Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 930385, 86 pages
doi:10.1155/2012/930385
Research Article
On the Parametric Stokes Phenomenon for
Solutions of Singularly Perturbed Linear Partial
Differential Equations
Stéphane Malek
UFR de Mathématiques, Université de Lille 1, 59655 Villeneuve d’Ascq Cedex, France
Correspondence should be addressed to Stéphane Malek, stephane.malek@math.univ-lille1.fr
Received 27 March 2012; Accepted 8 July 2012
Academic Editor: Roman Dwilewicz
Copyright q 2012 Stéphane Malek. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We study a family of singularly perturbed linear partial differential equations with irregular type
s k0 k1
t2 ∂t ∂Sz Xi t, z, t 1∂Sz Xi t, z, s,k0 ,k1 ∈S bs,k0 ,k1 z, t ∂t ∂z Xi t, z, in the complex
domain. In a previous work, Malek 2012, we have given sufficient conditions under which
the Borel transform of a formal solution to the above mentioned equation with respect to the
perturbation parameter converges near the origin in C and can be extended on a finite number
of unbounded sectors with small opening and bisecting directions, say κi ∈ 0, 2π, 0 ≤ i ≤ ν − 1
for some integer ν ≥ 2. The proof rests on the construction of neighboring sectorial holomorphic
solutions to the first mentioned equation whose differences have exponentially small bounds in the
perturbation parameter Stokes phenomenon for which the classical Ramis-Sibuya theorem can
be applied. In this paper, we introduce new conditions for the Borel transform to be analytically
continued in the larger sectors { ∈ C∗ /arg ∈ κi , κi1 }, where it develops isolated singularities
of logarithmic type lying on some half lattice. In the proof, we use a criterion of analytic
continuation of the Borel transform described by Fruchard and Schäfke 2011 and is based on a
more accurate description of the Stokes phenomenon for the sectorial solutions mentioned above.
1. Introduction
We consider a family of singularly perturbed linear partial differential equations of the form
t2 ∂t ∂Sz Xi t, z, t 1∂Sz Xi t, z, s,k0 ,k1 ∈S
bs,k0 ,k1 z, ts ∂kt 0 ∂kz1 Xi t, z, 1.1
for given initial conditions
j
∂z Xi t, 0, ϕi,j t, ,
0 ≤ i ≤ ν − 1, 0 ≤ j ≤ S − 1,
1.2
2
Abstract and Applied Analysis
where is a complex perturbation parameter, S is some positive integer, ν is some positive
integer larger than 2, and S is a finite subset of N3 with the property that there exists an
integer b > 1 with
S ≥ bs − k0 2 k1 ,
s ≥ 2k0
1.3
for all s, k0 , k1 ∈ S, and the coefficients bs,k0 ,k1 z, belong to O{z, } where O{z, } denotes
the space of holomorphic functions in z, near the origin in C2 . In this work, we make
the assumption that the coefficients of 1.1 factorize in the form bs,k0 ,k1 z, k0 bs,k0 ,k1 z, where bs,k0 ,k1 z, belong to O{z, }. The initial data ϕi,j t, are assumed to be holomorphic
functions on a product of two sectors T × Ei , where T is a fixed bounded sector centered at
0 and Ei , 0 ≤ i ≤ ν − 1, are sectors with opening larger than π centered at the origin whose
union form a covering of V \ {0}, where V is some neighborhood of 0. For all / 0, this family
belongs to a class of partial differential equations which have a so-called irregular singularity
at t 0 in the sense of 1.
In the previous work 2, we have given sufficient conditions on the initial data
ϕi,j t, , for the existence of a formal series
z, Xt,
Hk t, zk
k≥0
k!
∈ OT{z}
1.4
solution of 1.1, with holomorphic coefficients Hk t, z on T × D0, δ for some disc D0, δ,
with δ > 0, such that, for all 0 ≤ i ≤ ν − 1, the solution Xi t, z, of the problem 1.1,
on Ei . In
1.2 defines a holomorphic function on T × D0, δ × Ei which is the 1-sum of X
with respect to other words, for all fixed t, z ∈ T × D0, δ, the Borel transform of X
2
k
defined as BXs k≥0 Hk t, zs /k! is holomorphic on some disc D0, s0 and can be
analytically continued with exponential growth to sectors Gκi , centered at 0, with infinite
radius and with the bisecting direction κi ∈ 0, 2π of the sector Ei . But in general, due
to the fact that the functions Xi do not coincide on the intersections Ei ∩ Ei1 known as
the Stokes phenomenon, the Borel transform cannot be analytically extended to the whole
sectors Sκi ,κi1 {s ∈ C∗ / args ∈ κi , κi1 } for all 0 ≤ i ≤ ν − 1, where by convention κν κ0 ,
Eν E0 , and Xν X0 .
In this work, we address the question of the possibility of analytic continuation,
location of singularities, and behaviour near these singularities of the Borel transform within
the sector Sκi ,κi1 . More precisely, our goal is to give stronger conditions on the initial data
can be analytically continued to the fullϕi,j t, under which the Borel transform BXs
punctured sector Sκi ,κi1 except a half lattice of points λk/t, k ∈ N \ {0}, depending on t and
some well-chosen complex number λ ∈ C∗ and moreover develop logarithmic singularities
at λk/t Theorem 5.8.
In a recent paper of Fruchard and Schäfke, see 3, an analogous study has been
performed for formal WKB solutions yx, expx2 /2 − x3 /3/x−1/2 x − 1−1/2 v x, to the singularly perturbed Schrödinger equation
2 y x, x2 x − 12 yx, ,
1.5
Abstract and Applied Analysis
3
where v x, n≥0 yn xn is a formal series with holomorphic coefficients yn on some
domain avoiding 0 and 1. The authors show that the Borel transform of v with respect to converges near the origin and can be analytically continued along any path avoiding some
lattices of points depending on x2 /2 − x3 /3. We also mention that formal parametric Stokes
phenomenon for 1-dimensional stationary linear Schrödinger equation 2 y z Qzyz,
where Qz is a polynomial, has been investigated by several other authors using WKB
analysis, see 4–6. In a more general framework, analytic continuation properties related
with the Stokes phenomenon have been studied by several authors in different contexts. For
nonlinear systems of ODEs with irregular singularity at ∞ of the form yz fz, yz
and for nonlinear systems of difference equations yz 1 gz, yz, under nonresonance
conditions, we refer to 7, 8. For linearizations procedures for holomorphic germs of C, 0
in the resonant case, we make mention to 9, 10. For analytic conjugation of vector fields
in C2 to normal forms, we indicate 11, 12. For Hamiltonian nonlinear first-order partial
differential equations, we notice 13.
In the proof of our main result, we will use a criterion for the analytic continuation of
the Borel transform described by Fruchard and Schäfke in 3 Theorem FS in Theorem 5.8.
Following this criterion, in order to prove the analytic continuation of the Borel transform
BXs,
say, on the sector Sκ0 ,κ1 , for any fixed t, z ∈ T × D0, δ, we need to have a complete
description of the Stokes relation between the solutions X0 and X1 of the form
X1 t, z, − X0 t, z, m
iα
e−ah / Xh,0 t, z, O e−Ce /
1.6
h1
for all ∈ E0 ∩ E1 , for some integer m ≥ 1, where {ah }1≤h≤m is a set of aligned complex
numbers such that argah α ∈ κ0 , κ1 with |ak | < C for some C > 0 and Xh,0 t, z, ,
h ∈ OT × D0, δ on E0 . If the relation
h ≥ 1, are the 1-sums of some formal series G
1.6 holds, then BXs
can be analytically continued along any path in the punctured sector
Sκ0 ,κ1 ∩ D0, C \ {ah }1≤h≤m and has logarithmic growth as s tends to ah in a sector. Actually,
under suitable conditions on the initial data ϕi,j t, , we have shown that such a relation
holds for ak λk/t, for some well-chosen λ ∈ C∗ and for all k ≥ 1, see 5.145 in Theorem 5.8.
In order to establish such a Stokes relation 1.6, we proceed in several steps.
In the first step, following the same strategy as in 2, using the linear map T → T/ t,
we transform the problem 1.1 into an auxiliary regularly perturbed singular linear partial
differential equation which has an irregular singularity at T 0 and whose coefficients have
poles with respect to at the origin, see 4.9. Then, for λ ∈ C∗ , we construct a formal
transseries expansion of the form
Y T, z, exp−λh/T h≥0
h!
Yh T, z, 1.7
m
solution of the problem 4.9, 4.10, where each Yh T, z, m≥0 Yh,m z, T /m! is a
formal series in T with coefficients Yh,m z, , which are holomorphic on a punctured polydisc
D0, δ × D0, 0 \ {0}. We show that the Borel transform of each Yh T, z, with respect
to T , defined by Vh τ, z, m≥0 Yh,m z, τ m /m!2 , satisfies an integrodifferential Cauchy
problem with rational coefficients in τ, holomorphic with respect to τ, z near the origin
and meromorphic in with a pole at zero, see 4.20, 4.21. For well-chosen λ and suitable
4
Abstract and Applied Analysis
initial data, we show that each Vh τ, z, defines a holomorphic function near the origin with
respect to τ, z and on a punctured disc with respect to and can be analytically continued to
functions Vh,i τ, z, defined on the products Si ×D0, δ×Ei , where Si , 0 ≤ i ≤ ν−1 are suitable
open sectors with small opening and infinite radius. Moreover, the functions Vh,i τ, z, have
exponential growth rate with respect to τ, , namely, there exist A, B, K > 0 such that
sup |Vh,i τ, z, | ≤ Ah!Bh eK|τ|/||
z∈D0,δ
1.8
for all τ, z, in their domain of definition and all h ≥ 0 Proposition 4.12. In order to get
these estimates, we use the Banach spaces depending on two parameters β ∈ N and with
norms || · ||β, of functions vτ bounded by expKβ |τ|/|| for some bounded sequence Kβ
already introduced in 2. If one expands the functions Vh,i τ, z, β≥0 vh,i,β τ, zβ /β!
with respect to z, we show that the generating function h≥0,β≥0 ||vh,i,β τ, ||β, uh xβ /h!β!
can be majorized by a series Wi u, x which satisfies a Cauchy problem of Kowalevski type
4.47, 4.48 and is therefore convergent near the origin in C2 .
− 1 as Laplace
We construct a sequence of actual functions Yh,i T, z, , h ≥ 0, 0 ≤ i ≤ ν √
transform of the functions Vh,i τ, z, with respect to τ along a halfline Li R e −1γ ⊂ Si ∪{0}.
We show that the functions Xh,i t, z, Yh,i t, z, are holomorphic functions on the
domain T × D0, δ × Ei and that the functions Gh,i : Xh,i1 t, z, − Xh,i t, z, are
exponentially flat as tends to 0 on Ei1 ∩ Ei as OT × D0, δ-valued functions. In the
proof, we use, as in 2, a deformation of the integration’s path in Xh,i and the estimates
1.8. Using the Ramis-Sibuya theorem Theorem RS in Proposition 4.15, we deduce that
h ∈ OT × D0, δ on Ei , for
each Xh,i t, z, is the 1-sum of a formal series G
0 ≤ i ≤ ν − 1 Proposition 4.15. We notice that the functions X0,i t, z, actually coincide
with the functions Xi t, z, mentioned above solving the problem 1.1, 1.2. We deduce
that, for a suitable choice of λ, the function
Z0 t, z, exp−λh/t
h≥0
h!
Xh,0 t, z, 1.9
solves 1.1 on the domain T × D0, δ × E0 ∩ E1 .
In the second part of the proof, we establish the connection formula X0,1 t, z, Z0 t, z, which is exactly the Stokes relation 1.6 on T × D0, δ × E0 ∩ E1 Proposition 5.2.
The strategy we follow consists in expressing both functions X0,1 and Z0 as Laplace
transforms of objects that are no longer functions in general but distributions supported on R
which are called staircase distributions in the terminology of 8. We stress the fact that such
representations of transseries expansions as generalized Laplace transforms were introduced
for the first time by Costin in the paper 8. Notice that similar arguments have been used in
the work 14 to study the Stokes phenomenon for sectorial holomorphic solutions to linear
integro-differential equations with irregular singularity.
