}(δ) = {f : ∥f∥_{ν} ≤ δ} for some δ > 0. Assume N is eventually contractive in the following sense. There exist ν_{0}, ϵ > 0, α < 1, so that if ν ≥ ν_{0} and ∥f∥_{ν} + ∥g∥_{ν} ≤ ϵ then

∥N (f + g) − N (g)∥_{ν} ≤ α∥g∥_{ν}  (2.16)

Then N has a unique fixed point f_{0} ∈ F.

If N depends continuously (in the strong topology) on a parameter ϕ for ν > ν_{0} and if the constants ν_{0}, α and ϵ above do not depend on ϕ, then the fixed point f_{ϕ} is also continuous in ϕ. Furthermore, lim_{ν→∞} sup_{ϕ} ∥f_{ϕ}∥_{ν} = 0.

The proof is straightforward. To show existence, take ν > ν_{0} large enough so that, by (2.1) ∥N (0)∥_{ν} < (1 − α)ϵ. Then the closed ball B_{ν}(ϵ) is mapped by N into itself for any ϕ by (2.16) and N is contractive there. The fixed point obtained, for instance, as the limit of the (convergent, uniformly in ϕ) iteration ϕ_{n+1} = N (ϕ_{n}); ϕ_{0} = 0 is continuous in ϕ since N is. By construction ∥f_{0}∥_{ν} ≤ ϵ, for all ϕ.

For uniqueness, let f_{0} and f_{1} be fixed points of N; by (2.1) there is a ν > ν_{0} so that ∥f_{0,1}∥_{ν} < ϵ. Then by (2.16), ∥f_{0} − f_{1}∥_{ν} ≤ α∥f_{0} − f_{1}∥_{ν} and thus f_{0} = f_{1}.

.

Let ν_{0} > 0 and let {M}_{l} be a sequence of linear operators M_{l} : B_{ν} ↦ B_{ν}^{n} for ν ≥ ν_{0}. Assume that for some κ and all l, |l| ≥ 1,

∥M_{l}∥_{ν} ≤ C_{ν}κ^{|l|} and C_{1,ν} := lim_{ν→∞} max_{|l|=1} ∥M_{l}∥_{ν} = 0  (2.17)

Let F_{0} ∈ F^{n} and M : F^{n} ↦ F^{n} be defined by

M(Y) := F_{0} + ∑_{|l|≥1} M_{l} (Y^{∗l})  (2.18)

Lemma 9 M satisfies the assumptions of Lemma 8. M has therefore a unique fixed point in F^{n}.

Proof. We first need the following estimate.

Remark 10 Let V, W ∈ F^{n}. For |l| > 0 and any ν we have, with ∥∥ = ∥∥_{ν},

∥W_{l}∥ := ∥(V + W)^{∗l} − V^{∗l}∥ ≤ |l| (∥V∥ + ∥W∥)^{|l|−1} ∥W∥  (2.19)

This inequality is obtained by induction on l, with respect to ≺. For |l| = 1, (2.19) is trivial. Assume (2.19) holds for all l ≺ l_{1}; without loss of generality we may consider that l_{1} = l_{0} + e_{1}. We have:

∥(V + W)^{∗l_{1}} − V^{∗l_{1}}∥ = ∥(V + W)^{∗l_{0}} ∗ (V_{1} + W_{1}) − V^{∗l_{1}}∥

= ∥(V^{∗l_{0}} +W_{l_{0}}) ∗ (V_{1} + W_{1}) −V^{∗l_{1}}∥ = ∥V^{∗l_{0}} ∗W_{1} + W_{l_{0}} ∗V_{1} + W_{l_{0}} ∗W_{1}∥

≤ ∥V∥^{|l_{0}|}∥W∥ + ∥W_{l_{0}}∥∥V∥ + ∥W_{l_{0}}∥∥W∥

≤ ∥W∥ (∥V∥^{|l_{0}|} + |l_{0}|(∥V∥ + ∥W∥)^{|l_{0}|}) ≤ ∥W∥(|l_{0}| + 1)(∥V∥ + ∥W∥)^{|l_{0}|}

and (2.19) is proven.

For the sum in M to converge in ∥∥_{ν} it suffices to choose ν such that ∥V∥_{ν} < κ^{−1}. Let ϵ < κ^{−1} and V, W be such that ∥V∥ + ∥W∥ < ϵ. We have

∥M(V + W) − M(V)∥_{ν} ≤ (nC_{1,ν} + C_{ν} ∑_{|l|≥2} |l|(κϵ)^{|l|−1}) ∥W∥_{ν}

≤ (nC_{1,ν} + n2^{n}C_{ν}κϵ + nC_{ν} \frac{(2 − κϵ)(κϵ)^{1+2(n−1)}}{(1 − κϵ)^{n+1}} ) ∥W∥_{ν} = K_{ν}∥W∥_{ν} (2.20)

where we separated out the terms with |l| = 2, l_{i} = 0 or 1 and for the rest of the terms used the identity ∑_{l_{i}≥2} |l|e^{−γ|l|} = −\frac{d}{dγ} (∑_{l≥2} e^{−γl})^{n}. We see that, in fact, lim_{ν→∞} K_{ν} = 0.

2.1.5 Two lemmas on analytic structure

Lemma 11 Let f be analytic in the unit disc cut along the positive axis and let 0 < g(x) ∈ C^{1}[0, 1]. Assume that lim_{ϵ↓0} f(x ± iϵg(x)) = f^{±}(x) in L^{1}[0, 1] and

f^{+}(x) − f^{−}(x) = f_{δ}(x) = x^{r}A(x)  (2.21)

with ℜ(r) > −1, where A(ξ) extends to an analytic function for |ξ| < a ≤ 1. Then there exists a function B analytic in |ξ| < a so that

f(ξ) = \frac{1}{1 − e^{2πir}} ξ^{r}A(ξ) + B(ξ) (r ∉ ℕ)

f(ξ) = \frac{i}{2π} ln(ξ)ξ^{r}A(ξ) + B(ξ) (r ∈ ℕ)

(2.22)

If f^{+}(x) − f^{−}(x) is a linear combination ∑_{i=1}^{N} x^{r_{i}}A_{i}(x) (under the same assumptions on r_{i} and A_{i}), then f is given by the corresponding superposition of terms of the form (2.22).

The proof is given in § 2.3.4.

In the following, γ : ℝ^{+} ↦ ℂ will denote smooth curves in R′_{1}, γ_{k} denotes a curve that crosses through the interval (k, k + 1), γ_{ϵ} = ℜ(γ) + iϵℑ(γ) (cf.§ 1.2). Let a, b ∈ (0, π/2) and S_{0} = {p : arg(p) ∈ (ψ_{−}, 0)∪(0, ψ_{+})}. Let f be a function analytic in R′_{1} so that f ∘ γ_{ϵ} ∈ D′_{m,ν} has limits in D′_{m,ν} as ϵ ↓ 0. We denote the space of such functions by D′_{m,ν}(R_{1}). Let f_{0}^{±} = f^{±}, F_{j} = P^{(mj)}f, and for j > 0

f_{j}^{+}(z − j) = F_{j}^{−j+}(z) − F_{j}^{−(j−1)+}(z); f_{j}^{−}(z − j) = F_{j}^{+j−}(z) − F_{j}^{+(j−1)−}(z) (2.23)

By construction the f_{j}^{+} are in L^{1}[0, 1 − ϵ), analytic in a sectorial neighborhood of z = 0 and can be extended analytically for ℑ(z) > 0, ℜ(z) > 0 (this last property motivates the choice + for superscript, while the right shift is chosen in view of our application). Also by construction f_{j}^{±}(z) = 0 for z < 0; it is convenient to extend f_{j}^{±} by zero throughout ℜ(z) < 0. We have the “telescopic” decomposition

f^{∓ j ±}(z) = ∑_{i=0}^{j}(f_{i}^{±})^{(mi)}(z − i)  (2.24)

Relation (2.24) holds in D′ along the real axis, and as an equality of analytic functions for ℜ(z) > j, ℑ(z) > 0. For instance, for f = (z − 2)^{−1} ∈ D′_{1ν} we have f_{1}^{+} = 0, f_{2}^{+} = −2πiz for ℜ(z) > 0, and f_{2}^{+} = 0 otherwise.

Conversely, a decomposition of the form (2.24) together with analyticity in S_{0} implies analyticity in R′_{1}. More precisely, assume that f(t exp(iϕ)) ∈ D′_{m,ν}(ℝ^{+}) for ϕ ∈ (ψ_{−}, 0) ∪ (0, ψ_{+}) and that lim_{ϕ↓0} f(· exp(±iϕ) = f^{±} in D′_{m,ν}. Assume in addition that there exists the decomposition

f^{±} = ∑_{k=0}^{∞} [f_{k}^{∓}(p − k)χ_{[k,∞]}]^{(mk)}  (2.25)

where for each k, f_{k}^{∓} ∈ D′_{m,ν}(ℝ^{+}) (note: the f_{k}^{∓}χ are uniquely determined, cf. Remark 43). Assume in addition that f_{k}^{∓} extend analytically to S_{0}^{∓} in the following sense: there exist g_{k}^{∓} analytic in S_{0}^{∓}, with g_{k}^{∓}(∓t exp(iϕ)) ∈ D′_{m,ν}(ℝ^{+}) and such that lim_{ϕ↓0} g_{k}^{∓}(∓t exp(iϕ)) = f_{k}^{∓} in D′_{m,ν}(ℝ^{+}).

Lemma 12 (i) Under the above conditions, f extends analytically to R′_{1}. (ii) If for small argument f_{k} ∈ T_{β} then

f^{±}(p + k) = \frac{1}{1 − e^{2πiβ}} (f_{k}^{∓}(p))^{(mk)} + a(p) or \frac{i}{2π} (f_{k}^{∓}(p) ln p)^{(mk)} + a(p) (2.26)

according to whether β ∉ ℤ or β ∈ ℤ respectively, where a is analytic at zero. As in Lemma 11, if f_{k} = ∑_{i≤j} f_{ki} with f_{ki} ∈ T_{β_{i}} then f^{±}(p + k) is the corresponding superposition of terms of the form (2.26).

The proof is given in § 2.3.4.

2.1.6 Convolutions, analyticity and averaging

Define F(R′_{1}) as the functions in D′_{m,ν}(R′_{1}) such that, in the decomposition (2.24) ∥(f_{j}(p − j))^{(mj)}∥ ≤ K_{f}(ν)^{j} where lim_{ν→∞} K_{f}(ν) = 0.

If γ is a straight line in R′_{1} then AC_{γ}(f ∗ g) = AC_{γ}(f)_{γ} ∗ AC_{γ}(g) (if γ is not equivalent to a straight line, this equality is generally false [13]). Convolution does however commute with suitable averages of analytic continuations, as Proposition 13 below shows. In view of symmetry, we only need to look at the properties of the + decomposition.

For α ∈ ℂ, consider the operator A_{α} : F(R′_{1}) ↦ F_{m}(ℝ^{+}) given by

A_{α}(f) := f^{[α]}(p) = ∑_{i=0}^{∞} α^{i} (f_{i}^{+}(p − j))^{(mi)}  (2.27)

In our assumptions, convergence is ensured in D′_{m,ν} for large enough ν. An important case is the balanced average, α = 1/2. The operator A_{\frac{1}{2}} is similar to Écalle’s medianization, and is designed to substitute for analytic continuation along the singularity line ℝ^{+} in a way compatible with the ∗−algebra structure. As mentioned before, it can be shown that under our assumptions on (1.3), only for the choice α = 1/2 is the difference between the f = LA_{α}F and the optimally truncated asymptotic series of f always of the order of magnitude of the least term of the series [15]. Borel summability techniques and hyperasymptotic methods [22], [23] give, whenever they both apply, the same association between transseries and actual functions.

Proposition 13 i) If f, g ∈ F(R′_{1}) then f ∗ g defined for small argument by (1.8) extends to a function in F(R′_{1}). We have

(f ∗ g)_{j}^{±} = ∑_{s=0}^{j} f_{s}^{±} ∗ g_{j−s}^{±}  (2.28)

and K_{f∗g}(ν) ≤ K_{f}(ν) + K_{g}(ν). If h is analytic and bounded in the right half plane and f ∈ F(R′_{1}) then hf ∈ F(R′_{1}).

ii) If h is analytic in R′_{1} ∪ ℝ^{+} and f, g ∈ F(R′_{1}), a, b ∈ ℂ, then

A_{α}(af + bg) = aA_{α}(f) + bA_{α}(g)
A_{α}(hf) = hA_{α}(f)
A_{α}(1) = 1
A_{α}(f ∗ g) = A_{α}(f) ∗ A_{α}(g)  (2.29)

If M satisfies the hypothesis of Lemma 8 and in addition M_{l}(F(R′_{1})) ⊂ F(R′_{1}) and A_{α}M_{l} = M_{l}A_{α} then

A_{α}M = MA_{α}  (2.30)

In particular, if Y is a fixed point of M then so is A_{α}Y. An example is the case M_{l}(Y) = G_{l} ∗ Y^{∗l} with G_{l} analytic in R′_{1} ∪ ℝ^{+} and such that for some κ and all l we have |G_{l}(p)| ≤ exp(κp).

The proof and further details are given in Appendix 2.3.4.

Let now F_{r}(R′_{1}) ⊂ F(R′_{1}) consist in functions f whose only singularities are regular, in the sense that the elements f_{j} (cf. (2.23)) are of the form (∑_{i=1}^{N_{j}} p^{a_{i,j}}A_{i,j})^{(mj)} where A_{i,j} are analytic near p = 0.

Remark 14 F_{r}(R′_{1}) is stable with respect to convolution.

By construction, (Proposition 13) and (2.28), for small p, (f_{1} ∗ f_{2})_{j} is a sum of terms of the form

p^{a_{1}}A_{1} ∗ p^{a_{2}}A_{2}

and the proof follows without any difficulty from (2.6).

2.2 Main proofs

Proposition 15 i) For any κ > max{x_{0}, y_{0}^{−1}}, cf. (1.11), there is a constant K > 0 such that for all l ≻ 0

sup_{p∈ℂ} e^{−κ|p|}|G_{l}(p)| ≤ Kκ^{|l|}  (2.31)

(cf. (1.13)).

ii) Let F be one of the focusing algebras in § 2.1.2 and Y ∈ F. Let

D_{j} = ∑_{l≥j} \binom{l}{j} [G_{l} ∗ Y^{∗(l−j)} + g_{0,l} ∗ Y^{∗(l−j)}]

Then for large ν and some κ_{1} > 0, ∥D_{j}∥ ≤ κ_{1}^{|j|} while for |j| = 1, ∥D_{j}∥ = o(ν^{−M}).

Proof.

(i) From the analyticity of g it follows that

|g_{m,l}| < const. κ^{m+|l|}  (2.32)

where the constant is independent on m and l. Then, by (1.12),

|G_{l}(p)| < const. κ^{|l|+1} \frac{e^{κ|p|} − 1}{κ|p|} < const. κ^{|l|+1}e^{κ|p|}  (2.33)

The last claim is a direct consequence of (n5).

(ii) Note first that ∑_{|l|=l} 1 ≤ l^{n} ≤ 2^{nl} (since l_{i} ≤ l). Also, \binom{l_{i}}{j_{i}} ≤ 2^{l_{i}} so that \binom{l}{j} ≤ 2^{n|l|}. By (n5) we have g_{0,l} = 0 if |l| ≤ 1. Choosing δ < 2^{2nl+1}κ and ν such that ∥Y∥ ≤ δ, ∥D_{j}∥ is estimated by

∑_{l≥j} 2^{2nl}δ^{l−j}κ_{l}^{l}

where j = |j|, κ_{l} = ∥G_{l}∥ + |g_{0,l}| ≤ 2κ if l > 1 and κ_{l} = o(ν^{−M}) for l = 1, by (i), and the result follows.

Without loss of generality, we analyze (1.13) in a neighborhood of d_{1}, the Stokes line corresponding to λ_{1} = 1. (For the equations (1.16) we will need, in addition, to study a direction where p − λ_{i} + k · λ = 0.) Let ϵ and c_{0} be small and positive, V = {p : |p| < 1},

S_{c} = {p : arg(p) ∈ [ψ_{n} − 2π + c, −c] ∪ [c, ψ_{2} − c]}  (2.34)

S_{0} = ∪_{0<c<c_{0}}S_{c}, S_{c}^{±} = S_{c} ∩ {p : ± arg(p) > 0}, and let S′_{0}, S′_{c}, S_{c}^{±}′ be defined correspondingly, with ψ_{−} and ψ_{+} replacing ψ_{n} − 2π and ψ_{2}, respectively.

Lemma 16 i) For any ray d ⊂ S_{0}, there is a unique solution of (1.13) in D′_{m}(d, l) for any m ∈ ℕ, l ∈ (0, 1). This solution, Y_{0}, is in fact analytic in S_{0} ∪ V and, for large enough ν(d), Y_{0} ∈ L_{ν}^{1}(d).

ii) The function Y_{0}(· e^{iϕ}) is continuous in ϕ ∈ (ψ_{n}, 0] and (separately in) ϕ ∈ [0, ψ_{2}) in the D′_{ν,m}(ℝ^{+}, l) topology and sup_{ϕ∈[ψ_{1},ψ_{2}]} ∥Y_{0}(pe^{iϕ})∥_{m,ν} → 0 as ν → ∞.

iii) The description (1.17) holds for j = 1, l = 1.

iv) For a > 1, there is a one-parameter family of solutions of (1.13) in D′_{m}(0, a).

Note: the hyperfunctions Y_{0}(pe^{±0i}) = Y_{0}^{±} are different, in general.

Corollary 17 For k ∈ ℕ ∪ {0}, the function P^{mk}Y_{0}(pe^{iϕ}) is continuous in S_{k}^{−} := {p : 0 ≤ |p| < k + 1; arg(p) ∈ (ψ_{n}, 0]} and in S_{k}^{+} := {p : 0 ≤ |p| < k + 1; arg(p) ∈ [0, ψ_{2})} (and, of course, analytic in S_{0}).

Proof of Lemma 16.

We write (1.13) in the form:

Y = (Λ̂ − p)^{−1} (F_{0} − B̂PY + N (Y)) = M(Y)  (2.35)

Let d_{K} be an initial segment of d of length K < ∞. As the matrix Λ̂ − p is invertible in S_{c}, it is easy to see that the operator M in (2.35) satisfies the conditions of Lemma 9, in the spaces L_{ν}^{1}(d), A_{z,l}(S) (l ≤ M) (cf. Examples (1) through (4) in §2.1.2 and Remark 15). The conditions are in addition also satisfied in D′_{m,ν}(d); due to the special structure of this space, the proof the boundedness of the operator U = (Λ − p)^{−1} is more delicate, and is given in Lemma 42.

Thus, M has a unique fixed point in each of these spaces. The obvious inclusions between these spaces complete the proof of part (i). For the rest of Lemma 16 we need more results.

Proposition 18 The properties stated in Lemma 16 hold in S_{0}∩{p : |p| < 1+ϵ}

The proof is given in § 2.2.2.

Proposition 19 Let W_{0} be a solution of (1.13) in D′_{m}(0, a) with a > 1. For b ≥ a there exists a unique solution of (1.13) in D′_{m}(0, b), which agrees with W_{0} on D(0, a).

We use the decomposition D′_{m}(0, b) = D′_{m}(0, a) ⨁ D′_{m}(a, b), (a < b ≤ ∞) We identify W_{0} with an element of D′_{m}(0, b) by extending it with zero and define M_{1} on F_{m}(a, b) = ∪_{ν>ν_{0}}D′_{m,ν}(a, b) by

M_{1}(W_{1}) = M(W_{0} + W_{1}) − M(W_{0})  (2.36)

and (2.35) becomes

W_{1} = M_{1}(W_{1})  (2.37)

By Lemma 8, M is eventually contractive and by (2.36), clearly, so is M_{1}. By Lemma 8 then, M_{1} has a unique fixed point in F_{m}(a, b) (b ≤ ∞). In view of the inclusions between F_{m}(a, ∞) and F_{m}(a, b), the proof is complete. .

Proof of Lemma 16 (ii)

Let ϕ ∈ [0, ϵ] with ϵ small. We regard the space F_{m} as fixed and M_{1} as being dependent on ϕ through p and ∗_{ϕ} (cf. Remark 7). Firstly, W_{0}(ϕ) is continuous in ϕ ∈ [0, ϵ] and ∥W_{0}(pe^{iϕ})∥_{ν} = O(1/ν) uniformly in ϕ as follows from Proposition 18. Then the infinite sum in the definition of M_{1} is uniformly convergent in D′_{m,ν}(a, ∞) for ν large enough by (2.31). The operator U = (pe^{iϕ} − Λ̂)^{−1} = e^{−iϕ}(p − e^{−iϕ}Λ̂)^{−1} is strongly continuous in ϕ in F_{m}(a, ∞), a > 1, cf. Lemma 42. By Remark 7, M_{1} is ϕ-continuous and Lemma 8 applies.

We now study the convolution equations associated to the higher terms in the transseries, (1.16).

Lemma 20 i) Given the vector of constants C ∈ ℂ^{n_{1}} and and in addition given Y_{0} for d = ℝ^{+}, there is a unique solution of (1.16) in F_{m} with the (singular) initial condition

Y_{e_{j}}(p) = C_{j}Γ(β′_{j})^{−1}p^{β′_{j}}(e_{j} + o(1))  (p → 0, j = 1, 2, …, n_{1}) (2.38)

The general solution of (2.156), (1.16) is

C^{k}Y_{k}, C ∈ ℂ^{n_{1}}  (2.39)

where Y_{k} is the solution for C = (1, 1, …, 1).

ii) In a neighborhood of p = 0 we have

Y_{k}(p) = p^{kβ′−1}A_{k}(p)

with A_{k} analytic near the origin.

iii) The functions Y_{k}, k ⪰ 0 are analytic in R′_{1} and Y_{k}(xe^{iϕ}) are continuous in ϕ with respect to the D′_{m,ν} topology for ϕ ∈ (ψ_{−}, 0] and for ϕ ∈ [0, ψ_{+}).

iv) Each Y_{k} is in F(R′_{1}) (cf. § 2.1.6). Furthermore, there is a constant K and a function δ(ν) such that lim_{ν→∞} δ(ν) = 0 and in the decomposition (2.24) of Y_{k} we have Y_{k;j} ∈ T_{(k+je_{1})β′−1} and

sup_{ϕ,k} δ(ν)^{−|k|}∥Y_{k}∥_{D′_{m,ν}(e^{iϕ}ℝ^{+})} < K

sup_{ϕ,k,j} δ(ν)^{−|k|+j}∥Y_{k;j}∥_{D′_{m,ν}(e^{iϕ}ℝ^{+})} < K  (2.40)

where ϕ runs in (ψ_{−}, ψ_{+}).

v) The functions Y_{k}(· e^{iϕ}), ϕ ∈ (ψ_{−}, ψ_{+}), are simultaneously Laplace transformable in D′_{m,ν}(ℝ^{+}) and their Laplace transforms are solutions of (1.14). The expression

y^{±} = LY_{0}^{±} + ∑_{k≻0} x^{m·k}C^{k}e^{−k·λx}LY_{k}^{±}  (2.41)

is uniformly convergent for large enough x(C) in some open sector. In addition, (2.41) is a solution of (1.3) and LY_{k}^{±} ∼ ỹ_{k} for large x in the half plane ℜ(xp) > 0.

Without loss of generality, we analyze (1.16) in T_{{·}}(S_{c}^{+}′), D′_{m,ν}(d), with d ∈ S_{c}^{+}′, and in D′_{m,ν}(ℝ^{+}). We denote all the corresponding norms by ∥∥_{ν}.

Remark 21 Assume that for k′ ≺ k we have Y_{k′} ∈ T_{k′β′−1}. Then, in (2.162), we have T_{k}(Y_{0}, {Y_{k′}}) ∈ T_{kβ′−1}.

This follows immediately from Equations (2.6) and (2.7) and from the homogeneity of T implicit in the sum ∑_{(i_{mp};k)} (the notation is explained after (2.158)).

For |k| > 1 we take W_{k} := Y_{k} and R_{k} := T_{k} and write (1.16) as

(1 + J_{k})W_{k} = Q̂_{k}^{−1}R_{k}  (2.42)

with Q̂_{k} := (−Λ̂ + p + k · λ) (notice that for |k| > 1 and p ∈ S′_{0} we have det Q̂_{k}(p) ≠ 0).

(J_{k}W)(p) := Q̂_{k}^{−1} ((B̂ + m · k) ∫_{0}^{p} W(s)ds − ∑_{j=1}^{n} ∫_{0}^{p} W_{j}(s)D_{j}(p − s)ds)

(2.43)

The case |k| = 1 is special in that p = 0 is a singularity. The corresponding statements of Lemma 20 for |p| < ϵ with ϵ small are proven in Proposition 32: let W_{k}^{0} be the functions provided there. For the analytic part of Lemma 20 we need to show that W_{k}^{0} extend analytically to solutions of (1.16). To unify the treatment we derive equations of the form (2.42) for these continuations. Let δ ∈ S_{c}^{+}′, |δ| < ϵ. Using Proposition 32 and standard analyticity arguments, it suffices to show that W_{k}^{0} extend analytically in any sector ⊂ S_{c}^{+}′ centered at δ, and that the corresponding convolution equation is satisfied along the ray d_{δ} ∋ δ. We let a = W_{k}^{0}(δ), and with W_{k}^{1} = W_{k}^{0} for |p| < |δ| along d_{δ} and zero otherwise, we write Y_{k}(p) = W_{k}^{1}(p) + a + W_{k}(p − δ). For p ∈ d_{δ} we find that W_{k} must satisfy (2.42) where Q̂_{k} := (−Λ̂ + p + k · λ + δ) and R_{k}(p) is given by

(m · k + B̂) (∫_{0}^{δ} W_{k}^{0}(s)ds − ap) + ∑_{j=1}^{n} ∫_{0}^{δ}(W_{k}^{0})_{j}(s)D_{e_{j}}(p − s)ds − aD_{e_{j}}(p)

R_{k}(p) is manifestly analytic in S^{+}′. Since W_{k}(p) = W_{k}^{0}(p + δ) − a is already a solution of (2.42) for small p, and in this case the left side of (2.42) vanishes for p = 0, we have R_{k} ∈ T_{1}, for |k| = 1.

Combined with Remark 21 and induction on k, the following result completes the proof of Lemma 20, parts (i) and (ii).

Proposition 22 i) For large ν and constants K_{1} and K_{2}(ν) independent of k, with K_{2}(ν) = O(ν^{−1}) we have ∥Q_{k}^{−1}∥ ≤ \frac{K_{1}}{|k|} and

∥J_{k}∥ ≤ K_{2}(ν)  (2.44)

ii) For large ν, the operators (1 + J_{k}) defined in D′_{m,ν}, and also in T_{kβ′−1} for |k| > 1 and in T_{1} for |k| = 1 are simultaneously invertible. Given Y_{0} and C, the W_{k}, |k| ≥ 1 are uniquely determined. For any δ > 0 there is a large enough ν, so that

∥W_{k}∥ ≤ δ^{|k|}, k = 0, 1, ..  (2.45)

(in the D′_{m,ν} topology, (2.45) hold uniformly in ϕ ∈ [ψ_{−}+ϵ, 0] and ϕ ∈ [0, ψ_{+}−ϵ] for any small ϵ > 0).

Proof.

(i) For T_{kβ′−1}, this follows immediately from Remark 7, and the constants will depend on the parameter c in S_{c}^{+} ∩ S_{ℝ}. Given Y_{0}, the estimates (2.44) are also true in D′_{m,ν;ϕ} this time uniformly in ϕ, down to ϕ = 0. The proof of (2.44) in this case is given in Appendix 2.2.2, in Lemma 42, from which the continuity of J_{k} in ϕ also follows.

(ii) From (2.42) and (i) we get, for some K and j ≥ 1 ∥W_{k}∥ ≤ K∥R_{k}∥. We first show inductively that the W_{k} are bounded. Choosing a suitably large ν(ϵ) we can make max_{|k|≤1} ∥W_{k}∥_{ν} ≤ ϵ for any positive ϵ (uniformly in ϕ). We show by induction that ∥W_{k}∥_{ν} ≤ ϵ for all k. Using (2.45), (1.16), (2.160), Proposition 15 and the crude estimate \binom{a}{b} ≤ 2^{a} we get

∥W_{k}∥_{ν} ≤ K∥R_{k}∥_{ν} ≤ ∑_{l≤k} κ_{1}^{|l|}ϵ^{|k|} ∑_{(i_{mp})} 1 ≤ ϵ^{|k|} ∑_{s=0}^{|k|} κ_{1}^{s}2^{n_{1}(|k|+s)}2^{s+n_{1}} ≤ (C_{1}ϵ)^{|k|}

(2.46)

where C_{1} does not depend on ϵ, k. Choosing ϵ so that ϵ < C_{1}^{−2} we have, for |k| ≥ 2 (C_{1}ϵ)^{|k|} < ϵ completing the induction step . But as we now know that ∥W_{k}∥_{ν} ≤ ϵ, the same inequalities (2.46) show that in fact ∥W_{k}∥_{ν} ≤ (C_{1}ϵ)^{|k|}. Choosing ϵ small enough, the first part of Proposition 22, (ii) follows.