In Lemma 5.5, we show that Z0 can be written as a generalized Laplace transform
in the direction argλ of a staircase distribution Vr, z, β≥0 Vβ r, zβ /β! ∈ D σ, , δ,
which is a convergent series in z on D0, δ with coefficients Vβ r, in some Banach spaces of
on R depending on the parameters β and see Definition 2.3.
staircase distributions Dβ,σ,
We observe that the distribution Vr, z, solves moreover an integro-differential Cauchy
problem with rational coefficients in r, holomorphic with respect to z near the origin and
Abstract and Applied Analysis
5
meromorphic with respect to at zero, see 5.80, 5.81. The idea of proof consists in showing
that each function Xh,0 t, z, can be expressed as a Laplace transform in a sequence of
directions ζn tending to argλ of a sequence of staircase distributions Vh,n r, z, which
are actually convergent series in z with coefficients that are C∞ functions in r on R with
exponential growth. Moreover, each distribution Vh,n r, z, solves an integro-differential
Cauchy problem 5.37, 5.38, whose coefficients tend to the coefficients of an integrodifferential equation 5.39, 5.40, as n tends to ∞, having a unique staircase distribution
solution Vh,∞ r, z, . Under the hypothesis that the initial data 5.38 converge to 5.40 as
n → ∞, we show that the sequence Vh,n r, z, converges to Vh,∞ r, z, in the Banach
space D σ, , δ with precise norm estimates with respect to h and n Lemma 5.3. In order
to show this convergence, we use a majorazing series method together with a version of the
classical Cauchy-Kowalevski theorem Proposition 2.22 in some spaces of analytic functions
near the origin in C2 with dependence on initial conditions and coefficients applied to the
auxiliary problem 5.66, 5.68. Using a continuity property of the Laplace transform 3.5,
we show that each function Xh,0 t, z, can be actually expressed as the Laplace transform of
Vh,∞ r, z, in the direction argλ and finally that Z0 itself is the Laplace transform of some
staircase distribution Vr, z, solving 5.80, 5.81.
On the other hand, in Lemma 5.7, under suitable conditions on ϕ1,j t, , 0 ≤ j ≤ S − 1,
we can also write X0,1 t, z, as a generalized Laplace transform in the direction argλ of
the staircase distribution mentioned above Vr, z, solving 5.80, 5.81. Therefore, the
equality X0,1 t, z, Z0 t, z, holds on T × D0, δ × E0 ∩ E1 . The method of proof
consists again in showing that X0,1 t, z, can be written as Laplace transform in a sequence
of directions ξn tending to argλ of a sequence of staircase distributions Vn r, z, which
are actually convergent series in z with coefficients that are C∞ functions in r on R with
exponential growth. Moreover, each distribution Vn r, z, solves an integro-differential
Cauchy problem 5.98, 5.99, whose coefficients tend to the coefficients of the integrodifferential equation 5.80. Under the assumption that the initial data 5.99 converge to
the initial data 5.81, we show that the sequence Vn r, z, converges to the solution of
5.80, 5.81 i.e., Vr, z, in the Banach space D σ, , δ, as n → ∞, see Lemma 5.6.
This convergence result is obtained again by using a majorazing series technique which
reduces the problem to the study of some linear differential equation 5.106, 5.109, whose
coefficients and initial data tend to zero as n → ∞. Finally, by continuity of the Laplace
transform, X0,1 t, z, can be written as the Laplace transform of Vr, z, in direction argλ.
After Theorem 5.8, we give an application to the construction of solutions to some
specific singular linear partial differential equations in C3 having logarithmic singularities at
the points λk/t, t, z, for k ∈ N \ {0}. We show that under the hypothesis that the coefficients
turns out to solve the linear partial
bs,k0 ,k1 are polynomials in , the Borel transform BXs
differential equation 5.149. We would like to mention that there exists a huge literature
on the study of complex singularities and analytic continuation of solutions to linear partial
differential equations starting from the fundamental contributions of Leray in 15. Several
authors have considered Cauchy problems ax, Dux 0, where ax, D is a differential
operator of some order m ≥ 1, for initial data ∂hx0 u|x0 0 wh , 0 ≤ h < m. Under specific
hypotheses on the symbol ax, ξ, precise descriptions of the solutions of these problems are
given near the singular locus of the initial data wh . For meromorphic initial data, we may
refer to 16–18 and for more general ramified multivalued initial data, we may cite 19–23.
The layout of this work is as follows.
In Section 2, we introduce Banach spaces of formal series whose coefficients belong
to spaces of staircase distributions and we study continuity properties for the actions of
6
Abstract and Applied Analysis
multiplication by C∞ functions and integro-differential operators on these spaces. In this
section, we also exhibit a Cauchy Kowalevski theorem for linear partial differential problems
in some space of analytic functions near the origin in C2 with dependence of their solutions
on the coefficients and initial data which will be useful to show the connection formula 5.28
stated in Section 5.
In Section 3, we recall the definition of a Laplace transform of a staircase distribution
as introduced in 8 and we give useful commutation formulas with respect to multiplication
by polynomials, exponential functions, and derivation.
In Section 4, we construct formal and analytic transseries solutions to the singularly
perturbed partial differential equation with irregular singularity 1.1.
In Section 5, we establish the crucial connection formula relying on the analytic
transseries solution Z0 t, z, and the solution X0,1 t, z, of 1.1. Finally, we state the
main result of the paper which asserts that the Borel transform BXs
in the perturbation
parameter of the formal solution Xt, z, of 1.1 can be analytically continued along any
path in the punctured sector Sκ0 ,κ1 \ ∪h≥1 {λh/t} and has logarithmic growth as s tends to λh/t
in a sector for all h ≥ 1.
2. Banach Spaces of Formal Series with Coefficients in Spaces of
Staircase Distributions: A Cauchy Problem in
Spaces of Analytic Functions
2.1. Weighted Banach Spaces of Distributions
We define DR to be the space of complex valued C∞ -functions with compact support in
R , where R is the set of the positive real numbers x > 0. We also denote by D R the space
of distributions on R . For f ∈ D R , we write f k the k-derivative of f in the sense of
distribution, for k ≥ 0, with the convention f 0 f.
Definition 2.1. A distribution f ∈ D R is called staircase if f can be written in the form
f
∞
k
,
Δk f
2.1
k0
for unique integrable functions Δk f ∈ L1 R such that the support suppΔk f of Δk f
is in k, k 1 for all k ≥ 0.
Remark 2.2. Given a compact set K ∈ R , a general distribution Λ ∈ D R can always be
written as a k-derivative of a continuous function on R restricted to the test functions with
support in K, where k depends on K, see 24.
β
Definition 2.3. Let σ > 0 be a real number, b > 1 an integer and let rb β n0 1/n 1b for
all integers β ≥ 0. Let E be an open sector centered at 0 and let ∈ E. We denote by Lβ,σ, the
vector space of all locally integrable functions f ∈ L1loc R such that
fr
:
β,σ,
∞
0
fτ exp − σ rb β τ dτ
||
2.2
Abstract and Applied Analysis
7
the vector space of staircase distributions f is finite. We denote by Dβ,σ,
such that
∞
f β,σ,,d
k0
∞
k
k0 Δk f
σ k
Δk f rb β
β,σ,
||
2.3
is finite.
, then f ∈ Dβ ,σ, for all β ≥ β and
Remark 2.4. Let , σ, β such that || < σrb β. If f ∈ Dβ,σ,
we have that h → ||f||h,σ,,d is a decreasing sequence on β, ∞. Likewise, if f ∈ Dβ,
, then
σ ,
f ∈ Dβ,σ, for all σ ≥ σ and we have that σ → ||f||β,σ,,d is a decreasing sequence on σ , ∞.
Let H be the Heaviside one-step function defined by Hr 1, if r ≥ 0 and Hr 0,
if r < 0. Let P the operator defined on distributions T ∈ D R by PT H ∗ T . For a subset
A ⊂ R, we denote by 1A the function which is equal to 1 on A and 0 elsewhere.
The proofs of the following Lemmas 2.5 and 2.6, Propositions 2.7, 2.8, 2.9, and
Corollary 2.10 are given in the appendix of 25, see also 8.
Lemma 2.5. Let k ≥ 0 and f F k ∈ D R , where F ∈ L1 R and suppF ⊂ k, ∞. Then f
is a staircase distribution and the decomposition of f has the following terms Δ0 Δ1 · · · Δk−1 0, Δk F1k,k1 and for n ≥ 1, Δkn Gn 1kn,kn1 where Gn PGn−1 1kn,∞ and G0 F.
Lemma 2.6. Let f be as in Lemma 2.5 and , σ, β such that || < σrb β. Then, one has
Δkn β,σ, ≤
−n
σ rb β
||
Fβ,σ, ,
2.4
nn−1
Fβ,σ, .
n − 1!
2.5
if n 0, 1, 2 and for n ≥ 3,
Δkn β,σ, ≤ e2−nσ/||rb β
Proposition 2.7. Let f ∈ Lβ,σ/2, and , σ, β such that || < σrb β/2. Then f belongs to Dβ,σ,
and the decomposition 2.1 of f has the following terms Δn Gn 1n,n1 with Gn PGn−1 1n,∞ and G0 f, for n ≥ 0. Moreover, there exists a universal constant C1 > 0 such that ||f||β,σ,,d ≤
C1 ||f||β,σ/2, .
Proposition 2.8. The set DR of C∞ -functions with compact support in R is dense in Dβ,σ,
for
all β ≥ 0, σ > 0 and ∈ E.
, we have f ∗ f ∈ Dβ,σ,
.
Proposition 2.9. Let , σ, β such that || < σrb β. For all f, f ∈ Dβ,σ,
Moreover, there exists a universal constant C2 > 0 such that
f ∗ f
β,σ,,d
for all f, f ∈ Dβ,σ,
.
≤ C2 f β,σ,,d f
β,σ,,d
2.6
8
Abstract and Applied Analysis
∗k
In this paper, for all integers k ≥ 1, we will denote ∂−k
r fr the convolution H ∗ f for
∗k
all f ∈ Dβ,σ, where H stands for the convolution product of H with itself k − 1 times for
k ≥ 2 and with the convention that H∗1 H. From Propositions 2.7 and 2.9, we deduce the
following.
,
Corollary 2.10. Let , σ, β be such that || < σrb β and let k ≥ 1 be an integer. For all f ∈ Dβ,σ,
one has ∂−k
r fr ∈ Dβ,σ, . Moreover, there exists a universal constant C3 > 0 such that
−k
∂r fr
≤ C3
β,σ,,d
||
σrb β
k
fr
β,σ,,d
2.7
for all f ∈ Dβ,σ,
.
In the next proposition, we study norm estimates for the multiplication by bounded
analytic functions.
Proposition 2.11. Let σ and β ≥ 0 such that
3 σ 1−σ/||rb β
rb β e
< 1,
2 ||
|| < σrb β ,
2.8
and let h be a C∞ -function on R such that there exist constants Ch > 0, μ > 0 and ρ > ||/σrb β
such that
q!
q h r ≤ Ch q1
ρ rμ
2.9
, we have hrfr ∈ Dβ,σ,
. Moreover, there exists a constant
for all r ∈ R . Then, for all f ∈ Dβ,σ,
C4 > 0 (depending on μ, ρ) such that
hrfr
β,σ,,d
≤ C4 Ch fr
β,σ,,d
2.10
.
for all f ∈ Dβ,σ,
Proof. The proof can be found in 14 and is inspired from 25, Lemma 2.9.1, but for the sake
of completeness, we sketch it below. Without loss of generality, we can assume that f has the
k
following form ft Δk t, where Δk ∈ L1 R with suppΔk ∈ k, k 1, for k ≥ 1. Put
k−j
tΔk t. Then, suppgk,j t ⊂ k, k 1.
gk,j t h
From the Leibniz formula, we get the identity
k
htΔk t j
k
k!
j
gk,j t.
j0 j! k − j !
2.11
On the other hand, one can rewrite gk,j t Pk−j gk,j k , where suppPk−j gk,j ∈ k, ∞
and Pq denotes the qth iteration of P.
Abstract and Applied Analysis
9
j
j
Due to Lemma 2.5, gk,j can be written gk,j ∞ l
lk Δl,j , with Δl,j Gl,j 1l,l1 , Gl,j PGl−1,j 1l,∞ and Gk,j Pk−j gk,j .
Therefore, we get the following identity
k
htΔk t htΔk tk k−1
k−1
∞
k!
k!
l .
k Δ
Δ
k,j
l,j
j!
k
−
j
!
j!
k
−
j
!
j0
j0
lk1
2.12
First of all, we have
htΔk tk β,σ,,d
≤
k ∞
σrb β
|htΔk t|e−σrb βt/|| dt
||
0
Ch
ρμ
2.13
k
σrb β
Δk tβ,σ, ,
||
where Ch > 0 is given in 2.9. From Lemma 2.6, we have the estimates
Δkn,j β,σ,
Δl,j β,σ,
≤
||
σrb β
n
k−j
gk,j P
β,σ,
,
2.14
l − kl−k−1 k−j
≤ e2−l−kσ/||rb β
gk,j ,
P
β,σ,
l − k − 1!
for n 0, 1, 2 and all l ≥ k 3. Now, we give estimates for ||Pk−j gk,j ||β,σ, .
Using the Taylor formula with integral remainder and the hypothesis 2.9, we get
t
k−j !
t − sk−j−1
k−j
gk,j t ≤ Ch |Δk s|ds.
P
k − j − 1 ! k ρ s μ 1k−j
2.15
Hence, from the Fubini theorem and the identity
∞
e
−σrb β/||t
t − s
k−j−1
dt e
−σrb β/||s
s
k−j !
||
σrb β
k−j
,
2.16
we deduce
∞
e−σrb β/||t P k−j gk,j tdt
k
≤ Ch k − j
k−j !
||
σrb β
k−j ∞
k
|Δk s|
e−σrb β/||s 1k−j ds
ρ sμ
2.17
10
Abstract and Applied Analysis
and hence
k−j
gk,j t
P
β,σ,
k−j
Ch k − j k − j !
||
≤ Δk sβ,σ, .
1k−j
σrb β
ρ kμ
2.18
From 2.14 and 2.18, we obtain
k!
kn ≤ Ch Ak
Δ
t
kn,j
β,σ,,d
j0 j! k − j !
k−1
k
σrb β
Δk sβ,σ, ,
||
2.19
for n 0, 1, 2, all k ≥ 1, where
k−j
k! k − j
||
.
Ak 1k−j
σrb β
j0 j! ρ k μ
k−1
2.20
Now, we need to estimate Ak . Due to the Stirling formula, k! ∼ kk e−k 2πk1/2 as k tends to
infinity, there exists a universal constant C4,1 > 0 such that
j
k−1 k − j
k μ σrb β /|| ρ
1
1/2 −k
Ak ≤ C4,1 2πk e
k
k kμ ρ kμ
j! σrb β /|| ρ
j0
kk
2.21
for all k ≥ 1. Using the hypothesis σrb βρ/|| ≥ 1, we have
j j
k−1
j k−1
k−2 k μ j
σrb β /|| ρ
kμ
kμ
kμ
k
.
−μ
k−j
k−j
k ≤
j!
j!
j!
k − 1!
σrb β /|| ρ
j0
j0
2.22
k−1
j0
Using again the Stirling formula, we get a constant C4,μ > 0 depending on μ such that
k−1
kμ
k
≤ C4,μ k1/2 ek
k − 1!
2.23
k−2 k μ j
∞ k μ j
μ
≤μ
μekμ .
j!
j!
j0
j0
2.24
for all k ≥ 1. Moreover,
Hence,
j j σrb β /|| ρ
kμ
1/2
μ
ek
≤
C
k
μe
k−j
4,μ
k
j!
σrb β /|| ρ
k−1
j0
2.25
Abstract and Applied Analysis
11
for all k ≥ 1. Finally, we obtain a constant C4,μ,ρ > 0 depending only on ρ, μ such that
Ak ≤ C4,μ,ρ
2.26
for all k ≥ 1. From 2.14 and 2.18, we have
∞ k!
l k
≤ Ch Ak A
Δl,j β,σ,,d
j0 j! k − j ! lk3
k−1
k
σrb β
Δk sβ,σ, ,
||
2.27
where
k A
∞
lk3
l−k
h
l−k−1
∞
σrb β
σrb β
hh−1
2−l−kσrb β/|| l − k
.
e
e2−hσrb β/||
l − k − 1!
h − 1!
||
||
h3
2.28
k , k ≥ 1, is a bounded sequence. Again, by the Stirling formula,
Now, we show that A
we get a universal constant C4,2 > 0 such that
h−1
∞ 1
σ σ σ h
h
exp h 1 − rb β
rb β
Ak ≤ C4,2 exp 2 rb β
h−1
||
||
||
2πh − 11/2
h3
∞ h
3σ/||rb β
σ σ exp 1 − rb β
≤ C4,2 exp 2 rb β
.
2
||
||
h3
2.29
From the assumption 2.8 and the estimates that for all m1 , m2 > 0 two real numbers, we
have
supx
m1
x≥0
exp−m2 x m1
m2
m1
e−m1 ,
2.30
we get a constant 0 < δ < 1 such that
3
k ≤ C4,2 e
A
1−δ
3
3σ/||rb β
36 C4,2
σ exp − rb β
≤ 3
2
||
2 1 − δ
2.31
for all k ≥ 1.
Finally, from the equality 2.12 and estimates 2.13, 2.19, 2.26, 2.27 and 2.31,
k
we get a constant C4,μ,ρ,1 > 0 depending only on μ, ρ such that ||htΔk t||β,σ,,d ≤
k
Ch C4,μ,ρ,1 ||Δk t||β,σ,,d for all k ≥ 1. It remains to consider the case k 0.
12
Abstract and Applied Analysis
When k 0, let ft Δ0 t ∈ L1 R , with suppΔ0 ∈ 0, 1. By definition, we can
write
htΔ0 tβ,σ,,d htΔ0 tβ,σ,
σrb β
Ch
Ch
t dt ≤
|ht||Δ0 t| exp −
Δ0 tβ,σ, Δ0 tβ,σ,,d .
ρμ
ρμ
||
0
2.32
1
In the next proposition, we study norm estimates for the multiplication by
polynomials.
Proposition 2.12. Let σ and β ≥ 0 such that
3 σ 1−σ/||rb β
rb β e
< 1,
2 ||
|| < σ
2.33
, one has r s1 fr ∈ Dβ,σ,
. Moreover, there
and let s1 , k2 ≥ 1 be integers. Then, for all f ∈ Dβ−k
2 ,σ,
exists a constant C5 > 0 (depending on s1 ,σ) such that
s
bs r 1 fr
≤ C5 ||s1 β 1 1 fr
β−k2 ,σ,,d
β,σ,,d
2.34
for all f ∈ Dβ−k
.
2 ,σ,
Proof. The proof is an adaptation of Proposition 2.11. Without loss of generality, we can
k
assume that f has the following form ft Δk t where Δk ∈ L1 R with suppΔk ∈
s1
k, k 1, for k ≥ 1. We also put ht t . Let gk,j t hk−j tΔk t. Then, suppgk,j t ⊂
k, k 1. From the Leibniz formula, we get the identity
k
htΔk t k
k!
j
gk,j t.
j!
k
−
j
!
j0
2.35
j
On the other hand, one can rewrite gk,j t Pk−j gk,j k , where suppPk−j gk,j ∈ k, ∞
and Pq denotes the qth iteration of P.
j
j
l
Due to Lemma 2.5, gk,j can be written gk,j ∞
lk Δl,j , with Δl,j Gl,j 1l,l1 , Gl,j PGl−1,j 1l,∞ and Gk,j Pk−j gk,j . Therefore, we get the following identity as:
k
htΔk t htΔk tk k−1
k−1
∞
k!
k!
l .
k Δ
Δ
k,j
l,j
j!
k
−
j
!
j!
k
−
j
!
j0
j0
lk1
2.36
Abstract and Applied Analysis
13
1 We first give estimates for ||htΔk tk ||β,σ,,d . We write
htΔk tk β,σ,,d
k ∞
σrb β
σ τ s1 |Δk τ| exp − rb β τ dτ
||
||
0
k k
rb β
σrb β − k2
||
rb β − k2
k1
σ τ s1 exp −
rb β − rb β − k2 τ
||
k
σ × |Δk τ| exp − rb β − k2 τ dτ
||
k k1
σrb β − k2
σ ≤ A , β
|Δk τ| exp − rb β − k2 τ dτ,
||
||
k
2.37
×
where
⎛
⎞
k
rb β
σ
A , β sup⎝
rb β − rb β − k2 k ⎠.
k 1s1 exp −
||
rb β − k2
k≥1
2.38
Now, we gives estimates for A, β. We write
k
rb β
σ rb β − rb β − k2 k
k 1s1 exp −
||
rb β − k2
σ s1
k 1 exp −k
ψ rb β − ψ rb β − k2
||
σ s1 s1
ψ rb β − ψ rb β − k2
,
≤ 2 k exp −k
||
2.39
where ψx x − ||/σ logx for all k ≥ 1. From the Taylor formula applied to ψ on rb β −
k2 , rb β, we get that
ψ rb β − ψ rb β − k2 ≥
1−
k2
|| ||
rb β − rb β − k2 ≥ 1 −
σ
σ β 1b
2.40
for all β ≥ k2 . Now, we recall that for all m1 , m2 > 0 two real numbers, we have
sup xm1 exp−m2 x x≥0
m1
m2
m1
e−m1 .
2.41
14
Abstract and Applied Analysis
From 2.39, 2.40 and 2.41, we deduce that
A , β ≤ 2
s1
s1 e−1
1 − ||/σk2 σ
s 1
bs
||s1 β 1 1
2.42
for all β ≥ k2 . From 2.37 and 2.42, we deduce that
htΔk tk ≤2
s1
β,σ,,d
s1 e−1
1 − ||/σk2 σ
s 1
bs ||s1 β 1 1 ft
β−k2 ,σ,,d .
2.43
l,j ||β,σ, for all 0 ≤ j ≤ k − 1 and all l ≥ k. From Lemma 2.6,
2 We give estimates for ||Δ
we have the estimates
Δkn,j β,σ,
Δl,j β,σ,
≤
||
σrb β
n
k−j
gk,j P
β,σ,
,
l − kl−k−1 k−j
≤ e2−l−kσ/||rb β
gk,j ,
P
β,σ,
l − k − 1!
2.44
for n 0, 1, 2 and all l ≥ k 3. Now, we give estimates for ||Pk−j gk,j ||β,σ, . Using the Taylor
formula with integral remainder, we have that
t
k−j
gk,j t ≤ P
1
k−j −1 !
t − sk−j−1 hk−j sΔk sds,
2.45
k
and from the classical identity
∞
s
k−j !
σ σ k−j−1
exp − rb β t t − s
dt exp − rb β s k−j
||
||
σ/||rb β
2.46
we get from the Fubini theorem that
k−j
gk,j t
P
β,σ,
∞ σ k−j
gk,j t exp − rb β t dt
P
||
k
∞ ∞
σ t − sk−j−1
≤
exp − rb β t dt hk−j sΔk sds
||
k−j −1 !
k
s
1
σ/||rb β
k−j
k−j
∞
k
σ k−j
exp − rb β s h
sΔk sds.
||
2.47
Abstract and Applied Analysis
15
Again, we write
∞
k
σ k−j
exp − rb β s h
sΔk sds
||
∞
σ σ k−j rb β − rb β − k2 s |Δk s| exp − rb β − k2 s ds.
s exp −
h
||
||
k
2.48
From the expression of h, we have that
s
s1 !ss1
k−j s1 !s 1
s ≤ k−j ≤ k−j
h
s
k
2.49
for all s ≥ k, if 1 ≤ k − j ≤ s1 , and hk−j s 0, if k − j > s1 . Using 2.49 in the right-hand side
of the equality 2.48, we deduce from 2.47 that
k−j
gk,j t
P
β,σ,
k−j k1
σ ||
rb β − rb β − k2 s
≤ s1 ! k−j
×
ss1 exp −
||
σrb β
k
k
σ × |Δk s| exp − rb β − k2 s ds
||
2.50
k−j
if 1 ≤ k − j ≤ s1 , and ||Pk−j gk,j t||β,σ, 0 if k − j > s1 .
kn
3 We give estimates for k−1
j0 k!||Δkn,j ||β,σ,,d /j!k − j!, for n 0, 1, 2. From the
estimates 2.44 and 2.50, we get that
k!
kn ≤
Δ
kn,j β,σ,,d
j0 j! k − j !
k−1
k−j
k−j
k!
||
s1 ! k−j
σrb β
k
j≥0,j≥k−s1 j! k − j !
k−1
k k1
σrb β
σ rb β − rb β − k2 s
×
ss1 exp −
||
||
k
σ × |Δk s| exp − rb β − k2 s ds.
||
2.51
From 2.37 and 2.42, we deduce from 2.51 that
k!
kn ≤ Ak 2s1
Δ
kn,j β,σ,,d
j!
k
−
j
!
j0
k−1
s1 e−1
1 − ||/σk2 σ
s 1
bs ||s1 β 1 1 ft
β−k2 ,σ,,d ,
2.52
16
Abstract and Applied Analysis
where
k−1
k!s1 !
Ak j!
k
−
j − 1 !kk−j
jk−s1 ,j≥0
||
σrb β
k−j
2.53
for all k ≥ 1, and n 0, 1, 2. Now, we show that Ak , k ≥ 1, is a bounded sequence. We have
k!
Ak ≤ s1 ! k
k
s −1
1
kk−s1 m
k − s1 m!s1 − m − 1!
m0
2.54
for all k ≥ s1 . From the Stirling formula which asserts that k! ∼ kk e−k 2πk1/2 as k → ∞,
we get a universal constant C1 > 0 and a constant C2 > 0 depending on s1 , m such that
k!
≤ C1 e−k 2πk1/2 ,
kk
k!ek
ek
kk−s1 m
≤ C1
≤
C
2
k − s1 m!
k − s1 m!2πk1/2 ks1 −m
2πk1/2
2.55
for all k ≥ 1. From 2.54, 2.55, we get a constant C3 > 0 depending on s1 such that
Ak ≤ C3
2.56
for all k ≥ 1.