Proposition 23 i) Let Y_{0} be given by Lemma 16. We have, in D′_{m,ν}(ℝ^{+})

Y_{0}^{±} = Y_{0}^{∓} + ∑_{k=1}^{∞}(±S_{1})^{k} (Y_{ke_{1}}^{∓}(p − k)χ_{[k,∞]})^{(mk)}  (2.47)

(cf. (1.18)) and Y_{0} is analytic in R_{1}.

ii) The general solution of (1.13) in D′_{m,ν}(ℝ^{+}) is

Y_{0}^{+} + ∑_{k=1}^{∞} C^{k} (Y_{ke_{1}}^{+}(p − k)χ_{[k,∞]})^{(mk)}  (2.48)

with arbitrary C (a similar statement holds with Y_{0}^{−} replacing Y_{0}^{+}).

Proof. We start with (ii). Assuming first (2.48) is indeed a solution of (1.13), to see that there are no others, it suffices by Proposition 19 to check that (2.48) is the general solution on [0, 1 + ϵ). The latter part is immediate from Remark 31 and Proposition 32 below. Now Y_{0}^{+} is a solution of (1.13), by Lemma 16 (ii); the sum (2.48) is convergent in D′_{m,ν} by (2.46). Since (Y_{ke_{1}}(p − k)χ_{[k,∞]})^{(mk)} ∈ D′_{m,ν}(k, ∞), to show that (2.48) is a solution on ℝ^{+}, we check inductively on j

that H_{j} = ∑_{k=0}^{j} (Y_{ke_{1}}(p − k)χ_{[k,∞]})^{(mk)} solves (1.13) in D′[0, j+1). Assuming this for j′ < j and looking for a solution on [0, j + 1) in the form H_{j+1} =

H_{j} + (Ỹ_{j}(p − j)χ_{[j,∞]})^{(mj)} we obtain, by a straightforward calculation, using the induction hypothesis and (2.166)

(Λ̂ − p)Ỹ_{j}^{(mj)} + B̂PỸ_{j}^{(mj)} − ∑_{j=1}^{n}(Ỹ_{j})_{j}^{(mj)} ∗ D_{je_{j}}

= ∑_{|l|>1} d_{l} ∗ ∑_{Σs=j} ∗ ∏_{i=1}^{n} ∗ ∏_{j=1}^{l_{i}}(Ỹ_{s_{i,j}})_{i}^{(s_{i,j})} =: R_{j}(p) (2.49)

which integrated (mj) times is exactly the equation for Y_{je_{j}}, cf. also § 2.4. The claim now follows from Lemma 20, (iii). For (i), we note as before that Y_{0}^{±} are indeed solutions of (1.13). Applying (ii), we only need to identify C for which purpose we compare the left side with the right side on (1, 1 + ϵ), where all the terms except for k = 1 vanish and Remark 33 below applies. Lemma 12 completes the proof.

Proof of Lemma 20, (iv). We let Y_{j} = Y_{je_{j}}. By Lemma 12, D_{je_{j}} ∈ F(R′_{1}) since

D_{j}^{+} = ∑_{l≥j} \binom{l}{j}G_{l} ∗ (Y_{0}^{+})^{∗(l−j)} = ∑_{l≥j} G_{l} ∗ [Y_{0}^{−}

+ ∑_{s≥1} S^{s}(Y_{s}^{−}(p − s))^{(ms)}]^{∗(l−j)} = ∑_{k⪰0} (∑_{s≥1} S^{s}(Y_{s}^{−})^{(ms)})^{∗k} Q_{kj}

= ∑_{l=0}^{∞} S^{l}(D_{j;l}^{−})^{(ml)} (2.50)

where Q_{kj} = ∑_{l≥j+k} \binom{l}{j}\binom{l−j}{k}G_{l}(Y_{0}^{−})^{∗(l−k−j)} and

(D_{j;l}^{−})^{(ml)} = ∑_{\begin{matrix}∈ℕ^{n} \\ |k|∈[0,l] \end{matrix}} Q_{kj} ∗ ∑_{(i_{rs}:l)} \begin{matrix}n \\ ∗ & ∏ \\ r=1 \end{matrix} \begin{matrix}k_{r} \\ ∗ & ∏ \\ s=1 \end{matrix} (Y_{i_{rs}}^{−})_{r}^{(mi_{rs})}

= ( ∑_{\begin{matrix}∈ℕ^{n} \\ |k|∈[0,l] \end{matrix}} Q_{kj} ∗ ∑_{(i_{rs}:l)} \begin{matrix}n \\ ∗ & ∏ \\ r=1 \end{matrix} \begin{matrix}k_{r} \\ ∗ & ∏ \\ s=1 \end{matrix} (Y_{i_{rs}}^{−})_{r})^{(ml)}

(2.51)

and the notation ∑_{(i_{rs}:l)} is explained after Eq. (2.158)); in particular, (1) there are only finitely many terms in (2.51) and (2) by homogeneity, D_{j;l} ∈ T_{lβ′_{1}−1}.

In addition, it follows, as in (2.46), (noting that we only need the j with |j| = 1) that ∥Q_{k;j}∥ ≤ K(4κ)^{|k|} and ∥D_{j;l}∥ ≤ K(2^{2n+3}κC_{1}ϵ)^{l}. If we look for Y_{k}^{+} in the form Y_{k}^{−} + ∑_{l=1}^{∞}(Y_{k;l}^{−}(p − l))^{(ml)} then the equation for Y_{k;l}^{−}, l ≥ 1, reads

(−p + Λ̂ − k · λ − l) Y_{k;l}^{−} + (B̂ + m · k + m_{1}l) PY_{k;l}^{−}

+ ∑_{|j|=1} D_{j;0}^{−} ∗ (Y_{k;l}^{−})^{j} = (T_{k}(Y_{·;l}^{−})) − ∑_{s=1}^{l−1} ∑_{|j|=1} D_{j;s} ∗ (Y_{k;l−s}^{−})^{j} (2.52)

By induction, exactly as in (iii), it follows that Y_{k;l}(· e^{iϕ}) ∈ T_{kβ′+lβ′_{1}−1}, are ϕ− continuous in D′_{m,ν}(ℝ^{+}) and that, with ν large enough independent of k, l, ϕ, ∥Y_{k;l}∥_{D′_{m,ν}} ≤ Kδ^{|k|+l}. Analyticity in R′_{1} follows now from Lemma 12.

(v) Laplace transformability as well as the fact that y_{k} solve (1.14) follow immediately from (2.45) and Lemma 6. Uniform convergence follows from (2.40) and Lemma 6. Let y_{k}^{±} = LY_{k}^{±}. Now since y_{0}^{±} + ∑ x^{m·k}C^{k}e^{−k·λx}y_{k}^{±} formally solves (1.3) (by the very construction of (1.14), see §2.4) and is a uniformly convergent function series, the conclusion follows together with the fact that LY_{k} ∼ ỹ_{k} since by (ii), and (iv) LY_{k} have power series asymptotics which by construction must be formal solutions of (1.14).

Proof of Theorem 1, (ii) Modulo relabeling of the spatial directions and rescaling, the proofs of the properties of the solution Y_{0} of (1.13) are valid in a sector centered around any eigenvalue λ_{i}. The above proofs of the analytic properties work with virtually no change along any direction so that p + k′ · λ − λ_{i} ≠ 0 and the same is true with respect to the D′_{m,ν} properties of Y_{k} restricted to compact sets in ℂ. The analysis in D′_{m,ν}(e^{iϕ}ℝ^{+}) (i.e., along the infinite ray) and the associated norm estimates are not valid along directions so that ℜ(p − λ_{i} + k′ · λ) ≤ 0 (because of possible presence of small denominators in (1 + J_{k})^{−1}, the only difference in this case, but a significant one). However, we do not take Laplace transforms along such directions (cf. (c1)) and the associated infinite ray norms are not needed for our purposes. For the analytic properties in the neighborhood of −k′ · λ + λ_{i}, see Remark 28.

2.2.1 Higher resurgence relations

Proposition 24 i) Let y_{1} and y_{2} be solutions of (1.3) so that y_{1,2} ∼ ỹ_{0} for large x in an open sector S (or in some direction d); then y_{1} − y_{2} = ∑_{j} C_{j}e^{−λ_{i_{j}} x}x^{−β_{i_{j}}} (e_{i_{j}} +o(1)) for some constants C_{j}, where the indices run over the eigenvalues λ_{i_{j}} with the property ℜ(λ_{i_{j}}x) > 0 in S (or d). If y_{1} − y_{2} = o(e^{−λ_{i_{j}} x}x^{−β_{i_{j}}}) for all j, then y_{1} = y_{2}.

ii) Let y_{1} and y_{2} be solutions of (1.3) and assume that y_{1} − y_{2} has differentiable asymptotics of the form Ka exp(−ax)x^{b}(1 + o(1)) with ℜ(ax) > 0 and K ≠ 0, for large x. Then a = λ_{i} for some i.

iii) Let U_{k} ∈ T_{{·}} for all k, |k| > 1. Assume in addition that for large ν there is a function δ(ν) vanishing as ν → ∞ such that

sup_{k} δ^{−|k|} ∫_{d} |U_{k}(p)e^{−νp}| d|p| < K < ∞  (2.53)

Then, if y_{1}, y_{2} are solutions of (1.3) in S where in addition

y_{1} − y_{2} = ∑_{|k|>1} e^{−λ·kx}x^{m·k} ∫_{d} U_{k}(p) exp(−xp)dp  (2.54)

where λ, x are as in (c1), then y_{1} = y_{2}, and U_{k} = 0 for all k, |k| > 1.

Proof. (i) is a classical result (see [21] for the general treatment and [11] for a brief presentation of special cases and further references). However, what is actually needed for our purposes can be reduced to the more familiar linear asymptotic theory in the following way. Let d be a direction in the complex plane and let y_{0}, y_{1} be solutions of (1.3) such that y_{0,1} ∼ ỹ_{0} for large x along d. Then, by (n5), y_{0,1}(x) = O(x^{−M}) and for any j, g^{(e_{j})}(x, y_{0,1}(x)) = O(x^{−M}). If δ = y_{1} − y_{0} then by hypothesis δ(x) = o(x^{−l}) along d, for all l. The function δ is locally analytic and satisfies the equation

δ′ = −Λ̂δ − \frac{1}{x}B̂δ + ∑_{|k|=1} g^{(k)}(x, y_{0})δ^{k} + ∑_{|k|>1} g^{(k)}(x, y_{0})δ^{k} =

− Λ̂δ − \frac{1}{x}B̂δ + \frac{1}{x^{M}} ∑_{j=1}^{n}(δ)_{j}h_{e_{j}}(x) (2.55)

where h_{k}(x) are bounded along d. Obviously, because of the link between δ and h_{k}, the δ we started with might be the only solution of (2.55) which is also a difference of solutions of (1.3). The asymptotic characterization we need holds nevertheless for all decaying solutions solutions of (2.55): since no two eigenvalues are equal, there exists by the well-known linear asymptotic theory [11] a fundamental set {δ_{i}}_{1≤i≤n} of solutions of (2.55) such that δ_{i} ∼ e^{−λ_{i}x}x^{−β_{i}}(e_{i} + o(1)). Thus δ = ∑_{i=1}^{n} C_{i}δ_{i} = ∑_{i=1}^{n} C_{i}e^{−λ_{i}x}x^{−β_{i}}(e_{i} + o(1)). Since ℜ(−β_{i}) > 0 and the λ_{i} are distinct we must have C_{i} = 0 for all i for which ℜ(−λ_{i}x) ≥ 0, otherwise |δ(x)| would be unbounded for large x; the first part of (i) is proven. If on the other hand δ = o(e^{−λ_{i_{j}} x}x^{−β_{i_{j}}}) for all j, again because the λ_{i} are independent, it follows that C_{i} = 0 for all i = 1, 2, …, n, thus δ = 0.

(ii) is now obvious.

For (iii), note first that by (2.53) and (c1) the RHS of (2.54) converges uniformly for large x in some open sector. In addition, by an arbitrarily small change in ξ = arg(x), we can make the set {ℜ(xλ_{i})}_{i} ℤ-independent (the existence of k(ξ) ≠ 0 s.t. ℜ(e^{iξ}k · λ) = 0 for ξ in an interval of would imply the existence of a common k for a set of ξ with an accumulation point, giving kλ = 0). We choose such a ξ. Assume now there exist k so that U_{k} ≠ 0; among them let k_{0} have the least ℜ(xk · λ). By (2.53) for large x, y_{1} − y_{2} ∼ e^{−λ·k_{0}x}x^{m·k_{0}}L_{ϕ}U_{k_{0}}(1 + o(1)). Because U_{k_{0}} ∈ T_{{·}}, and by (2.53), L_{ϕ}U_{k_{0}} has a differentiable power series asymptotics which is the term-by-term Laplace transform of the Puiseux series at the origin of U_{k_{0}}, and thus nonzero. This contradicts (i) because with |k_{0}| > 1 we have λ · k_{0} ≠ λ_{j} for all j (ℤ−independence). Thus U_{k} = 0 for all k.

We let C ∈ (ℂ\{0})^{n_{1}} be an arbitrary constant vector. For x large enough, y^{+} defined in (2.41) is a solution of (1.3) in an open sector containing x. We now use Lemma 20 to write (2.41) in terms of functions analytic in the lower half plane:

y^{+} = LY_{0}^{−} + ∑_{j=1}^{∞} x^{mj}e^{−jx}LY_{0;j}^{−} + ∑_{k≻0,j≥0} x^{m·k+mj}C^{k}e^{−(k·λ+jλ_{1})x}LY_{k;j}^{−}

= LY_{0}^{−} + ∑_{k≻0} x^{mk}e^{−(k·λ)x} ∑_{k′;j:k′+je_{1}=k} C^{k′}LY_{k′;j}^{−} (2.56)

where, by (2.47) we have Y_{0;j} = S_{1}^{j}Y_{je_{1}}^{−}. On the other hand, the expression

y^{−} = LY_{0}^{−} + ∑_{k≻0} x^{m·k}C̃^{k}e^{−k·λx}LY_{k}^{−}  (2.57)

is, for any C̃, a solution of (1.3) as well. Choosing C̃_{1} = C_{1}+S_{1}; C̃_{i} = C_{i}; (i > 1) all the terms with |k| ≤ 1 in (2.57) and (2.56) coincide and thus

y^{+} − y^{−} = ∑_{|k|>1} x^{m·k}e^{−k·λx}LU_{k}

(2.58)

where

U_{k} = ∑_{k′+je_{1}=k} C^{k′}Y_{k′;j}^{−} − C̃^{k}Y_{k}^{−}  (2.59)

so that applying Proposition 24 (ii)

U_{k} = 0  (2.60)

Since C_{i} ≠ 0 we have, for any C_{1},

(C_{1} + S_{1})^{k_{1}}Y_{k}^{−} = ∑_{k′+je_{1}=k} C_{1}^{k′_{1}}Y_{k′;j}^{−}  (2.61)

with arbitrary C_{1} so that

Y_{k;j}^{−} = \binom{k_{1} + j}{j}S_{1}^{j}Y_{k+je_{1}}^{−}  (2.62)

Combined with the definition of Y_{k;j}^{−} this gives (1.26).