∞
l
4 We give estimates for k−1
j0 k!/j!k − j!
lk3 ||Δl,j ||β,σ,,d . From the estimates
2.44 and 2.50, we get that
∞ k!
l ≤
Δl,j β,σ,,d
j0 j! k − j ! lk3
k−1
k−1
k!
s1 !
k−j
j!
k
−
j
−
1
!
k
j≥0,j≥k−s1
||
σrb β
k−j
k k1
σrb β
σ rb β − rb β − k2 s
×
ss1 exp −
||
||
k
σ × |Δk s| exp − rb β − k2 s ds
||
∞ σ l − kl−k−1
σ l−k
exp 2 − l − k rb β
.
rb β
×
l − k − 1!
||
||
lk3
2.57
Again from 2.37 and 2.42, we deduce from 2.57 that
∞ k!
l ≤ Bk 2s1
Δl,j β,σ,,d
j!
k
−
j
!
j0
lk3
k−1
s1 e−1
1 − ||/σk2 σ
s 1
bs ||s1 β 1 1 ft
β−k2 ,σ,,d ,
2.58
Abstract and Applied Analysis
17
k and
where Bk Ak A
k A
∞ σ
lk3
||
∞ σ
h3
||
rb β
σ l − kl−k−1
exp 2 − l − k rb β
l − k − 1!
||
l−k
h
rb β
h−1
σ h
exp 2 − h rb β
h − 1!
||
2.59
k is a bounded sequence.
for all k ≥ 1. Now, we remind from 2.31 that A
Finally, from 2.31, 2.36, 2.43, 2.52, 2.56, and 2.58, we deduce a constant C5 >
0 depending on s1 , σ such that
k
htΔk t
β,σ,,d
bs k ≤ C5 ||s1 β 1 1 Δk β−k2 ,σ,,d
,
2.60
which gives the result. It remains to consider the case k 0.
When k 0, let ft Δ0 t ∈ L1 R , with suppΔ0 ∈ 0, 1. By definition, we can
write
htΔ0 tβ,σ,,d htΔ0 tβ,σ,
1
0
σ σ τ s1 exp −
rb β − rb β − k2 τ |Δ0 τ| exp − rb β − k2 τ dτ.
||
||
2.61
Using 2.41, we deduce from 2.61 that
htΔ0 tβ,σ,,d ≤
s1 e−1
σk2
s1 e−1
σk2
s1
bs
||s1 β 1 1
s1
||
1
σ |Δ0 τ| exp − rb β − k2 τ dτ
||
0
bs β 1 1 ft
β−k2 ,σ,,d .
s1 2.62
Hence there exists a constant C5,1 > 0 depending on s1 , σ such that
bs htft
≤ C5,1 ||s1 β 1 1 ft
β−k2 ,σ,,d ,
β,σ,,d
2.63
which yields the result.
Proposition 2.13. Let σ > σ > 0 be real numbers such that
3 σ 1−σ/||rb β
rb β e
< 1,
2 ||
|| < σ .
2.64
18
Abstract and Applied Analysis
, one has r s1 fr ∈ Dβ,σ,
. Moreover, there
Let s1 ≥ 0 be a nonnegative integer. Then, for all f ∈ Dβ,
σ ,
exists a constant C6 > 0 (depending on s1 , σ, σ ) such that
s
r 1 fr
≤ C6 ||s1 fr
β,σ ,,d
β,σ,,d
2.65
for all f ∈ Dβ,σ,
.
Proof. The line of reasoning will follow the proof of Proposition 2.12. We start from the
identity 2.36.
1 We first give estimates for ||htΔk tk ||β,σ,,d . We write
htΔk tk β,σ,,d
k ∞
σrb β
σ s1
τ |Δk τ| exp − rb β τ dτ
||
||
0
k k k1
σ rb β
σ
σ − σ s1
rb β τ
τ exp −
σ
||
||
k
σ × |Δk τ| exp − rb β τ dτ
||
k k1
σ rb β
σ ≤ A , β
|Δk τ| exp − rb β τ dτ,
||
||
k
2.66
where
k
σ
σ − σ s1
.
rb β k
k 1 exp −
σ
||
k≥1
, β sup
A
2.67
β. We write
Now, we give estimates for A,
k
rb β σ
σ − σ s1
s1
rb β k k 1 exp −k
ϕσ − ϕ
σ
k 1 exp −
σ
||
||
rb β s1 s1
≤ 2 k exp −k
ϕσ − ϕ
σ ,
||
2.68
where ϕx x − ||/rb β logx, for all k ≥ 1. From the Taylor formula applied to ϕ on
σ , σ, we get that
ϕσ − ϕ
σ ≥
||
1−
σ − σ .
σ
2.69
Abstract and Applied Analysis
19
From 2.68, 2.69, and 2.41, we deduce that
, β ≤ 2
A
s1
s1 e−1
σ σ − σ 1 − ||/
s 1
||s1 .
2.70
From 2.66 and 2.70, we get that
htΔk tk β,σ,,d
≤2
s1
s1 e−1
σ σ − σ 1 − ||/
s 1
||s1 ft
β,σ ,,d .
2.71
l,j ||β,σ, , for all 0 ≤ j ≤ k − 1, all l ≥ k. We start from the
2 We give estimates for ||Δ
formula 2.44 and 2.47. We write
∞
k
σ k−j
exp − rb β s h
sΔk sds
||
∞
σ σ − σ k−j rb β s |Δk s| exp − rb β s ds.
s exp −
h
||
||
k
2.72
We get that
k−j
gk,j t
P
β,σ,
≤ s1 !
k−j
kk−j
||
σrb β
k−j k1
k
σ σ − σ rb β s |Δk s| exp − rb β s ds
ss1 exp −
||
||
2.73
if 1 ≤ k − j ≤ s1 , and ||Pk−j gk,j t||β,σ, 0 if k − j > s1 .
kn
3 We give estimates for k−1
j0 k!||Δkn,j ||β,σ,,d /j!k − j!, for n 0, 1, 2. From the
estimates 2.44 and 2.73, we get that
k!
kn ≤
Δ
kn,j β,σ,,d
j0 j! k − j !
k−1
k−j
k−j
k!
||
s1 ! k−j
σrb β
k
j≥0,j≥k−s1 j! k − j !
k−1
k k1
σrb β
σ − σ rb β s
×
ss1 exp −
||
||
k
σ × |Δk s| exp − rb β s ds.
||
2.74
20
Abstract and Applied Analysis
From 2.66 and 2.70, we deduce from 2.74 that
k!
kn ≤ Ak 2s1
Δ
kn,j β,σ,,d
j!
k
−
j
!
j0
k−1
s1 e−1
σ σ − σ 1 − ||/
s 1
||s1 ft
β,σ ,,d ,
2.75
where Ak is the bounded sequence given in the proof of Proposition 2.12.
∞
l
We give estimates for k−1
j0 k!/j!k − j!
lk3 ||Δl,j ||β,σ,,d . From the estimates 2.44
and 2.73, we get that
∞ k!
l ≤
Δl,j β,σ,,d
j0 j! k − j ! lk3
k−1
k−1
s1 !
k!
k−j
j≥0,j≥k−s1 j! k − j − 1 ! k
||
σrb β
k−j
k k1
σrb β
σ − σ ×
ss1 exp −
rb β s
||
||
k
σ × |Δk s| exp − rb β s ds
||
∞
σ l − kl−k−1
σ l−k
.
exp 2 − l − k rb β
rb β
×
l − k − 1!
||
||
lk3
2.76
From 2.66 and 2.70, we deduce from 2.76, that
∞ k!
l ≤ Bk 2s1
Δl,j β,σ,,d
j0 j! k − j ! lk3
k−1
s1 e−1
σ σ − σ 1 − ||/
s 1
||s1 ft
β,σ ,,d ,
2.77
where Bk is the bounded sequence given in the proof of Proposition 2.12.
Finally, from 2.31, 2.36, 2.56, 2.71, 2.75, and 2.77, we deduce a constant C6 >
0 depending on s1 , σ, σ such that
k
htΔk t
β,σ,d
k ≤ C6 ||s1 Δk β,
σ ,,d
,
2.78
which gives the result. It remains to consider the case k 0.
When k 0, let ft Δ0 t ∈ L1 R , with suppΔ0 ∈ 0, 1. By definition, we can
write
htΔ0 tβ,σ,,d
htΔ0 tβ,σ, 1
0
σ σ − σ rb β τ |Δ0 τ| exp − rb β τ dτ.
τ s1 exp −
||
||
2.79
Abstract and Applied Analysis
21
Using 2.41, we deduce from 2.79 that
htΔ0 tβ,σ,,d ≤
s1 e−1
σ − σ
s1 e−1
σ − σ
s 1
s1
||
s 1
1
σ |Δ0 τ| exp − rb β τ dτ
||
0
|| ft
β,σ ,,d .
2.80
s1 Hence, there exists a constant C6,1 > 0 depending on s1 , σ, σ such that
htft
≤ C6,1 ||s1 ft
β,σ ,,d ,
β,σ,,d
2.81
which yields the result.
2.2. Banach Spaces of Formal Power Series with Coefficients in
Spaces of Distributions
Definition 2.14. Let δ > 0 be a real number. We denote by D σ, , δ the vector space of formal
for all β ≥ 0 and
series vr, z β≥0 vβ rzβ /β! such that vβ r ∈ Dβ,σ,
vr, zσ,,d,δ :
δβ
vβ r
β,σ,,d β!
β≥0
2.82
is finite. One can check that the normed space D σ, , δ, || · ||σ,,d,δ is a Banach space.
In the next proposition, we study some parameter depending linear operators acting
on the space D σ, , δ.
Proposition 2.15. Let s1 , s2 , k1 , k2 ≥ 0 be positive integers. Assume that the condition
k2 ≥ bs1
2.83
holds. Then, if
|| < σ,
3σ/||ζb 1−σ/||
e
< 1,
2
2.84
1 −k2
the operator τ s1 ∂−k
τ ∂z is a bounded linear operator from the space D σ, , δ, ||·||σ,,d,δ into itself.
Moreover, there exists a constant C7 > 0 (depending on b, s1 , k2 , σ) such that
s1 −k1 −k2
r ∂r ∂z vr, z
for all v ∈ D σ, , δ.
σ,,d,δ
≤ ||s1 k1 C7 δk2 vr, zσ,,d,δ
2.85
22
Abstract and Applied Analysis
Proof. Let vr, z ∈ D σ, , δ. By definition, we have
s1 −k1 −k2
r ∂r ∂z vr, z
σ,,d,δ
s1 −k1
r ∂r vβ−k2 r
δβ
.
β,σ,,d β!
β≥k2
2.86
From Corollary 2.10 and Proposition 2.12, we get a constant C3,5 > 0 depending on s1 ,σ
such that
s1 −k1 −k2
r ∂r ∂z vr, z
σ,,d,δ
≤ C3,5
bs1 β − k2 !
β1
β!
s1 k1 ||
β≥k2
δβ−k2
× vβ−k2 r
β−k2 ,σ,,d δk2 .
β − k2 !
2.87
From the assumptions 2.83, we get a constant Cb,s1 ,k2 > 0 depending on b, s1 , k2 such that
bs1 β − k2 !
β1
≤ Cb,s1 ,k2
β!
2.88
for all β ≥ k2 . Finally, from the estimates 2.87 and 2.88, we get the inequality 2.85.
In the next proposition, we study linear operators of multiplication by bounded
holomorphic and C∞ functions.
Proposition 2.16. For all β ≥ 0, let hβ τ be a C∞ function with respect to r on R such that there
exist A, B, ρ, μ > 0 with
β!q!
q hβ r ≤ AB−β q1
ρ rμ
2.89
for all r ∈ R . One consider the series
hr, z β≥0
hβ r
zβ
,
β!
2.90
which is convergent for all |z| < B, all r ∈ R . Let 0 < δ < B. Then, if
|| < σ,
|| < ρσ,
3σ/||ζb 1−σ/||
e
< 1,
2
2.91
Abstract and Applied Analysis
23
the linear operator of multiplication by hr, z is continuous from D σ, , δ, || · ||σ,,δ into itself.
Moreover, there exists a constant C8 (depending on μ, ρ, B) such that
hr, zvr, zσ,,d,δ ≤ C8 Avr, zσ,,d,δ
2.92
for all vr, z ∈ D σ, , δ satisfying 2.91.
Proof. Let vr, z β≥0
vβ rzβ /β! ∈ D σ, , δ. By definition, we have that
hτ, zvr, zσ,,d,δ
⎛
⎞
β
β!
hβ rvβ r
⎝
⎠δ .
≤
1
2
β,σ,,d β !β !
β!
1 2
β≥0
β β β
1
2.93
2
From Proposition 2.11 and Remark 2.4, we deduce that there exists C4 > 0 depending on
μ, ρ such that
hβ rvβ r
≤ C4 AB−β1 β1 !
vβ2 r
β,σ,,d ≤ C4 AB−β1 β1 !
vβ2 r
β2 ,σ,,d
1
2
β,σ,,d
2.94
for all β1 , β2 ≥ 0 such that β1 β2 β. From 2.93 and 2.94, we deduce that
⎛
hr, zvr, zσ,,d,δ ≤ C4 A⎝
δ β
β≥0
B
⎞
⎠vr, z
σ,,d,δ
2.95
which yields 2.92.