Solving for Y_{k+je_{1}}^{−}, (2.62) determines later series in the transseries in terms of earlier ones. The same arguments work of course with −/+ and +S_{1}/(−S_{1}) interchanged.

Theorem 3 part (iii) follows from the following.

Proposition 25 Any solution y of (1.3) so that y ∼ ỹ_{0} along some direction d ⊂ S_{x} is of the form (2.41), for a unique C^{+} (a similar statement holds with +/− interchanged). Alternatively, a solution y of (1.3) so that y ∼ ỹ_{0} along some direction d ⊂ S_{x} can be represented as (1.21) or more generally as (1.23) where Laplace integration is along ℝ^{+} (in distributions), for a unique C.

Proof. Let y be an arbitrary solution of (1.3) so that y ∼ ỹ_{0} along d ⊂ S_{x}. Then, by Proposition 24, y − y^{+} = ∑_{j} \overline{C}_{j}e^{−λ_{i_{j}} x}x^{−β_{i_{j}}} (e_{i_{j}} + o(1)) for some constant \overline{C}. Therefore y_{1} defined as the “+” solution in (2.41) with C = \overline{C} will have the property y_{1} − y = o(C_{j}e^{−λ_{i_{j}} x}x^{−β_{i_{j}}}) for all j, hence y = y_{1}. Formula (1.23). The last part, as well as the middle formula in (1.27), follow through a straightforward calculation from the first part, (2.62) using (2.40) to control convergence.

Proof of theorem 5. Let y^{±} be defined by (2.41) with C = (±\frac{1}{2}S_{1} + C)e_{1}, respectively. The same arguments leading to (2.60) show that y^{+} = y^{−} =: y. All the exponentials in the transseries of y are generated by construction by e^{−λx}. Choosing p in the path of integration above/below ℝ^{+} and consequently the +/− representation (2.41) of y we have by Lemma 20 that LY_{k}^{±} ∼ ỹ_{k} in (2.41), in the half plane ℜ(xp) > 0. By construction LY_{e_{1}} = x^{−β′_{1}}(e_{1} + o(1)) (cf. 2.89) while for j > 1 we have LY_{je_{1}} ∼ x^{−jβ′_{1}} by Lemma 20 (ii). The condition |x^{−β_{1}+1}e^{−xλ_{1}}| → 1 together with Lemma 20 guarantee the uniform convergence of the series (2.41). The conclusion is immediate.

2.2.2 Local analysis near p = 1.

We treat (1.13) near p = 1 as a perturbation of a differential equation having the same type of singularity. The associated differential equation splits the singularity, and our convolution equation is a regular perturbation of it, which is then solved by fixed point methods. Let Y_{0} be the unique solution in A_{z,l} of (1.13) and let ϵ > 0 be small. Define

H(p) := \{ \begin{matrix}Y_{0}(p) for p ∈ S_{0} ,|p| < 1 − ϵ \\ 0 otherwise \end{matrix}

and W(1 − p) := Y_{0}(p) − H(p)  (2.63)

(Y(p) − H(p) = W(p − 1) would be more “natural”, but would later complicate notations). In terms of W, for real z = 1 − p, z < ϵ, (1.13) becomes:

−(1 − z)W(z) = F_{1}(z) − Λ̂W(z) + B̂ ∫_{ϵ}^{z} W(s)ds + N (H + W) (2.64)

where

F_{1}(1 − s) := F_{0}(s) − B̂ ∫_{0}^{1−ϵ} H(s)ds

Proposition 26 i) For small ϵ, H^{∗l}(1 + z) extends to an analytic function in the disk 𝔻_{ϵ} := {z : |z| < ϵ}. Furthermore, for any δ there is an ϵ and a constant K_{1} := K_{1}(δ, ϵ) such that for z ∈ 𝔻_{ϵ}

|H^{∗l}(1 + z)| < K_{1}δ^{|l|}  (2.65)

ii) The equation (2.64) can be written as

−(1 − z)W(z) = F(z) − Λ̂W(z) + B̂ ∫_{ϵ}^{z} W(s)ds − ∑_{k=1}^{n} ∫_{ϵ}^{z} h_{j}(s)D_{j}(s − z)ds

(2.66)

where

F(z) := N (H)(1 − z) + F_{0}(z)  (2.67)

D_{j} = ∑_{|l|≥1} l_{j}G_{l} ∗ H^{∗l̄^{j}} + ∑_{|l|≥2} l_{j}g_{0,l}H^{∗l̄^{j}}; l̄^{j} := (l_{1}, l_{2}, ..(l_{j} − 1), ..l_{n}) (2.68)

extend to analytic functions in 𝔻_{ϵ}. Moreover, if H is a vector in L^{1}_{ν}(ℝ^{+}) then, for large ν, D_{j} ∈ L^{1}_{ν}(ℝ^{+}) and the functions F(z) and D_{j} extend to analytic functions in 𝔻_{ϵ}. Furthermore, D_{j} ∈ A_{z,M}.

iii) Near p = 1 we have (cf. Lemma 16)

P^{m+1}Y_{0} = (1 − p)^{β′}A + B (β ∉ ℤ)
P^{m+1}Y_{0} = (p − 1) (ln(p − 1)A(p) + B(p)) (β ∈ ℤ)  (2.69)

where A, B analytic at p = 1.

Proof.

Parts (i) and (ii), except for the last claim, are proven in [13], Propositions 18 and 19. To see that D_{j} ∈ A_{m} it is enough to remark again that A_{z,M} is a convolution ideal of A_{z,0} and that G_{j} ∈ A_{z,M} for |j| = 1, by (n5).

For (iii), consider again equation (2.66). Let Γ̂ = Λ̂ − (1 − z)ˆ1, where ˆ1 is the identity matrix. By construction Γ̂ and B̂ are diagonal, Γ̂_{11} = z and B̂_{11} = β_{1} =: β. We write this as Γ̂ = z ⊕ Γ̂_{c}(z) and similarly, B̂ = β ⊕ B̂_{c}, where Γ̂_{c} and B̂_{c} are (n − 1) × (n − 1) diagonal matrices. Γ̂_{c}(z) and Γ̂_{c}^{−1}(z) are analytic in 𝔻_{ϵ}. Let

Q := P_{ϵ}^{m+1}W  (2.70)

with P_{ϵ} := W ↦ (z ↦ ∫_{ϵ}^{z} W(s)ds). By Lemma 16, (i), Q is analytic in 𝔻_{ϵ} ∩ ({z : z + 1 ∈ S_{0}}). From (2.66) we obtain

(z ⊕ Γ̂_{c}(z))Q^{(m+1)}(z) − (β ⊕ B̂_{c})Q^{(m)}(z)

= F(z) − ∑_{j=1}^{n} ∫_{ϵ}^{z} D_{j}(s − z)Q_{j}^{(m+1)}(s)ds (2.71)

or, after m integrations by parts in the r.h.s. of (2.71), by Proposition 26 (ii), we get

(z ⊕ Γ̂_{c}(z))Q^{(m+1)}(z) − (β ⊕ B̂_{c})Q^{(m)}(z)

= F(z) − (−1)^{m} ∑_{j=1}^{n} ∫_{ϵ}^{z} D_{j}^{(m)}(s − z)Q′_{j}(s)ds (2.72)

so that with β′ = β′_{1}, B̂′_{c} = m_{1} + B̂_{c},

(z ⊕ Γ̂_{c}(z))Q′(z) − (β′ ⊕ B̂′_{c})Q(z)

= P_{ϵ}^{m}F(z) − (−P_{ϵ})^{m} ∑_{j=1}^{n} ∫_{ϵ}^{z} D_{j}^{(m)}(s − z)Q′_{j}(s)ds

= P(z) + ∑_{j=1}^{n} ∫_{ϵ}^{z} D′_{j}(s − z)Q_{j}(s)ds (2.73)

where P(z) = P_{ϵ}^{m}F(z). With the notation (Q_{1}, Q_{⊥}) := (Q_{1}, Q_{2}, .., Q_{n}) we write the system in the form

(z^{−β′}Q_{1}(z))′ = z^{−β′−1} (P_{1}(z) + ∑_{j=1}^{n} ∫_{ϵ}^{z} D′_{1j}(s − z)Q_{j}(s)ds)

(e^{Ĉ(z)}Q_{⊥})′ = e^{Ĉ(z)}Γ̂_{c}(z)^{−1} (P_{⊥} + ∑_{j=1}^{n} ∫_{ϵ}^{z} D′_{⊥}(s − z)Q_{j}(s)ds)

Ĉ(z) := − ∫_{0}^{z} Γ̂_{c}(s)^{−1}B̂′_{c}(s)ds
Q(ϵ) = 0  (2.74)

After integration we get:

Q_{1}(z) = R_{1}(z) + J_{1}(Q)
Q_{⊥}(z) = R_{⊥}(z) + J_{⊥}(Q)  (2.75)

with

J_{1}(Q) = z^{β′} ∫_{ϵ}^{z} t^{−β′−1} ∑_{j=1}^{n} ∫_{ϵ}^{t} Q_{j}(s)D′_{1j}(t − s)dsdt

J_{⊥}(Q)(z) := e^{−Ĉ(z)} ∫_{ϵ}^{z} e^{Ĉ(t)}Γ̂_{c}(t)^{−1} (∑_{j=1}^{n} ∫_{ϵ}^{z} D′_{⊥}(s − z)Q_{j}(s)ds) dt

R_{⊥}(z) := e^{−Ĉ(z)} ∫_{ϵ}^{z} e^{Ĉ(t)}Γ̂_{c}(t)^{−1}F_{⊥}(t)dt
R_{1}(z) = z^{β′} ∫_{ϵ}^{z} t^{−β′−1}P_{1}(t)dt  (β′ ≠ 1)
R_{1}(z) = P_{1}(0)z ln z + z ∫_{ϵ}^{z} \frac{P_{1}(s) − P_{1}(0)}{s}ds (β′ = 1)  (2.76)

Consider the space U_{β′} given by

U_{β′} = {Q analytic in {z : 0 < |z| < ϵ, arg(z) ≠ π} : Q = z^{β′}A(z) + B(z)} (2.77)

for β′ ≠ 1 and

U_{1} = {Q analytic in {z : 0 < |z| < ϵ, arg(z) ≠ π} : Q = z ln zA(z) + zB(z)}

where A, B are analytic in 𝔻_{ϵ}. (The decomposition of Q in (2.77) is unambiguous since z^{β′} and z ln z are not meromorphic in 𝔻_{ϵ}.)

The norm

∥Q∥ = sup {|A(z)|, |B(z)| : z ∈ 𝔻_{ϵ}}  (2.78)

makes U_{β′} a Banach space.

Proposition 27 The operator J := (Q ↦ (J_{1}Q, J_{⊥}Q) has norm O(ϵ), for small ϵ, in U_{β′} as well as in L^{1}[−ϵ, ϵ]. Along any segment d_{ϵ} originating at z = ϵ in the region |z| < ϵ, arg(z) ≠ π, Equation (2.73) has a unique solution in L_{ν}^{1}(d_{ϵ}). This solution belongs to T_{β}.

The proof uses the following elementary identities:

∫_{ϵ}^{z} A(s)s^{r}ds = const. + z^{r+1} ∫_{0}^{1}A(zt)t^{r}dt = const. + z^{r+1}Analytic(z)

∫_{0}^{z} s^{r} ln s A(s)ds = z^{r+1} ln z ∫_{0}^{1}A(zt)t^{r}dt + z^{r+1} ∫_{0}^{1}A(zt)t^{r} ln tdt

(2.79)

where the second equality is obtained by differentiating with respect to r the first equality.

Using (2.79) it is straightforward to check that the r.h.s. of (2.75) extends to a linear inhomogeneous operator on U_{β′} with image in U_{β′} and that the norm of J is O(ϵ) for small ϵ. For instance, one of the terms in J for β′ = 1,

z ∫_{0}^{z} t^{−2} ∫_{0}^{t} s ln s A(s)D′(t − s)ds

= z^{2} ln z ∫_{0}^{1} ∫_{0}^{1} σA(zτσ)D′(zτ − zτσ)dσdτ

+ z^{2} ∫_{0}^{1} dτ ln τ ∫_{0}^{1} dσ σ(1 + ln σ)A(zτσ)D′(zτ − zτσ) (2.80)

manifestly in U_{β′} if A is analytic in 𝔻_{ϵ}. Comparing with (2.77), the extra power of z accounts for a norm O(ϵ) for this term.

Therefore, in (2.74) (1 − J) is invertible and the solution Q ∈ U_{β′}. In view of the the uniqueness of the solution of (1.13) in S_{0}, (Lemma 16, (i)) the rest of the proof of Proposition 26, (iii) is immediate.

A short calculation shows that:

Remark 28 i) The equation for Y_{k} near z = 0 where z = −p − k · λ + λ_{i} can be written in the form (2.66), for a different F, and with B̂ + k · m instead of B̂. Thus Y_{k}(z) ∈ T_{{·}}.

ii) The equation for Y_{k} near z = 0 where z = −p − k′ · λ + λ_{i} where k′ ≺ k can be written as

(1 + J_{k})Y_{k}(z) = R_{k}(z)  (2.81)

where

J_{k}Y = (B̂ + m · k)M̂^{−1}PY + M̂^{−1} ∑_{|j|=1} \overline{D}_{j} ∗ Y^{j},

M̂ = z + Λ̂ − λ_{i} + (k′ − k) · λ, R_{k} = M̂^{−1}T_{k} and \overline{D}_{j} analytic for small z. Thus, arguments virtually identical to those for (2.75) imply that Y_{k} ∈ T_{{·}} near these points.

2.2.3 The solutions of (2.72) on [−ϵ, ϵ]

Let Q_{0} be the solution given by Proposition 27, take ϵ small enough and denote by O_{ϵ} a neighborhood in ℂ of width ϵ of the interval [0, 1 + ϵ]. We look for solutions of (2.73) in L^{1}[−ϵ, ϵ]. The main difference with respect to the previous section is that in integrating (2.73) to the analog of (2.75)for negative z, the constant of integration will now be undetermined leading to a one-parameter family of solutions. See also Remark 30 below.

Remark 29 . As ϕ → ±0, Q_{0}(ze^{iϕ}) → Q_{0}^{±}(z) in the sense of L^{1}([0, 1+ϵ]) and also in the sense of pointwise convergence for z ≠ 0, where

Q_{0}^{±}(z) := \{ \begin{matrix}Q_{0}(z) & z > 0 \\ |z|^{β′}e^{∓iπ(β′)}a_{1}(p) + a_{2}(p) & z < 0 \end{matrix} (β ≠ 1)
Q_{0}^{±} := \{ \begin{matrix}Q_{0}(z) & z > 0 \\ z(ln(|z|) − πi)a_{1}(z) + za_{2}(z) & z < 0 \end{matrix} (β′ = 1)  (2.82)

Moreover, Q_{0}^{±} are L_{loc}^{1} solutions of (27) on the interval [−ϵ, ϵ].