2.3. Cauchy Problems in Analytic Functions Spaces with
Dependence on Initial Data
In this section, we recall the well-know-Cauchy Kowalevski theorem in some spaces of
analytic functions for which the dependence on the coefficients and initial data can be
obtained.
The following Banach spaces were used in 26.
Definition 2.17. Let T, X be real numbers such that T, X > 0. We define a vector space GT, X
of holomorphic functions on a neighborhood of the origin in C2 . A formal series Ut, x ∈
Ct, x,
Ut, x ul,β
l,β≥0
tl x β
l! β!
2.96
belongs to GT, X if the series
ul,β T l Xβ
l
β
!
l,β≥0
2.97
24
Abstract and Applied Analysis
converges. We also define a norm on GT, X as
Ut, xT,X
ul,β T l Xβ .
l
β
!
l,β≥0
2.98
One can easily show that GT, X, || · ||T,X is a Banach space.
Remark 2.18. Let Ut, x be in GT0 , X0 for given T0 , X0 > 0. Then, Ut, x also belongs to
the spaces GT, X for all T ≤ T0 and X ≤ X0 . Moreover, the maps T → ||Ut, x||T,X and
X → ||Ut, x||T,X are increasing functions from 0, T0 resp., 0, X0 into R .
We depart from some preliminary lemma from 26. In the following, for ut, x ∈
x
Ct, x, we denote by ∂−1
x ut, x the formal series 0 ut, τdτ.
1
Lemma 2.19. Let h0 , h1 ∈ N such that h0 ≤ h1 . The operator ∂ht 0 ∂−h
is a bounded linear operator
x
from GT, X, || · ||T,X into itself. Moreover, there exists a universal constant C10 > 0 such that the
estimates
h0 −h1
∂t ∂x Ut, x
T,X
≤ C10 T −h0 X h1 Ut, xT,X
2.99
hold for all Ut, x ∈ GT, X.
l β
Lemma 2.20. Let At, x l,β≥0 al,β t x /l!β! be an analytic function on an open polydisc
containing D0, T × D0, X and let Ut, x be in GT, X. Then, the product At, xUt, x belongs
to GT, X. Moreover,
At, xUt, xT,X ≤ |A|T, XUt, xT,X
where |A|T, X l,β≥0
2.100
|al,β |T l X β /l!β!.
Proof. Let
Ut, x tl x β
.
l! β!
ul,β
l,β≥0
2.101
We have
At, xUt, x vl,β
l,β≥0
tl x β
,
l! β!
2.102
where
vl,β al1 ,β1 ul2 ,β2
β!l!
l1 !β1 ! l2 !β2 !
l l l β β β
1
2
1
2
2.103
Abstract and Applied Analysis
25
for all l, β ≥ 0. By definition, we have
|A|T, XU2 t, xT,X
At, xUt, xT,X
⎛
⎞
al ,β ul ,β 1 1
2 2
⎝
⎠T l X β ,
l
!β
!
l
β
2
2 !
l,β≥0
l1 l2 l β1 β2 β 1 1
al1 ,β1 ul2 ,β2 l!β! T l X β
l β! .
l
!β
!l
!β
!
1
1
2
2
l,β≥0 l1 l2 l β1 β2 β
2.104
On the other side, the next inequalities are well known:
lβ !
lβ !
l!β!
≤
≤
l1 !β1 !l2 !β2 !
l1 β1 ! l2 β2 ! l1 !β1 ! l2 β2 !
2.105
for all l1 , l2 ≥ 0 such that l1 l2 l and β1 , β2 ≥ 0 such that β1 β2 β.
Finally, from 2.105, we deduce that ||At, xUt, x||T,X converges and that the
estimates 2.100 hold.
Lemma 2.21. Let h1 , h2 ∈ N and let Ut, x be in GT0 , X0 for given T0 , X0 > 0. Then, there exist
T, X > 0 small enough (depending on T0 , X0 ) such that the formal series ∂ht 1 ∂hx2 Ut, x belongs to
GT, X. Moreover, there exists a constant C11 > 0 (depending on h1 , h2 ) such that
h1 h2
∂t ∂x Ut, x
T,X
≤ C11 T −h1 X −h2 Ut, xT0 ,X0 2.106
for all Ut, x ∈ GT0 , X0 .
Let C1 be a finite subset of N2 . For all l0 , l1 ∈ C1 , let cl0 ,l1 t, x l,β≥0 cl0 ,l1 ,l,β t
l β
x /l!β!
be analytic functions on some polydisc containing the closed polydisc D0, T0 × D0, X0 for
some T0 , X0 > 0. As in Lemma 2.20, we define
|cl0 ,l1 |t, x cl0 ,l1 ,l,β tl xβ
l,β≥0
l!β!
,
2.107
which converges on D0, T0 ×D0, X0 . We also consider dt, x ∈ GTd , Xd , for some Td , Xd >
0. The following proposition holds.
Proposition 2.22. Let S ≥ 1 be an integer. One make the following assumptions. For all l0 , l1 ∈ C1 ,
one has
S > l1 ,
S ≥ l0 l1 .
2.108
One consider the following Cauchy problem:
∂Sx Ut, x l0 ,l1 ∈C1
cl0 ,l1 t, x∂lt0 ∂lx1 Ut, x dt, x
2.109
26
Abstract and Applied Analysis
for the given initial conditions
j
∂x U t, 0 Uj t,
0 ≤ j ≤ S − 1,
2.110
which are analytic functions on some disc containing the closed discD0, T0 . If Uj t l
l
l≥0 Uj,l t /l!, we define |Uj |t l≥0 |Uj,l |t /l!, which converges for all t ∈ D0, T0 .
Then, there exist T1 > 0 with 0 < T1 < minT0 , Td (depending on T0 ,Td ,C1 ) and X1 > 0 with
0 < X1 < minX0 , Xd (depending on S, T0 , C1 , maxl0 ,l1 ∈C1 |cl0 ,l1 |T0 , X0 ) such that the problem
2.109, 2.110 has a unique formal solution Ut, x ∈ GT1 , X1 . Moreover, there exist constants
C12,1 , C12,2 , C12,3 > 0 (depending on S,T0 ,X0 ,C1 ) such that
Ut, xT1 ,X1 ≤ max Uj T0 C12,1 max |cl0 ,l1 |T0 , X0 C12,2 C12,3 dt, xTd ,Xd .
l0 ,l1 ∈C1
0≤j≤S−1
2.111
Proof. We denote by P the linear operator from Ct, x into itself defined by
PHt, x : ∂Sx Ht, x −
l0 ,l1 ∈C1
cl0 ,l1 t, x∂lt0 ∂lx1 Ht, x,
2.112
and A denotes the linear map from Ct, x into itself:
AHt, x :
l0 ,l1 ∈C1
cl0 ,l1 t, x∂lt0 ∂lx1 −S Ht, x
2.113
for all Ht, x ∈ Ct, x. By construction, we have that P ◦ ∂−S
x id − A, where id represents
the identity map H → H from Ct, x into itself.
Now, we show that for any given T1 > 0 such that 0 < T1 ≤ T0 , there exists XA,T1 > 0
with 0 < XA,T1 ≤ X0 depending on S, T1 , C1 , maxl0 ,l1 ∈C1 |cl0 ,l1 |T0 , X0 such that id − A is
an invertible map from GT1 , X into itself for all 0 < X ≤ XA,T1 . Moreover, the following
inequality
id − A−1 Ct, x
T1 ,X
≤ 2Ct, xT1 ,X
2.114
holds for all Ct, x ∈ GT1 , X, for any 0 < X ≤ XA,T1 . Indeed, from the assumption 2.108
and Lemmas 2.19 and 2.20, we get a universal constant C10,1 > 0 such that
⎛
ACt, xT1 ,X ≤ C10,1 ⎝
l0 ,l1 ∈C1
⎞
|cl0 ,l1 |T1 , XT1−l0 X S−l1 ⎠Ct, xT1 ,X
⎛
≤ C10,1 max |cl0 ,l1 |T0 , X0 ⎝
l0 ,l1 ∈C1
: NT1 ,XA,T1 Ct, xT1 ,X
l0 ,l1 ∈C1
⎞
S−l1 ⎠
T1−l0 XA,T
Ct, xT1 ,X
1
2.115
Abstract and Applied Analysis
27
for all Ct, x ∈ GT1 , X. Since S > l1 , for all l0 , l1 ∈ C1 , for the given T1 > 0 one can choose
XA,T1 small enough such that NT1 ,XA,T1 ≤ 1/2. Therefore, the inequality 2.114 holds.
j
Let wt, x S−1
j0 Uj tx /j!. From the hypothesis 2.110, we deduce that Pwt, x
and wt, x belong to GT1 , X0 , for some 0 < T1 < T0 depending on C1 , T0 . Indeed, from
Lemmas 2.20 and 2.21 we get constants C11,1 > 0, 0 < T1 < T0 depending on C1 , T0 such that
Pwt, xT1 ,X0 ≤
⎛
|cl0 ,l1 |T1 , X0 ⎝
l0 ,l1 ∈C1
≤ C11,1
l0
∂t Ujl1 t
j0
l0 ,l1 ∈C1
≤ C11,1
⎞
j
X0
⎠
T1 ,X0 j!
S−1−l
1
l0 ,l1 ∈C1
⎛
|cl0 ,l1 |T1 , X0 T1−l0 ⎝
S−1−l
1
Ujl
1
j0
⎛
|cl0 ,l1 |T1 , X0 T1−l0 ⎝
⎞
j
X0
⎠
T0 ,X0 j!
t
⎞
j
X0
Ujl T0 ⎠
1
j!
S−1−l
1
j0
2.116
≤ C11,1 max |cl0 ,l1 |T0 , X0 max Uj T0 l0 ,l1 ∈C1
×
l0 ,l1 ∈C1
wt, xT1 ,X0 0≤j≤S−1
⎛
T1−l0 ⎝
⎞
S−1−l
1 X0j
⎠
j0
j!
,
j
j
S−1
S−1
X0
X0 Uj t T1 ,X0 Uj T1 ≤
≤
j!
j!
j0
j0
S−1 X j
0
.
≤ max Uj T0 0≤j≤S−1
j!
j0
2.117
Now, for this constructed T1 > 0 satisfying 2.116 that we choose in such a way that T1 < Td
also holds, we select X1 > 0 such that 0 < X1 < minXA,T1 , Xd . From the estimates 2.116 and
Remark 2.4, we deduce that Pwt, x, wt, x, and dt, x belong to GT1 , X1 . From 2.114,
we deduce the existence of a unique Ht, x ∈ GT1 , X1 such that
P ◦ ∂−S
Ht, x −Pwt, x dt, x.
x
2.118
Now, we put Ut, x ∂−S
x Ht, x wt, x. By Lemma 2.19, we deduce that Ut, x ∈
GT1 , X1 and solves the problem 2.109, 2.110. Moreover, from 2.114 and 2.116, we
get constants C12,1 , C12,2 , C12,3 > 0 depending on S, T0 , X0 , C1 such that 2.111 holds, which
yields the result.
3. Laplace Transform on the Spaces D σ, , δ
We first introduce the definition of Laplace transform of a staircase distribution.
28
Abstract and Applied Analysis
Proposition 3.1. 1 Let β ≥ 0 be an integer, σ > 0 a real number, and ∈ E. Let
fr ∞
Δk rk ∈ Dβ,σ,
3.1
k0
and choose θ ∈ −π, π. Then, there exist ρθ > 0, ρ > 0 such that the function
k1 ∞
∞
eiθ
reiθ
Lθ f t dr
Δk f r exp −
t
t
0
k0
3.2
is holomorphic on the sector Sθ,ρθ ,||ρ {t ∈ C∗ /|θ − argt| < ρθ , |t| < ||ρ} for all ∈ E. Moreover,
for all compacts K ⊂ Sθ,ρθ ,||ρ , there exists CK > 0 (depending on K and σ) such that
Lθ f t ≤ CK f β,σ,,d
3.3
for all t ∈ K.
2 Let δ > 0 and let fr, z β≥0 fβ rzβ /β! ∈ D σ, , δ. We define the Laplace transform
of fr, z in direction θ ∈ −π, π to be the function
Lθ fβ tzβ
,
Lθ fr, z t β!
β≥0
3.4
which defines a holomorphic function on Sθ,ρθ ,||ρ × D0, δ, for some ρθ > 0, ρ > 0, for all ∈ E.
Moreover, for all compacts K ⊂ Sθ,ρθ ,||ρ , there exists CK > 0 (depending on K and σ) such that
Lθ fr, z t ≤ CK fr, z
σ,,d,δ
3.5
for all t, z ∈ K × D0, δ.