The proof is immediate from Propositions 26 and 27.

Remark 30 For any λ ∈ ℂ the combination Q_{λ} = λQ_{0}^{+} + (1 − λ)Q_{0}^{−} is a solution of (2.72) in L^{1}[−ϵ, ϵ].

Proof. Follows from Remark 29 as (2.72) is linear.

*

Let now Q_{0} be any solution of (2.72) in L^{1}[−ϵ, ϵ]. We search for other solutions in the form Q = Q_{0} + q. Since (2.72) is linear and Q_{0} is already a solution we have

(z ⊕ Γ̂_{c}(z))q′(z) − (β′ ⊕ B̂′_{c})q(z) = ∑_{j=1}^{n} ∫_{ϵ}^{z} D′_{j}(s − z)q_{j}(s)ds

(2.83)

and, by the uniqueness of Q_{0} for z > 0 we have q = 0 for q < 0 and the equation becomes

(z ⊕ Γ̂_{c}(z))q′(z) − (β′ ⊕ B̂′_{c})q(z) = ∑_{j=1}^{n} ∫_{0}^{z} D′_{j}(s − z)q_{j}(s)ds

(2.84)

with the initial condition q(0) = 0. Changing variables to z = −p (p > 0 now corresponds to going beyond the singularity) and q(z) = Y(−p) we have

(p ⊕ Γ̂_{c}(−p))Y′(p) − (β′ ⊕ B̂′_{c})Y(p) + ∑_{j=1}^{n} ∫_{0}^{p} D′_{j}(p − t)Y_{j}(t)dt = 0

(2.85)

We recognize in (2.85) the equation for PY_{e_{1}}:

Remark 31 Equation (2.85) is at the same time the equation for PY_{e_{1}} and for the difference P^{m+1}(Y_{0}^{[1]} − Y_{0}^{[2]}) where Y_{0}^{[1,2]} are any solutions of (1.13).

Proposition 32 Let ϵ be small. In T_{β′}({|p| < ϵ}) as well as in D′_{m,ν}(0, ϵe^{iϕ}) for any ϕ, there is a unique solution of (2.85) W_{0} such that, for small p, W_{0} = Γ(β′)^{−1}p^{β′}(e_{1} + o(1)). The general solution of (2.85) is Y = CW_{0}, with C ∈ ℂ arbitrary.

Notes: (1) Modulo relabeling of the spatial directions, the statement and proof hold for any of the equations for Y_{e_{j}}.

(2) The point p = 0 is singular, and so is the “initial condition” W_{0} ∼ Γ(β′)^{−1}p^{β′}e_{1}.

Proof. We have

(p^{−β′}Y_{1}(z))′ = −p^{−β′−1} ∑_{j=1}^{n} ∫_{0}^{p} D′_{1j}(p − t)Y_{j}(t)dt

(e^{Ê(p)}Y_{⊥})′ = −e^{Ê(p)}Γ̂_{c}(−p)^{−1} ∑_{j=1}^{n} ∫_{0}^{p} D′_{⊥}(p − t)Y_{j}(t)dt

Ê(p) := − ∫_{0}^{p} Γ̂_{c}(−t)^{−1}B̂′_{c}(t)ds
Y(0) = 0  (2.86)

After integration we get:

(1 + J_{1})Y_{1}(z) = CR_{1}(p)
(1 + J_{⊥})Y_{⊥}(p) = 0  (2.87)

with C ∈ ℂ arbitrary and

J_{1} V = p^{β′} ∫_{0}^{p} t^{−β′−1} ∑_{j=1}^{n} ∫_{0}^{t} Y_{j}(s)D′_{1j}(t − s)dsdt

J_{⊥} V := e^{−Ê(p)} ∫_{0}^{p} e^{Ê(t)}Γ̂_{c}(−t)^{−1} (∑_{j=1}^{n} ∫_{0}^{t} D′_{⊥}(t − s)Y_{j}(s)ds) dt

R_{1}(p) = p^{β′}

(2.88)

In a region |p| < ϵ, for small ϵ, the norm of the operator J defined on T_{β′} is O(ϵ), exactly as in Proposition 27. Given C the solution of the system (2.87) is unique and can be written as

Y = CW_{0}; W_{0} := Γ(β′)^{−1}(ˆ1 + J)^{−1}R ≠ 0  (2.89)

(The prefactor Γ(β′)^{−1} was introduced so that the coefficient of the leading power in the asymptotic series of LW_{0} is one).

The proof is essentially the same if we consider (2.87) in L_{ν}^{1}(0, ϵe^{iϕ}), which coincides with D′_{m,ν}(0, ϵe^{iϕ}) for small ϵ.

Remark 33 On (1, 1 + ϵ), (Y_{0}^{+} − Y_{0}^{−})(p) = S_{1}Y_{e_{1}}^{(m)}(p − 1).

The existence of some S_{1} is obvious from Remark 31 and Proposition 32. Its value follows by comparing (2.69), (2.62) and (2.89).

Proposition 34 i) Let Y_{k}^{+}, k ≥ 0 solve (1.13), (1.14) in T_{{·}}(S^{+}′). Then Y_{k}^{ba}, cf. (1.20) solve (1.13), (1.14) in D′_{m,ν}(ℝ^{+})

ii) For any of the functions Y_{k}, interchanging + with − in (1.20) does not change the balanced average.

Proof (i) The fact that Y_{0}^{ba} is a solution of (1.13) follows from Proposition 23. From Proposition 23 and Proposition 13 we see that D_{j}^{ba} is obtained by simply replacing Y_{0}^{+} by Y_{0}^{ba} in (2.50) (notice that on any finite interval, there are finitely many terms in the expression of D_{j}^{ba} − D_{j}^{+}.) The rest of the proof merely consists in inductively applying A_{α} to the equations (1.14), noting that each contains finitely many convolutions, and applying the commutation relation (2.29).

(ii) This is true for Y_{0} as an immediate verification shows that the + and − averages coincide on (0, 2) (where they consist in two terms). Thus by Proposition 19 they have to coincide on ℝ^{+}. With this, for the rest of the Y_{k} the property follows by an obvious induction from Proposition 13.

2.3 Appendix

2.3.1 The C^{∗}–algebra D′_{m,ν}

Let D be the space of test functions (compactly supported C^{∞} functions on (0, ∞)) and D(0, x) be the test functions on (0, x).

We say that f ∈ D′ is a staircase distribution if for any k = 0, 1, 2, ... there is an L^{1} function on [0, k + 1] so that f = F_{k}^{(km)} (in the sense of distributions) when restricted to D(0, k + 1) or

F_{k} := P^{mk}f ∈ L_{1}(0, k + 1)  (2.90)

(since f ∈ L_{loc}^{1}[0, 1−ϵ] and by Remark 43, Pf is well defined). With this choice we have

F_{k+1} = P^{m}F_{k} on [0, k] and F_{k}^{(j)}(0) = 0 for j ≤ mk − 1  (2.91)

We denote these distributions by D′_{m} (D′_{m}(0, k) respectively, when restricted to D(0, k)) and observe that ⋃_{m>0} D′_{m} ⊃ S′, the distributions of slow growth. The inclusion is strict since any element of S′ is of finite order.

Let f ∈ L^{1}. Taking F = P^{j}f ∈ C^{j} we have, by integration by parts and noting that the boundary terms vanish,

(F ∗ F)(p) = ∫_{0}^{p} F (s)F (p − s)ds = ∫_{0}^{p} F^{(j)}(s)P^{j}F (p − s)  (2.92)

so that F ∗ F ∈ C^{2j} and

(F ∗ F)^{(2j)} = f ∗ f  (2.93)

This motivates the following definition: for f, f̃ ∈ D′_{m} let

f ∗ f̃:= (F_{k} ∗ F̃_{k})^{(2km)} in D′(0, k + 1)  (2.94)

We first check that the definition is consistent in the sense that

(F_{k+1} ∗ F_{k+1})^{(2m(k+1))} = (F_{k} ∗ F_{k})^{(2mk)}

on D(0, k + 1). For p < k + 1 integrating by parts and using (2.91) we obtain

\frac{d^{2m(k+1)}}{dp^{2m(k+1)}} ∫_{0}^{p} F_{k}(s)P^{2m}F̃_{k}(p − s)ds = \frac{d^{2mk}}{dp^{2mk}} ∫_{0}^{p} F_{k}(s)F̃_{k}(p − s)ds

(2.95)

The same argument shows that the definition is compatible with the embedding of D′_{m} in D′_{m′} with m′ > m. Convolution is commutative and associative: with f, g, h ∈ D′_{m} and identifying (f ∗g) and h by the natural inclusion with elements in D′_{2m} we obtain (f ∗ g) ∗ h = ((F ∗ G) ∗ H)^{(4mk)} = f ∗ (g ∗ h).

The following staircase decomposition exists in D′_{m}.

Lemma 35 . For each f ∈ D′_{m} there is a unique sequence {Δ_{i}}_{i=0,1,..} such that Δ_{i} ∈ L^{1}(ℝ^{+}), Δ_{i} = Δ_{i}χ_{[i,i+1]} and

f = ∑_{i=0}^{∞} Δ_{i}^{(mi)}  (2.96)

Also (cf. (2.91)),

F_{i} = ∑_{j≤i} P^{m(i−j)}Δ_{i} on [0, i + 1)  (2.97)

Note that the infinite sum is D′−convergent since for a given test function only a finite number of distributions are nonzero.

Proof We start by showing (2.97). For i = 0 we take Δ_{0} = F_{0}χ[0, 1] (where F_{0}χ[0, 1] := ϕ ↦ ∫_{0}^{1} F_{0}(s)ϕ(s)ds). Assuming (2.97) holds for i < n we simply note that

Δ_{n} := χ_{[0,n+1]} (F_{n} − ∑_{j≤n−1} P^{m(n−j)}Δ_{j})

= χ_{[0,n+1]} (F_{n} − P^{m}(F_{n−1}χ_{[0,n]})) = χ_{[n,n+1]} (F_{n} − P^{m}(F_{n−1}χ_{[0,n]})) (2.98)

(with χ_{[n,∞]}F_{n} defined in the same way as F_{0}χ[0, 1] above) has, by the induction hypothesis and (2.91) the required properties. Relation (2.96) is immediate. It remains to show uniqueness. Assuming (2.96) holds for the sequences Δ_{i}, Δ̃_{i} and restricting f to D(0, 1) we see that Δ_{0} = Δ̃_{0}. Assuming Δ_{i} = Δ̃_{i} for i < n we then have Δ_{n}^{(mn)} = Δ̃_{n}^{(mn)} on D(0, n + 1). It follows from Remark 43 that Δ_{n}(x) = Δ̃_{n}(x) + P (x) on [0, n + 1) where P is a polynomial (of degree < mn). Since by definition Δ_{n}(x) = Δ̃_{n}(x) = 0 for x < n we have Δ_{n} = Δ̃_{n}(x).

The expression (2.94) hints to decrease in regularity, but this is not the case. In fact, we check that the regularity of convolution is not worse than that of its arguments.

Remark 36

(· ∗ ·) : D_{n} ↦ D_{n}  (2.99)

Since

χ_{[a,b]} ∗ χ_{[a′,b′]} = (χ_{[a,b]} ∗ χ_{[a′,b′]})χ_{[a+a′,b+b′]}  (2.100)

we have

F ∗ F̃ = ∑_{j+k≤⌊p⌋} P^{m(i−j)}Δ_{j} ∗ P^{m(i−k)}Δ̃_{k} = ∑_{j+k≤⌊p⌋} Δ_{j} ∗ P^{m(2i−j−k)}Δ̃_{k}

(2.101)

which is manifestly in C^{2mi−m(j+k)}[0, p) ⊂ C^{2mi−m⌊p⌋}[0, p).

2.3.2 Norms on D′_{m}

For f ∈ D′_{m} define

∥f∥_{ν,m} := c_{m} ∑_{i=0}^{∞} ν^{im}∥Δ_{i}∥_{L_{ν}^{1}}  (2.102)

(the constant c_{m}, immaterial for the moment, is defined in (2.112). When no confusion is possible we will simply write ∥f∥_{ν} for ∥f∥_{ν,m} and ∥Δ∥_{ν} for ∥Δ_{i}∥_{L_{ν}^{1}} (no other norm is used for the Δ’s). Let D′_{m,ν} be the distributions in D′_{m} such that ∥f||_{ν} < ∞.

Remark 37 ∥ · ∥_{ν} is a norm on D′_{m,ν}.

If ∥f∥_{ν} = 0 for all i, then Δ_{i} = 0 whence f = 0. In view of Lemma 35 we have ∥0∥_{ν} = 0. All the other properties are immediate.

Remark 38 D′_{m,ν} is a Banach space. The topology given by ∥ · ∥_{ν} on D′_{m,ν} is stronger than the topology inherited from D′.

Proof. If we let D′_{m,ν}(k, k+1) be the subset of D′_{m,ν} where all Δ_{i} = 0 except for i = k, with the norm (2.102), we have

D′_{m,ν} = ⨁_{k=0}^{∞} D′_{m,ν}(k, k + 1)  (2.103)

and we only need to check completeness of each D′_{m,ν}(k, k + 1) which is immediate: on L^{1}[k, k + 1], ∥ · ∥_{ν} is equivalent to the usual L^{1} norm and thus if f_{n} ∈ D′_{m,ν}(k, k + 1) is a Cauchy sequence then Δ_{k,n} →^{L_{ν}} Δ_{k} (whence weak

convergence) and f_{n} →^{D′_{m,ν}(k,k+1)} f where f = Δ_{k}^{(mk)}.

Lemma 39 The space D′_{m,ν} is a C^{∗} algebra with respect to convolution.

Proof. Let f, f̃ ∈ D′_{m,ν} with

f = ∑_{i=0}^{∞} Δ_{i}^{(mi)} , f̃ = ∑_{i=0}^{∞} Δ̃_{i}^{(mi)}

Then

f ∗ f̃ = ∑_{i,j=0}^{∞} Δ_{i}^{(mi)} ∗ Δ̃_{j}^{(mj)} = ∑_{i,j=0}^{∞} (Δ_{i} ∗ Δ̃_{j})^{m(i+j)}  (2.104)

and the support of Δ_{i} ∗Δ̃_{j} is in [i+j, i+j +2] i.e. Δ_{i} ∗Δ̃_{j} = χ_{[i+j,i+j+2]}Δ_{i} ∗Δ̃_{j}.

We first evaluate the norm in D′_{m,ν} of the terms (Δ_{i} ∗ Δ̃_{j})^{m(i+j)}.

I. Decomposition formula. Let f = F^{(mk)} ∈ D′(ℝ_{+}), where F ∈ L^{1}(ℝ_{+}), and F is supported in [k, k + 2] i.e., F = χ_{[k,k+2]}F (k ≥ 0). Then f ∈ D′_{m} and the decomposition of f (cf. (2.96)) has the terms:

Δ_{0} = Δ_{1} = ... = Δ_{k−1} = 0 , Δ_{k} = χ_{[k,k+1]}F  (2.105)

and

Δ_{k+n} = χ_{[k+n,k+n+1]}G_{n}, where G_{n} = P^{m} (χ_{[k+n,∞)}G_{n−1}) , G_{0} = F (2.106)

Proof of Decomposition Formula. We use first line of (2.98) of the paper

Δ_{j} = χ_{[j,j+1]} (F_{j} − ∑_{i=0}^{j−1} P^{m(j−i)}Δ_{i})  (2.107)

where, in our case, F_{k} = F, F_{k+1} = P^{m}F, ..., F_{k+n} = P^{mn}F, ....

The relations (2.105) follow directly from (2.107). Formula (2.106) is shown by induction on n. For n = 1 we have

Δ_{k+1} = χ_{[k+1,k+2]} (P^{m} F − P^{m}Δ_{k})

= χ_{[k+1,k+2]}P^{m} (χ_{[k,∞)}F − χ_{[k,k+1]}F ) = χ_{[k+1,k+2]}P^{m} (χ_{[k+1,∞)}F )

Assume (2.106) holds for Δ_{k+j}, j ≤ n−1. Using (2.107), with χ = χ_{[k+n,k+n+1]} we have

Δ_{k+n} = χ (P^{mn}F − ∑_{i=k}^{n−1} P^{m(n−i)}Δ_{i}) = χP^{m} (G_{n−1} − Δ_{n−1})

= χP^{m} (χ_{[k+n−1,∞)}G_{n−1} − χ_{[k+n−1,k+n]}G_{n−1}) = χP^{m} (χ_{[k+n,∞)}G_{n−1})

II. Estimating Δ_{k+n}. For f as in I, we have

||Δ_{k+1}||_{ν} ≤ ν^{−m}||F ||_{ν} , ||Δ_{k+2}||_{ν} ≤ ν^{−2m}||F ||_{ν}  (2.108)

and, for n ≥ 3

||Δ_{k+n}||_{ν} ≤ e^{2ν−nν}(n − 1)^{nm−1} \frac{1}{(nm − 1)!}||F ||_{ν}  (2.109)

Proof of estimates of Δ_{k+n}. (A) Case n = 1.

||Δ_{k+1}||_{ν} ≤ ∫_{k+1}^{k+2} dt e^{−νt}P^{m} (χ_{[k+1,∞)}|F |) (t)

= ∫_{k+1}^{k+2} dt e^{−νt} ∫_{k+1}^{t} ds_{1} ∫_{k+1}^{s_{1}} ds_{2} ... ∫_{k+1}^{s_{m−1}} ds_{m}|F (s_{m})|
≤ ∫_{k+1}^{k+2} ds_{m}|F (s_{m})| ∫_{s_{m}}^{∞} ds_{m−1} ... ∫_{s_{2}}^{∞} ds_{1} ∫_{s_{1}}^{∞} dt e^{−νt}

= ∫_{k+1}^{k+2} ds_{m}|F (s_{m})|e^{−νs_{m}}ν^{−m} ≤ ν^{−m}||F ||_{ν} (2.110)

(B) Case n = 2:

||Δ_{k+1}||_{ν} ≤ ∫_{k+2}^{k+3} dt e^{−νt}P^{m} (χ_{[k+2,∞)}P^{m} (χ_{[k+1,∞)}|F |))

= ∫_{k+2}^{k+3} dt e^{−νt} ∫_{k+2}^{t} dt_{1} ∫_{k+2}^{t_{1}} dt_{2} ... ∫_{k+2}^{t_{m−1}} dt_{m} ∫_{k+1}^{t_{m}} ds_{1} ∫_{k+1}^{s_{1}} ds_{2} ... ∫_{k+1}^{s_{m−1}} ds_{m}|F (s_{m})|

≤ ∫_{k+2}^{k+3} ds_{m}|F (s_{m})| ∫_{s_{m}}^{∞} ds_{m−1} ... ∫_{s_{2}}^{∞} ds_{1} ∫_{max{s_{1},k+2}}^{∞} dt_{m} ∫_{t_{m}}^{∞} dt_{m−1} ... ∫_{t_{1}}^{∞} dt e^{−νt}

= ∫_{k+2}^{k+3} ds_{m}|F (s_{m})| ∫_{s_{m}}^{∞} ds_{m−1} ... ∫_{s_{2}}^{∞} ds_{1}e^{−ν max{s_{1},k+2}}ν^{−m−1}

≤ ∫_{k+2}^{k+3} ds_{m}|F (s_{m})| ∫_{s_{m}}^{∞} ds_{m−1} ... ∫_{s_{3}}^{∞} ds_{2}e^{−νs_{2}}ν^{−m−2} = ∫_{k+2}^{k+3} ds_{m}|F (s_{m})|e^{−νs_{m}}ν^{−2m}

(C) Case n ≥ 3. We first estimate G_{2}, ..., G_{n}:

|G_{2}(t)| ≤ P^{m} (χ_{[k+2,∞)}P^{m} (χ_{[k+1,∞)}|F |)) (t)

= ∫_{k+2}^{t} dt_{1} ∫_{k+2}^{t_{1}} dt_{2} ... ∫_{k+2}^{t_{m−1}} dt_{m} ∫_{k+1}^{t_{m}} ds_{1} ∫_{k+1}^{s_{1}} ds_{2} ... ∫_{k+1}^{s_{m−1}} ds_{m}|F (s_{m})|

and using the inequality

|F (s_{m})| = |F (s_{m})|χ_{[k,k+2]}(s_{m}) ≤ |F (s_{m})|e^{−νs_{m}}e^{ν(k+2)}

we get

|G_{2}(t)| ≤ e^{ν(k+2)}||F ||_{ν} ∫_{k+1}^{t} dt_{1} ∫_{k+1}^{t_{1}} dt_{2} ... ∫_{k+1}^{t_{m−1}} dt_{m} ∫_{k+1}^{t_{m}} ds_{1} ∫_{k+1}^{s_{1}} ds_{2} ... ∫_{k+1}^{s_{m−2}} ds_{m−1}

= e^{ν(k+2)}||F ||_{ν}(t − k − 1)^{2m−1} \frac{1}{(2m − 1)!}

The estimate of G_{2} is used for bounding G_{3}:

|G_{3}(t)| ≤ P^{m} (χ_{[k+3,∞)}|G_{2}|) ≤ P^{m} (χ_{[k+1,∞)}|G_{2}|)

≤ e^{ν(k+2)}||F ||_{ν}(t − k − 1)^{3m−1} \frac{1}{(3m − 1)!}

and similarly (by induction)

|G_{n}(t)| ≤ e^{ν(k+2)}||F ||_{ν}(t − k − 1)^{nm−1} \frac{1}{(nm − 1)!}

Then

||Δ_{k+n}||_{ν} ≤ e^{ν(k+2)}||F ||_{ν} \frac{1}{(nm − 1)!} ∫_{k+n}^{k+n+1} dt e^{−νt}(t − k − 1)^{nm−1}

and, for ν ≥ m the integrand is decreasing, and the inequality (2.109) follows. III. Final Estimate. Let ν_{0} > m be fixed. For f as in I, we have for any ν > ν_{0},

||f|| ≤ c_{m}ν^{km}||F ||_{ν}  (2.111)

for some c_{m}, if ν > ν_{0} > m. Proof of Final Estimate

||f|| = ∑_{n≥0} ν^{km+kn}||Δ_{k+n}||_{ν} ≤ ν^{km}||F ||_{ν} [3 + ∑_{n≥3} ν^{nm}e^{2ν−nν} \frac{(n − 1)^{nm−1}}{(nm − 1)!} ]

and, using n − 1 ≤ (mn − 1)/m and a crude Stirling estimate we obtain

||f|| ≤ ν^{km}||F ||_{ν} [3 + me^{2ν−1} ∑_{n≥3} (e^{m−ν}ν^{m}/m^{m})^{n}] ≤ c_{m}ν^{km}||F ||_{ν} (2.112)

Thus (2.111) is proven for ν > ν_{0} > m. End of the proof. From (2.104) and (2.111) we get

||f ∗ f̃|| ≤ ∑_{i,j=0}^{∞} || (Δ_{i} ∗ Δ̃_{j})^{m(i+j)} ||

≤ ∑_{i,j=0}^{∞} c_{m}^{2}ν^{m(i+j)}||Δ_{i} ∗ Δ̃_{j}||_{ν} ≤ c_{m}^{2} ∑_{i,j=0}^{∞} ν^{m(i+j)}||Δ_{i}||_{ν} ||Δ̃_{j}||_{ν} = c_{m}^{2}||f|| ||f̃||

Remark 40 Let f ∈ D′_{m,ν} for some ν > ν_{0} where ν_{0}^{m} = e^{ν_{0}}. Then f ∈ D′_{m,ν′} for all ν′ > ν and furthermore,

∥f∥_{ν} ↓ 0 as ν ↑ ∞  (2.113)

Proof. We have

ν^{mk} ∫_{k}^{k+1} |Δ_{k}(s)|e^{−νs}ds = (ν^{m}e^{−ν})^{k} ∫_{0}^{1} |Δ_{k}(s + k)|e^{−νs}ds  (2.114)

which is decreasing in ν. The rest follows from the monotone convergence theorem.

2.3.3 Embedding of L_{ν}^{1} in D′_{m}

Lemma 41 i) Let f ∈ L_{ν_{0}}^{1} (cf. Remark 40). Then f ∈ D′_{m,ν} for all ν > ν_{0}. ii) D(ℝ^{+}\ℕ) ∩ L_{ν}^{1}(ℝ^{+}) is dense in D_{m,ν} with respect to the norm ∥∥_{ν}.

Proof.

Note that if for some ν_{0} we have f ∈ L_{ν_{0}}^{1}(ℝ^{+}) then

∫_{0}^{p} |f(s)|ds ≤ e^{ν_{0}p} ∫_{0}^{p} |f(s)|e^{−ν_{0}s}ds ≤ e^{ν_{0}p}∥f∥_{ν_{0}}  (2.115)

to which, application of P^{k−1} yields

P^{k}|f| ≤ ν_{0}^{−k+1}e^{ν_{0}p}∥f∥_{ν_{0}}  (2.116)

Also, Pχ_{[n,∞)}e^{ν_{0}p} ≤ ν_{0}^{−1}χ_{[n,∞)}e^{ν_{0}p} so that

P^{m}χ_{[n,∞)}e^{ν_{0}p} ≤ ν_{0}^{−m}χ_{[n,∞)}e^{ν_{0}p}  (2.117)

so that, by (2.98) (where now F_{n} and χ_{[n,∞]}F_{n} are in L_{loc}^{1}(0, n + 1)) we have for n > 1,

|Δ_{n}| ≤ ∥f∥_{ν_{0}}e^{ν_{0}p}ν_{0}^{1−mn}χ_{[n,n+1]}  (2.118)

Let now ν be large enough. We have

∑_{n=2}^{∞} ν^{mn} ∫_{0}^{∞} |Δ_{n}|e^{−νp}dp ≤ ν_{0}∥f∥_{ν_{0}} ∑_{n=2}^{∞} ∫_{n}^{n+1} e^{−(ν−ν_{0})p} (\frac{ν}{ν_{0}} )^{p} dp

= \frac{e^{−2(ν−ν_{0}−ln(ν/ν_{0}))}}{ν − ν_{0} − ln(ν/ν_{0})}ν_{0}∥f∥_{ν_{0}} (2.119)

For n = 0 we simply have ∥Δ_{0}∥ ≤ ∥f∥, while for n = 1 we write

∥Δ_{1}∥_{ν} ≤ ∥1^{∗(m−1)} ∗ |f|∥_{ν} ≤ ν^{m−1}∥f∥_{ν}  (2.120)

Combining the estimates above, the proof of (i) is complete. To show (ii), let f ∈ D′_{m,ν} and let k_{ϵ} be such that c_{m} ∑_{i=k_{ϵ}}^{∞} ν^{im}∥Δ_{i}∥_{ν} < ϵ. For each i ≤ k_{ϵ} we take a function δ_{i} in D(i, i + 1) such that ∥δ_{i} − Δ_{i}∥_{ν} < ϵ2^{−i}. Then ∥f − ∑_{i=0}^{k_{ϵ}} δ_{i}^{(mi)}∥_{m,ν} < 2ϵ.

Proof of continuity of f(p) ↦ pf(p). If f(p) = ∑_{k=0}^{∞} Δ_{k}^{(mk)} then pf = ∑_{k=0}^{∞}(pΔ_{k})^{(mk)} − ∑_{k=0}^{∞} mkP(Δ_{k}^{(mk)})= ∑_{k=0}^{∞}(pΔ_{k}^{(mk)}) − 1 ∗ ∑_{k=0}^{∞}(mkΔ_{k})^{(mk)}. The rest is obvious from continuity of convolution, the embedding shown above and the definition of the norms.

2.3.4 Laplace transforms

Proof of Lemma 6 . Let ν > ν_{0} (cf. Remark 40). Equation (2.14) follows most easily from the corresponding well property of Laplace transforms on D, from the continuity of L and Lemma 41 (ii). For the second notice that by the definition of D′_{m}, f′ ∈ D′_{m} implies f ∈ AC(0, 1 − ϵ) and the property follows by density from the D identity L(Pg) = x^{−1}L(g). The third equality also follows by density. The rest of the properties, except injectivity, follow immediately from the definitions and the topology used.

To show injectivity assume that Ld(x) = 0 where d ∈ D′_{m,ν}, ν < x_{0} < x ∈ ℝ^{+}. By analyticity, Ld(x) = 0 in (say) S_{2} := {z : |z| > 2x_{0} : | arg(z)| < π/4}. Using dominated convergence, assuming x_{0} is large enough, we have

|∑_{k=1}^{∞} x^{m(k−1)}e^{−(k−1)x} ∫_{0}^{1} e^{−sx}|Δ_{k}(s + k)|ds| ≤ 1

in S_{2}. Thus |f(x)| = |∫_{0}^{1} e^{px}Δ_{0}(1 − p)dp| ≤ |x|^{m} in S_{2}. The function f^{(m)}(x) is entire, of exponential order less than 1 + ϵ for any ϵ and, using the previous inequality in Cauchy’s formula we see that |f^{(m)}(x)| < const. in (x_{0}, ∞). Since for ϕ ∈ (π/2, 3π/2) we obviously also have f^{(m)}(re^{iϕ}) → 0 as r → ∞, an elementary instance of the Phragmén-Lindelöf principle [28] implies that f^{(m)} is bounded in ℂ, therefore constant, so f itself is a polynomial that decays in the left half plane, thus f = 0. Therefore ∫_{0}^{1} e^{−px}Δ_{0}(p)dp = LΔ_{0} = 0 so that Δ_{0} = 0. Inductively and in the same way, we see that Δ_{k} = 0, k ∈ ℕ.