Proof. We prove part 1. The second part 2 is a direct application of 1. We have
∞
∞
σrb β
1
Lθ f t ≤
Δk f r exp −
r
k1
||
0
k0 |t|
cos θ − argt
σ dr.
− rb β
× exp −r
|t|
||
3.6
We choose δ1 > 0 and ρθ > 0 such that cosθ − argt > δ1 for all t ∈ Sθ,ρθ ,||ρ . Moreover, we
choose 0 < δ2 < δ1 and ρ > 0 such that
|t| < ||
δ 1 − δ2
,
σrb β
||e−δ2 /|t|
<1
|t|σrb β
3.7
Abstract and Applied Analysis
29
for all t ∈ Sθ,ρθ ,||ρ . Let k ≥ 0 an integer, for r ∈ k, k 1, we get that
exp −r
cos θ − argt
kδ2
σ ≤ exp −
− rb β
.
|t|
||
|t|
3.8
We deduce that for k 0,
1
|t|
∞
0
σr β
cos θ − argt
σ Δ0 f r exp − b
r × exp −r
− rb β
dr
||
|t|
||
1
≤ Δ0 fr
β,σ,
|t|
3.9
and for k ≥ 1,
1
|t|k1
∞
0
1
≤
|t|
σrb β
cos θ − argt
σ
Δk f r exp −
dr
r × exp −r
− rb β
||
|t|
||
||e−δ2 /|t|
|t|σrb β
k σ rb β
||
k
Δk fr
3.10
β,σ,
||e−δ2 /|t| σ k Δk f r
≤ 2
.
rb β
β,σ,
||
|t| σrb β
From the estimates 3.9 and 3.10 we get the inequality 3.3.
In the next proposition, we show that if f is a function, then the Laplace transform of
f introduced in Proposition 3.1 coincides with the classical one.
. The
Proposition 3.2. Let fr ∈ Lβ,σ/2, . Then, from Proposition 2.7, one knows that f ∈ Dβ,σ,
Laplace transform Lθ ft coincides with the classical Laplace transform of f in the direction θ defined
by
eiθ
Tθ f t t
∞
0
reiθ
fr exp −
t
dr
3.11
for all t ∈ Sθ,ρθ ,||ρ .
k
Proof. From Proposition 2.7, the staircase decomposition of f has the
k≥0 Δk f
following form Δk r Gk r1k,k1 , with Gk PGk−1 1k,∞ and G0 r fr for all k ≥ 0.
We have to compute the integrals
Ak eiθ
k1 k1
tk1
k
reiθ
Δk r exp −
t
dr
3.12
30
Abstract and Applied Analysis
for all k ≥ 0. For k 0, we have that
eiθ
A0 t
1
0
reiθ
fr exp −
t
3.13
dr.
For k 1, by one integration by parts, we get that
2
eiθ
eiθ 2
reiθ
reiθ
G1 r exp −
dr,
fr exp −
A1 −
t
t
t 1
t
3.14
1
and using successive integrations by parts, we get that
Ak k
−
m1
eiθ
t
m reiθ
Gm r exp−
t
k1
k
eiθ
t
k1
k
reiθ
fr exp −
t
dr
3.15
for all k ≥ 1. On the other hand, from the hypothesis that fr ∈ Lβ,σ/2, and from the fact that
Gm r 0 for all r ≤ m, we have that the next telescopic sum
∞
k1
−
eiθ
t
m reiθ
Gm r exp−
t
k1
3.16
k
is convergent and equal to zero for all m ≥ 1. Finally, we deduce that
k≥0
Ak Tθ ft.
In the next proposition, we describe the action of multiplication by a polynomial and
derivation on the Laplace transform.
. Then, the following relations
Proposition 3.3. Let fr ∈ Dβ,σ,
Lθ eiθ ∂−1
f
t tLθ f t,
r
Lθ eiθ rfr t t2 ∂t t Lθ f t
3.17
hold for all t ∈ Sθ,ρθ ,||ρ . Let s, k0 ≥ 0 be two integers such that s ≥ 2k0 . Then, there exist a finite subset
0
Os,k0 ⊂ N2 such that for all q, p ∈ Os,k0 , q p s − k0 and integers αs,k
q,p ∈ Z, for q, p ∈ Os,k0
(depending on s, k0 ) such that
⎛
ts ∂kt 0 Lθ f t Lθ ⎝eis−k0 θ
⎞
−p
0 q
⎠t
αs,k
q,p r ∂r fr
3.18
q,p∈Os,k0
for all t ∈ Sθ,ρθ ,||ρ .
Proof. First of all, we have to check that the relations 3.17 and 3.18 hold when f ∈ DR .
, || · ||β,σ,,d , from the inequality 3.3 and with the help of
Since DR is dense in Dβ,σ,
.
Corollary 2.10 and Proposition 2.12, we will get that 3.17 and 3.18 hold for all f ∈ Dβ,σ,
Abstract and Applied Analysis
31
Now, let f ∈ DR . The first relation of 3.17 is obtained by integrating once by parts and
the second formula of 3.17 is a consequence of the equality
∂t
eiθ
t
∞
0
eiθ
− 2
t
reiθ
fr exp −
t
∞
dr
reiθ
fr exp −
t
0
e2iθ
dr 3
t
∞
0
reiθ
rfr exp −
t
3.19
dr
for all t ∈ Sθ,ρθ ,||ρ . To get the formula 3.18, we first show the following relation:
∂t Lθ fr t Lθ e−iθ r∂2r ∂r fr t
3.20
for all t ∈ Sθ,ρθ ,||ρ . Indeed, using one integration by parts, we get that
Lθ e
−iθ
r∂2r
eiθ ∞
reiθ
dr.
∂r fr t 2
∂r frr exp −
t
t 0
3.21
By a second integration by parts on the right-hand side of 3.21 and by comparison with
3.19, we get 3.20. Now, let s, k0 ∈ N be such that s ≥ 2k0 . Applying the first relation of
3.17 and 3.20, we get that
k0 2
r∂
∂
fr
ts ∂kt 0 Lθ f t Lθ eis−k0 θ ∂−s
t.
r
r
r
3.22
Now, we recall a variant of Lemmas 5 and 6 in 2.
Lemma 3.4. For all k0 ≥ 1, there exist constants ak,k0 ∈ N, k0 ≤ k ≤ 2k0 such that
r∂2r ∂r
k 0
ur 2k0
ak,k0 r k−k0 ∂kr ur
3.23
kk0
for all C∞ functions u : R → C.
Lemma 3.5. Let a, b, c ≥ 0 be positive integers such that a ≥ b and a ≥ c. We put δ a b − c.
b c
Then, for all C∞ function u : R → C, the function ∂−a
r r ∂r ur can be written in the form
r b ∂cr ur ∂−a
r
b ,c ∈Oδ
αb ,c r b ∂cr ur,
3.24
where Oδ is a finite subset of Z2 such that for all b , c ∈ Oδ , b −c δ, b ≥ 0, c ≤ 0, and αb ,c ∈ Z.
Finally, we observe that the relation 3.18 follows from 3.22 and Lemmas 3.4 and
3.5.
32
Abstract and Applied Analysis
The next proposition can be found in 25, Appendix A, see also 8.
with || < σrb β. Then, for every l ≥ 0, the
Proposition 3.6. Let α ≥ 1 and fr ∈ Dβ,σ,
. Moreover, there exist a universal constant A > 0
expression ft − αl1αl,∞ l belongs to Dβ,σ,
and Bσ, b, > 0 (depending on σ, b, ) such that
l ft − αl1αl,∞ β,σ,,d
≤ ABσ, b, l fr
β,σ,,d ,
3.25
with Bσ, b, → 0 when → 0.
In the forthcoming proposition, we explain the action of multiplication by an
exponential function on the Laplace transform.
Proposition 3.7. Let α ≥ 1 and fr ∈ Dβ,σ,
with || < σrb β. From the latter proposition, one
. The following formula
knows that Fl r fr − αl1αl,∞ l belongs to Dβ,σ,
Lθ Fl t eiθ
t
l
αleiθ
exp −
t
Lθ f t
3.26
holds for all t ∈ Sθ,ρθ ,||ρ .
Proof. Since DR is dense in Dβ,σ,
, it is sufficient to prove that
Lθ Fl t eiθ
t
l
αleiθ
exp −
t
Tθ f t
3.27
for all f ∈ DR , all t ∈ Sθ,ρθ ,||ρ . Then, we get the inequality 3.26 by using 3.3 and
Proposition 3.6. Now, let f ∈ DR . We write
fτ − αl1αl,∞
l
lr
∂−r
,
τ fτ − αl1αl,∞
3.28
where r ≥ 0 is an integer chosen such that αl ∈ l r, l r 1. From our assumption, we
have that τ → fτ − αl1αl,∞ belongs to L1 R and that suppfτ − αl1αl,∞ ⊂ l r, ∞.
h τ
By Lemma 2.5, we deduce that fτ − αl1αl,∞ lr is a staircase distribution h≥0 Δ
h,l
h,l τ are constructed as follows:
where the functions Δ
j,l τ 0,
Δ
for 0 ≤ j ≤ l r − 1,
lr,l τ fτ − αl1αl,∞ 1lr,lr1 ,
Δ
3.29
lrn,l τ Gn τ1lrn,lrn1 where
and for all n ≥ 1, we have Δ
Gn τ ∂−1
τ Gn−1 τ1lrn,∞ ,
G0 fτ − αl1αl,∞ .
3.30
Abstract and Applied Analysis
33
By definition, we have
Lθ
fτ − αl1αl,∞
lr h1 ∞
∞
eiθ
t t
h0
iθ
h,l τ exp − τe
Δ
t
0
dτ.
3.31
∞
h,l τ exp−τeiθ /tdτ for all h ≥ 0.
Now, we will compute the integrals Ah,l eiθ /th1 0 Δ
By construction, we have that Ah,l 0 for all 0 ≤ h ≤ l r − 1. For h l r, we get
Alr,l eiθ
t
eiθ
t
lr1 lr1
τeiθ
fτ − αl exp −
t
αl
lr1
αleiθ
exp −
t
1−αlr1
0
dτ
seiθ
fs exp −
t
3.32
ds.
For h l r 1, by one integration by parts, we get that
Alr1,l
⎡ ⎤lr2
lr1
iθ
iθ
τe
e
G1 τ⎦
⎣−
exp −
t
t
lr1
eiθ
t
lr1 lr2
lr1
τeiθ
fτ − αl exp −
t
dτ
3.33
⎡ ⎤lr2
lr1
iθ
iθ
τe
e
⎣−
G1 τ⎦
exp −
t
t
lr1
eiθ
t
lr1
αleiθ
exp −
t
1−αlr2
1−αlr1
seiθ
fs exp −
t
ds.
For h l r n, with n ≥ 1, by successive integrations by parts, we get that
Alrn,l
⎡ ⎤lrn1
lrq
n
iθ
iθ
τe
e
⎣−
Gq τ⎦
exp −
t
t
q1
3.34
lrn
eiθ
t
lr1
exp −
αleiθ
t
1−αlrn1
1−αlrn
fs exp −
seiθ
t
ds.
Since Gq l r q 0, for all q ≥ 1, we deduce that the telescopic sum
⎤lrn1
⎡ lrq
∞
iθ
iθ
e
τe
⎣
Gq τ⎦
exp −
t
t
nq
lrn
3.35
34
Abstract and Applied Analysis
is equal to 0. From the formula 3.32, 3.33, 3.34, and 3.35, we get that
Lθ
fτ − αl1αl,∞
lr t ∞
Ah,l
h0
eiθ
t
lr
αleiθ
exp −
t
eiθ
t
∞
0
seiθ
fs exp −
t
ds.
3.36
From Proposition 3.3, we have that
−r lr Lθ Fl t tr eiθ Lθ fτ − αl1αl,∞
t.
3.37
Finally, from 3.36 and 3.37, we get the equality 3.27.
4. Formal and Analytic Transseries Solutions for a Singularly
Perturbed Cauchy Problem
4.1. Laplace Transform and Asymptotic Expansions
We recall the definition of Borel summability of formal series with coefficients in a Banach
space, see 27.
Definition 4.1. A formal series
Xt
∞ a
j
j0
j!
tj ∈ Et
4.1
with coefficients in a Banach space E, · E is said to be 1-summable with respect to t in the
direction d ∈ 0, 2π if
i there exists ρ ∈ R such that the following formal series, called formal Borel
of order 1,
transform of X
∞ a τj
j
τ B X
2 ∈ Eτ
j0 j!
4.2
is absolutely convergent for |τ| < ρ;
ii there exists δ > 0 such that the series BXτ
can be analytically continued with
∗
respect to τ in a sector Sd,δ {τ ∈ C : |d − argτ| < δ}. Moreover, there exist C > 0
and K > 0 such that
4.3
BXτ
≤ CeK|τ|
E
for all τ ∈ Sd,δ . We say that BXτ
has exponential growth of order 1 on Sd,δ .
Abstract and Applied Analysis
35
If this is so, the vector valued Laplace transform of order 1 of BXτ
in the direction
d is defined by
t t−1
τe−τ/t dτ
B X
Ld B X
Lγ
4.4
along a half-line Lγ R eiγ ⊂ Sd,δ ∪ {0}, where γ depends on t and is chosen in such a way
that cosγ − argt ≥ δ1 > 0, for some fixed δ1 , for all t in a sector
Sd,θ,R
θ
∗
,
t ∈ C : |t| < R, d − argt <
2
4.5
where π < θ < π 2δ and 0 < R < δ1 /K. The function Ld BXt
is called the 1-sum
is a holomorphic and
of the formal series Xt
in the direction d. The function Ld BXt
has the formal
a bounded function on the sector Sd,θ,R . Moreover, the function Ld BXt
series Xt as Gevrey asymptotic expansion of order 1 with respect to t on Sd,θ,R . This means
that for all 0 < θ1 < θ, there exist C, M > 0 such that
n−1 a
d
p p
L B X
≤ CMn n!|t|n
t −
t
p!
p0
4.6
E
for all n ≥ 1, all t ∈ Sd,θ1 ,R .
In the next proposition, we recall some well-known identities for the Borel transform
that will be useful in the sequel.
n≥0 bn tn /n! be formal series in Et. One
Proposition 4.2. Let Xt
n≥0 an tn /n! and Gt
has the following equalities as formal series in Eτ:
τ∂2τ ∂τ
τ B ∂t Xt
B X
τ,
τ B tXt
B X
∂−1
τ,
τ
τ B t2 ∂t t Xt
τB X
τ.
4.7
4.2. Formal Transseries Solutions for an Auxiliary Singular Cauchy Problem
Let S ≥ 1 be an integer. Let S be a finite subset of N3 and let
bs,k0 ,k1 z, bs,k0 ,k1 ,β zβ
β≥0
β!
4.8
36
Abstract and Applied Analysis
be holomorphic and bounded functions on a polydisc D0, ρ × D0, 0 , for some ρ, 0 > 0,
with 0 < 1, for all s, k0 , k1 ∈ S. We consider the following singular Cauchy problems:
T 2 ∂T ∂Sz Y T, z, T 1∂Sz Y T, z, s,k0 ,k1 ∈S
bs,k0 ,k1 z, k0 −s T s ∂kT0 ∂kz1 Y T, z, , 4.9
for given formal transseries initial conditions
exp−hλ/T j
ϕh,j T, ,
∂z Y T, 0, h!
h≥0
where ϕh,j T, m≥0
0 ≤ j ≤ S − 1,
4.10
ϕh,j,m T m /m! ∈ CT for all ∈ E and λ ∈ C∗ .
Proposition 4.3. The problem 4.9, 4.10 has a formal transseries solutions
Y T, z, exp−hλ/T h≥0
h!
Yh T, z, ,
4.11
where the formal series Yh T, z, ∈ CT, z, for all ∈ E, all h ≥ 0, satisfy the following singular
Cauchy problems:
T 2 ∂T ∂Sz Yh T, z, T 1 λh∂Sz Yh T, z, ⎛
bs,k0 ,k1 z, ⎝k0 −s T s ∂k0 ∂kz1 Yh T, z, T
s,k0 ,k1 ∈S
⎞
k01
2
k0 ! k01
q k0 −s s−k01 q k0 k1 cq hλ T
∂T ∂z Yh T, z, ⎠
k1 !k2 !
k1 k2 k0 ,k1 ≥1 0 0 q1
0
0
0
4.12
with initial conditions
j
∂z Yh T, 0, ϕh,j T, ,
0 ≤ j ≤ S − 1,
4.13
k1
for some real numbers cq 0 , for 1 ≤ q ≤ k01 and 1 ≤ k01 ≤ k0 .
Proof. We have that
hλ hλ hλ
∂T exp −
Yh T, z, exp −
Yh T, z, ∂T Yh T, z, ,
T
T
T2
4.14
Abstract and Applied Analysis
37
and from the Leibniz rule we also have
∂kT0
hλ exp −
Yh T, z, T
2
k0 ! k01
hλ
k
exp −
∂
∂T0 Yh T, z, .
1 2 T
T
k !k !
k1 k2 k0 0 0
0
4.15
0
On the other hand, by the Faa Di Bruno formula we have, for all k01 ≥ 1, that
k01
∂T
k01
λh
hλ
exp −
exp −
T
T λ
q1
1 ,...,λk 1 ∈Aq,k 1
0
0
k01 !
λi
k01
−1i1 hλ/T i1
λi !
i1
4.16
⎛
⎞
k01 1
q
hλ ⎝
k0 hλ
⎠,
cq
exp −
1
T
T k0 q
q1
1
where Aq,k01 {λ1 , . . . , λk01 ∈ Nk0 /
k01
λi q,
k01
k1
iλi k01 } and cq 0 ∈ R, for all q 1, . . . , k01 .
Using the expressions 4.14, 4.15, 4.16, by plugging the formal expansion Y T, z, into the problem 4.9, 4.10 and by identification of the coefficients of exp−hλ/T we get
that Yh satisfies the problem 4.12, 4.13.
i1
i1
4.3. Formal Solutions to a Sequence of Regular Cauchy Problems
Proposition 4.4. One makes the assumption that
s ≥ 2k0
S > k1 ,
4.17
for all s, k0 , k1 ∈ S. Then, the problem 4.12, 4.13 has a unique formal solution Yh T, z, ∈
CT, z for all ∈ E. Let
Yh T, z, Yh,m z, T m
m≥0
m!
,
4.18
where Yh,m z, ∈ Cz, be the formal solution of 4.12, 4.13 for all ∈ E. One denotes by
Vh τ, z, Yh,m z, m≥0
τm
m!2
4.19
38
Abstract and Applied Analysis
the formal Borel transform of Yh with respect to T . Then, for all h ≥ 0, Vh τ, z, satisfies the problem
τ 1 λh∂Sz Vh τ, z, ⎛
⎜
bs,k0 ,k1 z, ⎝k0 −s
s,k0 ,k1 ∈S
−p
1
r,p∈Os−k
0
α1r,p τ r ∂τ ∂kz1 Vh τ, z, 1
k0
k0 ! k1
cq 0 hλq k0 −s
1 2
k !k !
k1 k2 k0 ,k1 ≥1 0 0 q1
0
0
0
×
2
r,p∈Os−k
⎞
2,q
−p
⎟
αr,p τ r ∂τ ∂kz1 Vh τ, z, ⎠
0 −q
4.20
with initial data
τm
j
ϕh,j,m ∈ Cτ,
∂z Vh τ, 0, vh,j τ, m!2
m≥0
0 ≤ j ≤ S − 1,
4.21
1
1
2
is a finite subset of N2 such that r, p ∈ Os−k
implies r p s − k0 and Os−k
is a
where Os−k
0
0
0 −q
2,q
2
implies r p s − k0 − q, and α1r,p , αr,p are integers.
finite subset of N2 such that r, p ∈ Os−k
0 −q
Proof. The proof follows by direct computation on the problems 4.12 and 4.13, using
Proposition 4.2 and the following two lemmas from 2.
Lemma 4.5. For all k0 ≥ 1, there exist constants ak,k0 ∈ N, k0 ≤ k ≤ 2k0 , such that
τ∂2τ ∂τ
k 0
uτ 2k0
ak,k0 τ k−k0 ∂kτ uτ
4.22
kk0
for all holomorphic functions u : Ω → C on an open set Ω ⊂ C.
Lemma 4.6. Let a, b, c ≥ 0 be positive integers such that a ≥ b and a ≥ c. We put δ a b − c. Then,
b c
for all holomorphic functions u : Ω → C, the function ∂−a
τ τ ∂τ uτ can be written in the form
∂−a
τ b ∂cτ uτ τ
b ,c ∈Oδ
αb ,c τ b ∂cτ uτ,
4.23
where Oδ is a finite subset of Z2 such that for all b , c ∈ Oδ , b − c δ, b ≥ 0, c ≤ 0, and αb ,c ∈ Z.
4.4. An Auxiliary Cauchy Problem
We denote by Ω1 an open star-shaped domain in C meaning that Ω1 is an open subset of
C such that for all x ∈ Ω1 , the segment 0, x belongs to Ω1 . Let Ω2 be an open set in C∗
Abstract and Applied Analysis
39
contained in the disc D0, 0 . We denote by Ω Ω1 × Ω2 . For any open set D ⊂ C, we denote
by OD the vector space of holomorphic functions on D.
β
Definition 4.7. Let b > 1 a real number and let rb β n0 1/n 1b for all integers β ≥ 0.
Let ∈ Ω2 and σ > 0 be a real number. We denote by Eβ,,σ,Ω the vector space of all functions
v ∈ OΩ1 such that
vτβ,,σ,Ω : sup|vτ| 1 τ∈Ω1
|τ|2
||2
σ exp −
rb β |τ|
2||
4.24
is finite.
Proposition 4.8. One makes the assumption that
S > k1 ,
s ≥ 2k0
4.25
for all s, k0 , k1 ∈ S. Moreover, one makes the assumption that there exists c , δ > 0 such that
|τ 1 hλ| ≥ c |τ 1| > δ ,
∀τ ∈ Ω1 , ∀h ∈ N.
4.26
For all h ≥ 0, all ∈ Ω2 , the problem 4.20 with initial conditions
j
∂z Vh τ, 0, vh,j τ, ∈ OΩ1 ,
0≤j ≤S−1
4.27
has a unique formal series
Vh τ, z, zβ
vh,β τ, ∈ OΩ1 z,
β!
β≥0
4.28
where vh,β τ, satisfies the following recursion:
τ 1 hλvh,βS τ, ⎛
bs,k0 ,k1 ,β1 k0 −s ⎜
β!
⎝
β1 !
β
s,k0 ,k1 ∈S β1 β2
1
⎞
v
τ,
−p h,β2 k1
⎟
α1r,p τ r ∂τ
⎠
β2 !
1
r,p∈Os−k
0
k0
bs,k0 ,k1 ,β1 k01
k0 ! cq hλq
β!
1 2
β
!
k
!k
!
1
1
2
1
k0 k0 k0 ,k0 ≥1 0 0 q1 β1 β2 β
⎞
⎛
v
τ,
2,q
−p h,β2 k1
⎟
⎜
× k0 −s ⎝
αr,p τ r ∂τ
⎠
β
!
2
2
r,p∈Os−k
for all τ ∈ Ω1 , all ∈ Ω2 .
0 −q
4.29
40
Abstract and Applied Analysis
Proposition 4.9. One makes the assumption that
s ≥ 2k0
S > k1 ,
4.30
for all s, k0 , k1 ∈ S. Let also the assumption 4.26 holds. Let us assume that
vh,j τ, ∈ Ej,,σ,Ω ,
∀h ≥ 0, ∀0 ≤ j ≤ S − 1, ∀ ∈ Ω2 .
4.31
Then, one has that vh,β τ, ∈ Eβ,,σ,Ω for all β ≥ 0, all h ≥ 0, all ∈ Ω2 . We put vh,β ||vh,β τ, ||β,,σ,Ω , for all h ≥ 0, all β ≥ 0, and all ∈ Ω2 . Then, the following inequalities hold: there
1
2
, C18
> 0 (depending on S,σ,S) such that
exist two constants C18
vh,βS ≤
bs,k ,k ,β 0 1 1
β!
β1 !
β
s,k0 ,k1 ∈S β1 β2
1
× C18
bs−k0 2 vh,β2 k1 bs−k0 βS1
βS1
β2 !
k01
bs,k ,k ,β k1 k0 ! 0 1 1
cq 0 hq |λ|q
β!
1 2
β
!
k
!k
!
1
1
2
1
q1
β
β
β
k k k0 ,k ≥1 0 0
1
2
0
0
4.32
0
2
× ||−q C18
bs−k0 −q2 vh,β2 k1 bs−k0 −q βS1
βS1
β2 !
for all h ≥ 0, all β ≥ 0.
Proof. The proof follows by direct computation using the recursion 4.29 and the next
lemma. We keep the notations of Proposition 4.8.
Lemma 4.10. There exists a constant C18 > 0 (depending on s, σ, S, k0 , k1 ) such that
r −p
τ ∂τ vh,β2 k1 τ, ≤ ||
rp
C18
βS,,σ,Ω
brp2 brp vh,β k τ, βS1
βS1
2
1
β2 k1 ,,σ,Ω
4.33
for all h ≥ 0, and all β ≥ 0, 0 ≤ β2 ≤ β, all r, p ∈ N2 with r p ≤ s − k0 .
Proof.
We follow the proof of Lemma 1 from 2. By definition, we have that ∂−1
τ vh,β2 k1 τ, τ
v
τ
,
dτ
for
all
τ
∈
Ω
.