Proof of Lemma 11. Take first r ∉ ℤ. Choose a_{1}, a_{2} so that 0 < a_{1} < a_{2} < a and consider the closed contour C going along the upper cut from ξ = 0 to ξ = a_{2}, continuing towards the lower cut anticlockwise along the circle C(a_{2}) of radius a_{2} centered at origin, and finally coming from ξ = a_{2} back to ξ = 0 along the lower cut. For |ξ| < a_{1} we have, by the assumptions of the lemma,

2πif(ξ) = ∮_{C} \frac{f(s)}{s − ξ} ds = ∮_{C(a_{2})} \frac{f(s)}{s − ξ} ds + ∫_{0}^{a_{2}} \frac{s^{r}A(s)}{s − ξ}ds

(2.121)

On the other hand, defining z^{r}A(z) in the interior of C(a) cut along the positive axis (with the usual convention arg(z) = 0 on the upper cut), we have, for the same contour as above and ξ ∈ V_{a_{1}}

2π iξ^{r}A(ξ) = ∮_{C(a_{2})} \frac{f(s)}{s − ξ} ds + (1 − e^{2πir}) ∫_{0}^{a_{2}} \frac{s^{r}A(s)}{s − ξ}ds

(2.122)

Comparing (2.121) to (2.122) we get:

f(ξ) = \frac{1}{1 − e^{2πir}} ξ^{r}A(ξ)

−\frac{1}{2πi(1 − e^{2πir})} ∮_{C(a_{2})} \frac{A(s)}{s − ξ} ds + \frac{1}{2πi} ∮_{C(a_{2})} \frac{f(s)}{s − ξ} ds

(2.123)

As integrals of analytic functions with respect to complex absolutely continuous measures (A(s)ds and f(s)ds), the last two terms in (2.123) are analytic in ξ for |ξ| < a_{1}. Since a_{1} can be chosen arbitrarily close to a, the case r ∉ ℤ is proven. For r ∈ ℤ the argument is essentially the same, in terms of A(ξ)ξ^{r} ln ξ instead of ξ^{r}A(ξ). The proof generalizes immediately to linear combinations of ξ^{r}A(ξ).

Proof of Lemma 12. On the interval (k, k + 1) we have f^{+} = ∑_{i=1}^{k}(f_{i}^{−})^{(mi)} or

P^{mk+1}f^{+} = ∑_{i=1}^{k} P^{m(k−i)+1}f_{i}^{−}  (2.124)

Let ϵ be small and positive. Since f(te^{iϕ}) and g_{i}^{−}(te^{iϕ}) converge as ϕ → 0 in D′_{m,ν} we have that P^{m(k−i)+1}f(te^{iϕ}) and P^{m(k−i)+1}g_{i}^{−}(e^{iϕ}t) converge on [0, k + 1 − ϵ] uniformly to P^{mk+1}f^{+} and P^{m(k−i)+1}f_{i}^{−} respectively. The left side of (2.124) is the limit on I = [k + ϵ, k + 1 − ϵ] of a function analytic in a neighborhood in the upper half plane of I and continuous on I while the right side is the limit of a function analytic in a neighborhood in the lower half plane of I and continuous on I. The equality of their continuous limits on I implies in particular that P^{mk+1}f(te^{iϕ}) extends analytically through I in the lower half plane, and its continuation is analytic where ∑_{i=1}^{k} P^{m(k−i)+1}g_{i}^{−}(te^{iϕ}) is. A corresponding statement is true for the upper plane continuation of P^{mk+1}f(te^{−iϕ}) and (i) follows. Part (ii) now follows also, as an immediate application of Lemma 11.

Proof of Proposition 13.

The fact that multiplication by a bounded analytic function is well defined on F(R′_{1}) is immediate. Since

2f ∗ g = (f + g) ∗ (f + g) − f ∗ f − g ∗ g  (2.125)

we may take f = g. With h = P^{mk+1}f ∈ F(R′_{1}) it suffices to show for every k that h ∗ h (defined near zero by (1.8) and which equals P^{2mk+2}(f ∗ f) there) extends analytically to R′_{1} for ℜ(x) < k. Since f is analytic in R′_{1} so is h. In particular h can be analytically continued along any ray d ⊂ R′_{1} other than the real line, and we have, by analyticity and with ∗_{d} meaning convolution along d,

AC(h ∗ h) = AC(h) ∗_{d} AC(h)  (2.126)

Also, by (2.24)

h^{−}(p) = h^{+}(p) + ∑_{j=1}^{∞}(h_{j}^{+}(p − j))^{(mj)}  (2.127)

Let H_{0} = h^{+} and H_{j}(p) = (h_{j}^{+}(p))^{(mj)}. By construction H′_{j} have L^{1} boundary values on [0, k − j + 1) as ℜ(z) > 0, ℑ(z) ↓ 0 and so H_{j} extend continuously to the strip 0 < ℜ(z) < k − j + 1, ℑ(z) ≥ 0. We have, by (2.24) and continuity

h^{−}(z) = ∑_{i=0}^{j} H_{i}(z − j)  (2.128)

for ℜ(z) ∈ [0, j) and ℑ(z) ≥ 0 since H_{i}(x) = 0 in the left half plane, by definition. For the same reason we have, with p′ = p − i − j and i + j ≥ 1,

∫_{0}^{p} H_{i}(x − i)H_{j}(p − x − j)dx

= \{ \begin{matrix}∫_{0}^{p′} H_{i}(x)H_{j}(p′ − x)dx = J_{i,j}(p′) & (ℜ(p)′ > i + j) \\ 0 & (ℜ(p′) < i + j) \end{matrix} (2.129)

As both H_{i} and H_{j} are analytic in an open strip S in the first quadrant and continuous on [0, k + 1 − ϵ] we see from (2.129) that J_{ij}(p′) are also analytic in S and continuous on [0, k + 1 − ϵ]. For p ∈ (0, l + 1), l ≤ k we have

(H^{−} ∗ H^{−})(p) = ∑_{i=0}^{l} ∑_{j=0}^{i} J_{ij}(p − i)

(2.130)

Now, by (2.126) and using the continuity of H and of convolution, we note that the left side of (2.130) represents the continuous limit along (l, l+1) of (H ∗H)^{−}, a function analytic in a domain in the lower half plane while the right side is the limit of a function analytic in the upper half plane and (l, l + 1) is contained in the common boundary. As in the proof of Lemma 12 we conclude that h ∗ h, thus f ∗ f extend analytically in R′_{1}.

Going back to the definition of H we get on (0, l + 1),

(f^{−} ∗ f^{−})(p) = (f^{+} ∗ f^{+})(p) + ∑_{i=1}^{l} ∑_{j=0}^{i}(f_{j}^{+})^{(mj)} ∗ (f_{i−j}^{+})^{m(i−j)}(p − i)

(2.131)

where f_{j} ∗ f_{i−j} = (H_{j} ∗ H_{i−j})^{(2mk+2)} is the convolution in D′_{m,ν} and in our case gives a function analytic in the open region S. By comparing with (2.24) and (2.128) we get by induction (f ∗ f)_{j} = ∑_{s=0}^{j} f_{s} ∗ f_{j−s} or, using (2.125) we get (2.24).

Since by assumption f_{s} and g_{s} belong to D′_{m,ν} and the sum in (2.28) only contains a finite number of terms, it follows that all analytic continuations of (f ∗ g) also belong to D′_{m,ν}. Furthermore, it follows immediately that K(f ∗ g, ν) ≤ 2K(f, ν)K(g, ν).

Only the last equality in (2.29) needs a proof; we have

(A_{α}(f))^{∗2} = (∑_{i=0}^{∞} α^{i}(f_{i}(p − i))^{(mi)})^{∗2} = ∑_{k=0}^{∞} α^{k} ∑_{j=0}^{k}(f_{j} ∗ f_{k−j})^{(mk)}(p − k)

= ∑_{k=0}^{∞} α^{k}((f ∗ f)_{k})^{(mk)} = A_{α}(f ∗ f) (2.132)

We have ∥A_{C}(f)∥_{m,ν} ≤ ∥f∥_{m,ν} ∑_{j=0}^{∞} C^{j}K(f, ν)^{j} = (1 − KC)^{−1}∥f∥_{m,ν} so that if Y ∈ F_{r} then ∥A_{C}(Y^{∗l})∥_{m,ν} = ∥(A_{C}Y)^{∗l}∥_{m,ν} ≤ (∥Y∥_{m,ν}/(1 − KC))^{|l|} so that if ν is large enough the sum involved in the expression of M is uniformly convergent in D′_{m,ν}(ℝ^{+}) and (2.28) follows.

Lemma 42 (i) Let k_{0} ≥ 0 and let λ be such that ℜ(λ) < α_{1} < k_{0} and |ℑ(λ)| < α_{2}. Alternatively, let k_{0} ≥ 0 and λ be such that 0 < α_{1} < |ℑ(λ)| < α_{2}. There exists a constant C(α_{1}, α_{2}) independent of k_{0}, ν and λ so that

∥U∥_{D′_{m,ν}(k_{0},∞)↦D′_{m,ν}(k_{0},∞)} ≤ C(α_{1}, α_{2})(1 + |λ|)^{−1}  (2.133)

(ii) In both cases in (i), U is strongly continuous in λ.

Proof

The impediments in the proof come on the one hand from having to estimate quotients of the form ∫ |P^{n}Uf|/ ∫ |P^{n}f| and on the other hand from the nonlocal character of the action of U in our space.

In view of Eq. (2.103) it is enough to find a k−independent upper bound for the norms of the restrictions of U to D′_{m,ν}(k, k + 1), U : D′_{m,ν}(k, k + 1) ↦ D′_{m,ν}. We are interested in ℜ(λ) < 1 in the cases (a) b > |ℑ(λ)| > a > 0, (b) λ < −a < 0 real, (c) λ ∈ ℝ^{+} or λ complex, |ℑ(λ)| < b but with supp(f) ∈ (k_{1}, ∞) with k_{1} > a > ℜ(λ), k_{1} ∈ (a, a + 1). We let k_{1} = 0 in (a) and (b). We have the following identity

\frac{f^{(r)}}{p − λ} = (r(p − λ)^{r−1} ∫_{k}^{p} \frac{f(s)}{(s − λ)^{r+1}} ds + \frac{f(p)}{p − λ})^{(r)}  (2.134)

which is proved by straightforward differentiation of the r.h.s. or by writing f^{(r)} = (p − λ)g^{(r)} = (pg − λg)^{(r)} − rg^{(r−1)} so that f = (p − λ)(Pg)′ − rPg and solving for Pg. We take k ∈ ℕ with k + 1 > k_{1}, a distribution f with supp(f) ∈ (k_{0}, k + 1) where k_{0} = max{k, k_{1}} and we let

c = ∫_{k_{0}}^{k+1} \frac{f(s)}{(s − λ)^{r+1}} ds (r := mk)  (2.135)

For ϵ small (to be made zero in the end), we write the decomposition

\frac{f^{(r)}}{p − λ}

= (r(p − λ)^{r−1} ∫_{k_{0}}^{p} \frac{f(s)}{(s − λ)^{r+1}} ds + \frac{f(p)}{p − λ} − cr(p − λ)^{r−1}χ_{[k+1−ϵ,∞]})^{(r)}

+ (cr(p − λ)^{r−1}χ_{[k+1−ϵ,∞]})^{(r)} = f_{1} + f_{2} = f_{1} + c r g_{2}^{(r)} (2.136)

where by construction f_{1} ∈ D′_{m,ν}(k, k + 1) whence, for f ∈ D(k_{0}, k + 1) we have

∥\frac{f^{(r)}}{p − λ}∥_{m,ν} ≤ ∥f_{1}∥_{m,ν} + ∥f_{2}∥_{m,ν} = ∥f_{1}∥_{D′_{m,ν}(k,k+1)} + ∥f_{2}∥_{m,ν}

≤ ∥f_{1} − f_{2}∥_{D′_{m,ν}(k,k+1)} + 2∥f_{2}∥_{m,ν} (2.137)

and then, for some C_{1}

∥f_{1} − f_{2}∥_{D′_{m,ν}(k,k+1)}

≤ ν^{r} ∫_{k_{0}}^{k+1} |r(p − λ)^{r−1} ∫_{k_{0}}^{p} \frac{f(s)}{(s − λ)^{r+1}} ds + \frac{f(p)}{p − λ}| e^{−νp}dp

≤ sup_{∗}|p − λ|^{−1}∥f∥_{m,ν} + sup_{∗} |\frac{(p − λ)^{r−1}}{(s − λ)^{r+1}} | ∥f∥_{m,ν}

≤ C_{1}sup_{∗}|p − λ|^{−1}∥f∥_{m,ν} (2.138)

where the supremum is taken over {k ∈ ℕ, p, s ∈ [k, k + 1] ∩ (k_{0}, ∞)}. For the constant c in (2.136) we have, for some C_{2} depending on a and otherwise independent of λ, k the estimate

|c| ≤ \frac{C_{2}e^{ν(k+1)}}{|k_{0} − λ|^{r+1}} ∫_{k_{0}}^{k+1} |f(s)|e^{−νs}ds = ν^{−r} \frac{C_{2}e^{ν(k+1)}}{|k_{0} − λ|^{r+1}} ∥f∥_{m,ν}  (2.139)

Let k′ = k + 1 − ϵ. For some C_{3} = C_{3}(a) ≤ exp [(k_{0} + 1)|k_{0} − λ|^{−1}] we have ∥g_{2}∥_{m,ν,k} = ν^{r} ∫_{k′}^{k+1} e^{−νx}|x − λ|^{r−1}dx ≤ C_{3}ν^{r−1}|k_{0} − λ|^{r−1}e^{−νk′}

⇒ c r ∥g_{2}∥_{m,ν,k} ≤ ν^{−r}mk \frac{e^{ν(k+1)}}{|k_{0} − λ|^{r+1}} ∥f∥_{m,ν}C_{2}C_{3}ν^{r−1}|k_{0} − λ|^{r−1}e^{−νk′}

= \frac{C_{4}mke^{νϵ}}{ν|k_{0} − λ|^{2}} ∥f∥_{m,ν} (2.140)

For n ≥ k + 1 we write (2.98) as

Δ_{n}(g_{2}^{(r)}) = χ_{[n,n+1]}P^{m} (χ_{[n,∞)}P^{m(n−k−1)}g_{2})  (2.141)

(cf. (2.98)). For λ complex we take K_{1}(a) = sup_{s≥k_{0}}(s − λ)^{−1}(s + 1 + |λ|). We let λ̃ = λ if λ is real and λ̃ = −1 − |λ| otherwise and write q = m(n − k − 1). Noting that K_{1}^{mk_{0}+m−1} ≤ C_{5}(a, b) = K we have

Γ(q)P^{q}|g_{2}| ≤ K ∫_{k′}^{x} (x − s)^{q−1}(s − λ̃)^{r−1}ds
≤ K ∫_{0}^{x} (x − s)^{q−1}(s − λ̃)^{r−1}ds = \frac{(x − λ̃)^{q+r−1}Γ(q)Γ(r)}{Γ(q + r)} (2.142)

The estimate above is true for λ̃ ≤ 1 but is “optimal” only when the maximum of the integrand is inside the region of integration i.e. when λ̃ > −(2 + m^{−1})k(n − k)^{−1} + m^{−1}. If this is not the case we prefer to simply estimate the integral in terms of the maximum of the integrand over the region of integration. So for, say, λ̃ < −3k we use the inequality

Γ(q)P^{q}|g_{2}| ≤ K(x − k′)^{q}(k′ − λ̃)^{r−1}  (2.143)

Now, for λ̃ > −3k, using (2.142) and (2.141)

P^{m}(χ_{[n,∞)}P^{m(n−k−1)}|g_{2}|) ≤ \frac{Γ(r)}{Γ(q + r)Γ(m)} ∫_{n}^{x}(x − s)^{m−1}(s − λ̃)^{q+r−1}ds

≤ \frac{Γ(r)}{Γ(q + r)Γ(m)}(x − n)^{m}(x − λ̃)^{q+r−1}

(2.144)

(as m is fixed we do not lose too much by this evaluation which has the advantage of preserving the behavior near x = n). Further, we have

∫_{n}^{n+1} e^{−νx}(x − n)^{m}(x − λ̃)^{q+r−1}dx ≤ (n + 1 − λ̃)^{q+r−1} ∫_{n}^{n+1} e^{−νx}(x − n)^{m}dx

≤ \frac{Γ(m)}{ν^{m}}(n + 1 − λ̃)^{q+r−1} (2.145)

and

∑_{n=k+1}^{∞} ν^{mn}∥Δ∥_{m,ν} ≤ Γ(mk)K ∑_{n=k+1}^{∞} \frac{ν^{m(n−1)}e^{−νn}(n + 1 − λ̃)^{mn−1−m}}{Γ(m(n − 1))}

(2.146)

The ratio of two successive terms s_{n+1}/s_{n} of the infinite series above is estimated by:

ν^{m}e^{−ν}e^{\frac{mn−1−m}{n+1−λ̃}} (\frac{n + 2 − λ̃}{mn − m})^{m} ≤ \frac{1}{2}  (2.147)

when ν > C_{1} for some C_{1} independent of k, n, λ̃ in the region k > 1, n > k, λ̃ ∈ (−3k, 1). This means that

∑_{n=k+1}^{∞} ν^{mn}∥Δ∥_{m,ν} ≤ 2Kν^{mk}e^{−ν(k+1)}(k + 2 − λ̃)^{mk−1}

(2.148)

and combining with (2.139) and (2.137) we have

2∥f_{2}∥ ≤ 4Kν^{−mk} \frac{e^{ν(k+1)}}{(k − λ̃)^{mk+1}} ∥f∥_{m,ν}ν^{mk}e^{−ν(k+1)}(k + 2 − λ̃)^{mk−1}

≤ 4Ke^{2 \frac{(mk+1)}{k−λ̃}} \frac{∥f∥_{m,ν}}{(k − λ̃)^{2}} ≤ 4Ke^{\frac{4m}{k_{0}λ̃}} \frac{∥f∥_{m,ν}}{(k_{0} − λ̃)^{2}} ≤ \frac{C_{6}(a, b)∥f∥_{m,ν}}{|k_{0} − λ|^{2}} (2.149)

If we started with (2.143) we would have obtained in the same way, for λ̃ < −3k, instead of (2.144),

\frac{1}{Γ(q)Γ(m)}(x − n)^{m}(x − k′)^{q}(k′ − λ̃)^{r−1}  (2.150)

and the calculations are similar from this point on. Condition (2.147) is of the same type, with k′ replacing −λ̃ and final estimate is

C_{7}∥f∥_{m,ν} \frac{(k′ − λ̃)^{r−1}}{(k − λ̃)^{r+1}} ≤ C_{8} \frac{∥f∥_{m,ν}}{(k_{0} − λ̃)^{2}}  (2.151)

Finally we take the limit ϵ → 0 and noting that K → 0 as λ → ∞, (i) is proved. For (ii), merely notice that U(λ_{2}) − U(λ_{1}) = (λ_{2} − λ_{1})U(λ_{1})U(λ_{2}).

Remark 43 Let ψ ∈ L^{1}[0, 1] with the property ∫_{0}^{1} ψ(t)ϕ^{(m)}(t)dt = 0 for all ϕ ∈ D(0, 1). Then ψ is a polynomial of degree at most m − 1.

This is a well-known property. We sketch an elementary proof for m = 1 (for general m the proof is similar). Let x ∈ (0, 1), and consider a sequence χ_{n} in D(0, x) L^{1}–convergent to χ_{[0,x]}. Then ϕ_{n}(t) := χ_{n}(t) − κ^{−1}χ_{n}(κ(1 − t)) with κ = x(1 − x)^{−1} converges to χ_{[0,x]} − κχ_{[x,1]}. Furthermore, since ∫_{0}^{1} ϕ_{n}(t)dt = 0 we have Φ_{n}(t) := ∫_{0}^{t} ϕ_{n}(s)ds ∈ D(0, 1). Since Φ′_{n} + κ^{−1} → (1 + κ^{−1})χ_{[0,x]} it follows that ∫_{0}^{x}(ψ − C) = 0, where C = ∫_{0}^{1} ψ(t)dt. Thus ψ = C a.e.

2.4 Derivation of the equations for the transseries.

Consider first the scalar equation

y′ = f_{0}(x) − λy − x^{−1}By + g(x, y) = −y + x^{−1}By + ∑_{k=1}^{∞} g_{k}(x)y^{k} (2.152)

For x → +∞ we take

y = ∑_{k=0}^{∞} y_{k}e^{−kx}  (2.153)

where y_{k} will be either formal series x^{−s_{k}} ∑_{n=0}^{∞} a_{kn}x^{−n}, with a_{k,0} ≠ 0 or actual functions with the condition that (2.153) converges uniformly. As a transseries, (2.153) can be also understood as a well ordered double sequence t_{kn} = x^{p_{kn}}e^{−kx}, with p_{k n+1} < p_{kn}. (The order relation is x^{p}e^{−kx} ≫ x^{p′}e^{−k′x} as x → +∞ iff k < k′ or k = k′ and p > p′; thus a strictly increasing sequence of terms of a transseries necessarily terminates.) Power series are a special case of transseries, with y_{1} = y_{2} = … = 0. Two transseries ∑_{k=0}^{∞} y_{k}e^{−kx} coincide iff all corresponding component power series y_{k} coincide. Transseries of this type are closed under addition, multiplication and infinite sums of the form involved in (2.152) (this last aspect will become clear in the calculation leading to (2.155) below). Note that well-ordering plays an important part in defining multiplication of transseries; in contrast, for the unrestricted formal expansion S = ∑_{k=−∞}^{∞} x^{k}, no immediate meaning can be given to S^{2}. Let y_{0} be the first term in (2.153) and δ = y − y_{0}. We have

y^{k} − y_{0}^{k} − ky_{0}^{k−1}δ = ∑_{j=2}^{k} \binom{k}{j}y_{0}^{k−j}δ^{j} = ∑_{j=2}^{k} \binom{k}{j}y_{0}^{k−j} ∑_{i_{1},... ,i_{j}=1}^{∞} ∏_{s=1}^{j} (y_{i_{s}}e^{−i_{s}x})

= ∑_{m=1}^{∞} e^{−mx} ∑_{j=2}^{k} \binom{k}{j}y_{0}^{k−j} ∑_{(i_{s})}^{(m;j)} ∏_{s=1}^{j} y_{i_{s}} (2.154)

where ∑_{(i_{s})}^{(m;j)} means the sum over all positive integers i_{1}, i_{2}, …, i_{j} with the restriction i_{1}+i_{2}+⋯+i_{j} = m. Let d_{1} = ∑_{k≥1} kg_{k}y_{0}^{k−1}. Introducing y = y_{0}+δ in (2.152) and equating the coefficients of e^{−lx} we get, by separating the terms containing y_{l} for l ≥ 1 and interchanging the j, k orders of summation,

y′_{l} + (λ(1 − l) + x^{−1}B − d_{1}(x))y_{l} = ∑_{j=2}^{∞} ∑_{(i_{s})}^{(l;j)} ∏_{s=1}^{j} y_{i_{s}} ∑_{k≥\{2,j\}} \binom{k}{j}g_{k}y_{0}^{k−j}

= ∑_{j=2}^{l} ∑_{(i_{s})}^{(l;j)} ∏_{s=1}^{j} y_{i_{s}} ∑_{k≥\{2,j\}} \binom{k}{j}g_{k}y_{0}^{k−j} =: ∑_{j=2}^{l} d_{j}(x) ∑_{(i_{s})}^{(l;j)} ∏_{s=1}^{j} y_{i_{s}} (2.155)

where for the middle equality we note that the infinite sum terminates because i_{s} ≥ 1 and ∑_{s=1}^{j} i_{s} = l. The fact mentioned before that ∑_{k=1}^{∞} g_{k}(x)y_{k} is well defined when y_{k} are formal series is now visible: collecting the coefficient of x^{p}e^{−kx}, only finite sums of coefficients appear.

For a vectorial equation like (1.3) we first write

y′ = f_{0}(x) − Λ̂y + x^{−1}B̂y + ∑_{k≻0} g_{k}(x)y^{k}  (2.156)

with y^{k} := ∏_{i=1}^{n_{1}}(y)_{i}^{k_{i}}. The formal operations and ordering extend naturally to the vectorial general transseries (1.4), under the restriction ℜ(k · λx) > 0 As with (2.155), we introduce the transseries (1.4) in (2.156) and equate the coefficients of exp(−k · λx). Let v_{k} = x^{−k·m}y_{k} and

d_{j}(x) = ∑_{l≥j} \binom{l}{j}g_{l}(x)v_{0}^{l−j}  (2.157)

Noting that, by assumption, k · λ = k′ · λ ⇔ k = k′ we obtain, for k ∈ ℕ^{n_{1}}, k ≻ 0

v′_{k} + (Λ̂ − k · λÎ+ x^{−1}B̂) v_{k} + ∑_{|j|=1} d_{j}(x)(v_{k})^{j}

= ∑_{{j≤k \atop |j|≥2}} d_{j}(x) ∑_{(i_{mp}:k)} ∏_{m=1}^{n} ∏_{p=1}^{j_{m}} (v_{i_{mp}})_{m} = t_{k}(v)  (2.158)

where \binom{l}{j} = ∏_{j=1}^{n} \binom{l_{i}}{j_{i}}, (v)_{m} means the component m of v, and ∑_{(i_{mp}:k)} stands for the sum over all vectors i_{mp} ∈ ℕ^{n}, with p ≤ j_{m}, m ≤ n, such that i_{mp} ≻ 0 and ∑_{m=1}^{n} ∑_{p=1}^{j_{m}} i_{mp} = k. We use the convention ∏_{∅} = 1, ∑_{∅} = 0. With m_{i} = 1 − ⌊β_{i}⌋ we obtain for y_{k}

y′_{k} + (Λ̂ − k · λÎ+ x^{−1}(B̂ + k · m)) y_{k} + ∑_{|j|=1} d_{j}(x)(y_{k})^{j} = t_{k}(y) (2.159)

There are clearly finitely many terms in t_{k}(y). To find a (not too unrealistic) upper bound for this number of terms, we compare with ∑_{(i_{mp})′} which stands for the same as ∑_{(i_{mp})} except with i ≥ 0 instead of i ≻ 0. Noting that \binom{k+s−1}{s−1} = ∑_{a_{1}+...+a_{s}=k} 1 is the number of ways k can be written as a sum of s integers, we have

∑_{(i_{mp})} 1 ≤ ∑_{(i_{mp})′} 1 = ∏_{l=1}^{n_{1}} ∑_{(i_{mp})_{l}} 1 = ∏_{l=1}^{n_{1}} \binom{k_{l} + |j| − 1}{|j| − 1} ≤ \binom{|k| + |j| − 1}{|j| − 1}^{n_{1}} (2.160)

Remark 44 Equation (2.158) can be written in the form (1.15)

Proof. The fact that only predecessors of k are involved in t(y_{0}, ·) and the homogeneity property of t(y_{0}, ·) follow immediately by combining the conditions ∑ i_{mp} = k and i_{mp} ≻ 0. The formal inverse Laplace transform of (2.159) is then

(−p + Λ̂ − k · λ) Y_{k} + (B̂ + k · m) PY_{k} + ∑_{|j|=1} D_{j} ∗ (Y_{k})^{j} = T_{k}(Y)

(2.161)

with

T_{k}(Y) = T (Y_{0}, \{Y_{k′}\}_{0≺k′≺k}) = ∑_{j≤k; |j|>1} D_{j}(p) ∗ ∑_{(i_{mp};k)} \begin{matrix}n_{1} \\ ∗ & ∏ \\ m=1 \end{matrix} \begin{matrix}j_{m} \\ ∗ & ∏ \\ p=1 \end{matrix} (Y_{i_{mp}})_{m}

(2.162)

and

D_{j} = ∑_{l≥m} \binom{l}{m}G_{l} ∗ Y_{0}^{∗(l−m)} + ∑_{l≥m;|l|≥2} \binom{l}{m}g_{0,l}Y_{0}^{∗(l−m)}  (2.163)

2.5 Useful formulas

A straightforward computation shows that

B(\frac{1}{x^{n}}) = \frac{p^{n−1}}{Γ(n)} or L(p^{n}) = \frac{Γ(n + 1)}{x^{n+1}}  (2.164)

p^{q} ∗ p^{r} = \frac{Γ(q + 1)Γ(r + 1)}{Γ(q + r + 2)}p^{q+r+1}  (2.165)

Also, with f_{1,2}(p) := p ↦ H(p − k_{1,2})g_{1,2}(p − k_{1,2}) we have

(f_{1} ∗ f_{2})(p) = H(p − k_{1} − k_{2})(g_{1} ∗ g_{2})(p − k_{1} − k_{2})  (2.166)

3 Acknowledgments

Special thanks are due to Professors Martin Kruskal and Jean Écalle for their many in-depth comments, and to Professor B. L. J. Braaksma for carefully reading the manuscript and pointing out an error in a formula. The author is grateful to Professors Michael Berry and Percy Deift for very interesting discussions.

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