Using
the
parametrization
τ
h
τ
with
0
≤
h1 ≤ 1, we get
h,β
k
1
1
1
1
1
2
1
0
that
∂−1
τ vh,β2 k1 τ, τ
1
0
vh,β2 k1 h1 τ, M1 h1 dh1 ,
4.34
Abstract and Applied Analysis
41
where M1 h1 1. More generally, for all p ≥ 2, we have by definition:
−p
∂τ vh,β2 k1 τ, τ τ1
0
···
τp−1
0
vh,β2 k1 τp , dτp dτp−1 · · · dτ1
4.35
0
for all τ ∈ Ω1 . Using the parametrization τj hj τj−1 , τ1 h1 τ, with 0 ≤ hj ≤ 1, for 2 ≤ j ≤ p,
we can write
−p
∂τ vh,β2 k1 τ, τ
p
1
0
···
1
vh,β2 k1 hp · · · h1 τ, Mp h1 , . . . , hp dhp dhp−1 · · · dh1 ,
4.36
0
where Mp h1 , . . . , hp is a monomial in h1 , . . . , hp whose coefficient is equal to 1. Using these
latter expressions, we now write
r −p
τ ∂τ vh,β2 k1 τ, 1 1
hp · · · h1 τ 2
σ rp
hp · · · h1 τ τ
exp
−
r
· · · vh,β2 k1 hp · · · h1 τ, 1 k
β
b
2
1
2||
||2
0
0
exp σ/2||rb β2 k1 hp · · · h1 τ ×
Mp h1 , . . . , hp dhp . . . dh1 .
2
2
1 hp · · · h1 τ /||
4.37
Therefore,
σ |τ|2
r −p
rb β S |τ|
τ ∂τ vh,β2 k1 τ, 1 2 exp −
2||
||
σ |τ|2
rp
1 2 exp −
rb β S − rb β2 k1 |τ| .
≤ vh,β2 k1 τ, β2 k1 ,,σ,Ω |τ|
2||
||
4.38
By construction of rb β, we have
rb β S − rb β2 k1 βS
β − β2 S − k1
S − k1
≥ b ≥ b
βS1
βS1
nβ2 k1 1 n 1
1
b
4.39
for all β ≥ 0. From 4.38 and 4.39, we get that
σ |τ|2
r −p
rb β S |τ|
τ ∂τ vh,β2 k1 τ, 1 2 exp −
2||
||
σ
S − k1
|τ|2
rp
1 2 exp −
≤ vh,β2 k1 τ, β2 k1 ,,σ,Ω |τ|
|τ|
2|| β S 1b
||
4.40
42
Abstract and Applied Analysis
for all β ≥ 0. From 2.41, we deduce that
|τ|
rp
1
|τ|2
σ
S − k1
exp −
|τ|
2|| β S 1b
||2
⎛ −1 rp
brp
rp ⎝ 2 r p e
≤ ||
βS1
σS − k1 4.41
⎞
rp2
2 r p 2 e−1
brp2
⎠
βS1
σS − k1 for all τ ∈ Ω1 . From the estimates 4.40 and 4.41, we deduce the inequality 4.33.
Proposition 4.11. Assume that the conditions 4.26 and 4.31 hold. Assume moreover, that
S ≥ bs − k0 2 k1 ,
s ≥ 2k0
4.42
for all s, k0 , k1 ∈ S and that the following sums converge near the origin in C,
Wj u :
uh
∈ C{u},
sup
vh,j τ, j,,σ,Ω
h!
h≥0 ∈Ω2
0 ≤ j ≤ S − 1.
4.43
One make also the hypothesis that for all s, k0 , k1 ∈ S, one can write
bs,k0 ,k1 z, k0 bs,k0 ,k1 z, ,
4.44
where bs,k0 ,k1 z, β≥0 bs,k0 ,k1 ,β zβ /β! is holomorphic for all ∈ D0, 0 on D0, ρ. Then, the
problem 4.20 with initial data
j
∂z Vh τ, 0, vh,j τ, ,
0≤j ≤S−1
4.45
has a unique solution Vh τ, z, which is holomorphic with respect to τ, z ∈ Ω1 × D0, x1 /2 for
all ∈ Ω2 .
The constant x1 is such that 0 < x1 < ρ and depends on S, u0 (which denotes a
common radius of absolute convergence of the series 4.43), S, b, σ, |λ|, maxs,k0 ,k1 ∈S |b|s,k0 ,k1 x0 ,
s,k ,k x0 , where x0 < ρ and |b|s,k ,k , |b|
s,k ,k are defined below.
maxs,k0 ,k1 ∈S |b|
0 1
0 1
0 1
Abstract and Applied Analysis
43
Moreover, the following estimates hold: there exists a constant u1 such that 0 < u1 < u0
(depending on u0 , S, and b, σ) and a constant C19 > 0 (depending on max0≤j≤S−1 Wj u0 (where Wj
s,k ,k x0 , S, u0 , x0 , S, b) such that
are defined above), |λ|, maxs,k0 ,k1 ∈S |b|s,k0 ,k1 x0 , maxs,k0 ,k1 ∈S |b|
0 1
−1
h 2
σ
C19
|τ|2
1 2
ζb|τ|
h!
exp
|Vh τ, z, | ≤
1 − 2|z|/x1
u1
2||
||
4.46
for all τ, z ∈ Ω1 × D0, x1 /2, all ∈ Ω2 , and all h ≥ 0.
Proof. We consider the following Cauchy problem
∂Sx Wu, x s,k0 ,k1 ∈S
1
C18
x∂x S 1bs−k0 x∂x S 1bs−k0 2
|b|s,k0 ,k1 x∂kx1 Wu, x
k01
1
k0 ! 2 k0 C18 cq |λ|q
1 2
k !k !
k1 k2 k0 ,k1 ≥1 0 0 q1
0
0
4.47
0
× x∂x S 1bs−k0 −q x∂x S 1bs−k0 −q2
× b
s,k0 ,k1
xu∂u q ∂kx1 Wu, x
for given initial data
uh
j
∂x W u, 0 Wj u ∈ C{u},
supvh,j h!
h≥0 ∈Ω2
0 ≤ j ≤ S − 1,
4.48
where
|b|s,k0 ,k1 x xβ
sup bs,k0 ,k1 ,β ,
β!
β≥0 ∈D0,0 b
s,k0 ,k1
x xβ
sup bs,k0 ,k1 ,β β!
∈D0,
0
β≥0
4.49
are convergent series near the origin in C with respect to x. From the assumption 4.42 and
the fact that b > 1, we also deduce that
S ≥ b s − k0 − q 2 q k 1
4.50
for all s, k0 , k1 ∈ S and all 0 ≤ q ≤ k0 . Since the initial data 4.48 and the coefficients
4.47 are analytic near the origin, we get that all the hypotheses of the classical Cauchy
Kowalevski theorem from Proposition 2.22 are fulfilled. We deduce the existence of U1 with
0 < U1 < U0 , where U0 denotes a common radius of absolute convergence for the series
4.48, which depends on U0 , S and b, and X1 with 0 < X1 < ρ depending on S, U0 , S, b,
44
Abstract and Applied Analysis
s,k ,k X0 , where X0 < ρ such that there exist a
σ, |λ|, maxs,k0 ,k1 ∈S |b|s,k0 ,k1 X0 , maxs,k0 ,k1 ∈S |b|
0 1
unique formal series Wu, x ∈ GU1 , X1 which solves the problem 4.47, 4.48.
Now, let Wu, x h,β≥0 wh,β uh /h!xβ /β! be its Taylor expansion at 0, 0. Then,
by construction the sequence wh,β satisfies the following equalities:
wh,βS β!
sup∈D0,0 bs,k0 ,k1 ,β1 β1 !
s,k0 ,k1 ∈S β1 β2 β
×
1
C18
bs−k0 2 wh,β2 k1
bs−k0 βS1
βS1
β2 !
k01
b
sup
k1 ∈D0,0 s,k0 ,k1 ,β1
k0 ! cq 0 hq |λ|q
β!
β !
k1 !k2 !
2
× C18
1
0 q1 β1 β2 β
k01 k02 k0 ,k01 ≥1 0
4.51
bs−k0 −q2 wh,β2 k1
bs−k0 −q βS1
βS1
β2 !
for all h ≥ 0 and all β ≥ 0, with
wh,j supvh,j ,
∀h ≥ 0, ∀0 ≤ j ≤ S − 1.
4.52
∈Ω2
Using the inequality 4.32 and the equality 4.51, with the initial conditions 4.52, one gets
that
supvh,β ≤ wh,β
4.53
∈Ω2
for all h ≥ 0, all β ≥ 0. Using the fact that Wu, x ∈ GU1 , X1 and the estimates 2.111, we
deduce from 4.53 that there exist a constant C19 > 0 depending on max0≤j≤S−1 Wj U0 ,|λ|,
s,k ,k X0 , S, U0 , X0 , S, b, σ such that
maxs,k0 ,k1 ∈S |b|s,k0 ,k1 X0 , maxs,k0 ,k1 ∈S |b|
0 1
vh,β τ, ≤ C19 h β !
≤ C19 h!β!
2
U1
1
U1
h h 2
X1
1
X1
β 1
β 1
for all τ ∈ Ω1 , all ∈ Ω2 , all h ≥ 0, and all β ≥ 0.
|τ|2
||2
−1
|τ|2
||2
−1
exp
σ rb β |τ|
2||
σ rb β |τ|
exp
2||
4.54
Abstract and Applied Analysis
45
4.5. Analytic Solutions for a Sequence of Singular Cauchy Problems
Assume that the conditions 4.42 and 4.44 hold. We consider the following problem:
T 2 ∂T ∂Sz Yh,Sd ,E T, z, T 1 λh∂Sz Yh,Sd ,E T, z, ⎛
bs,k0 ,k1 z, ⎝k0 −s T s ∂kT0 ∂kz1 Yh,Sd ,E T, z, s,k0 ,k1 ∈S
⎞
k01
1
2
k0 !
1
k
k
cq 0 hλq k0 −s T s−k0 q ∂T0 ∂kz1 Yh,Sd ,E T, z, ⎠
1 2
k
!k
!
k1 k2 k0 ,k1 ≥1 0 0 q1
0
0
0
4.55
with initial conditions
j
∂z Yh,Sd ,E T, 0, ϕh,j,Sd ,E T, ,
0 ≤ j ≤ S − 1.
4.56
The initial conditions ϕh,j,Sd ,E T, , 0 ≤ j ≤ S − 1 are defined as follows. Let Sd be an open
sector centered at 0, with infinite radius and bisecting direction d ∈ 0, 2π, D0, τ0 an open
disc centered at 0 with radius τ0 > 0, and E an open sector centered at 0 contained in the
disc D0, 0 . We make the assumption that the condition 4.26 holds for the set Ω1 Sd ∪
D0, τ0 . We consider a set of functions vh,j τ, ∈ Ej,,σ,D0,τ0 ×D0,0 \{0} for all ∈ D0, 0 \
{0} such that
Wj,τ0 ,0 u :
uh
vh,j τ, ∈ C{u},
j,,σ,D0,τ0 ×D0,0 \{0} h!
h≥0 ∈D0,0 \{0}
sup
0 ≤ j ≤ S − 1.
4.57
We also assume that for all h ≥ 0 and all 0 ≤ j ≤ S − 1, vh,j τ, has an analytic continuation
denoted by vh,j,Sd ,E τ, ∈ Ej,,σ,Sd ∪D0,τ0 ×E for all ∈ E such that
Wj,Sd ,E u :
uh
∈ C{u},
sup
vh,j,Sd ,E τ, j,,σ,Sd ∪D0,τ0 ×E
h!
h≥0 ∈E
0 ≤ j ≤ S − 1.
4.58
Let
vh,j τ, ϕh,j,m m≥0
τm
m!2
4.59
be the convergent Taylor expansion of vh,j with respect to τ on D0, τ0 for all ∈ D0, 0 \{0}.
We consider the formal series
ϕh,j T, ϕh,j,m m≥0
Tm
m!
4.60
46
Abstract and Applied Analysis
for all ∈ D0, 0 \ {0}. We define ϕh,j,Sd ,E T, as the 1-sum in the sense of Definition 4.1
of ϕj,h T, in the direction d. From the hypotheses, we deduce that T → ϕh,j,Sd ,E T, defines
a holomorphic function for all T ∈ Ud,θ,ι|| , for all ∈ E, where
Ud,θ,ι|| θ
T ∈ C∗ : |T | < ι||, d − argT <
2
4.61
for some θ > π and some constant ι > 0 independent of for all 0 ≤ j ≤ S − 1.
Proposition 4.12. Assume that the conditions 4.26, 4.31, 4.42, and 4.44 hold.
Then, the problem 4.55, 4.56 has a solution T, z → Yh,Sd ,E T, z, which is holomorphic
and bounded on the set Ud,θ,ι|| × D0, x1 /4, for some ι > 0 (independent of ), for all ∈ E, where
0 < x1 < ρ depends on S, u0 (which denotes a common radius of absolute convergence of the series
s,k ,k x0 , where x0 < ρ.
4.57, 4.58), S, b, σ, |λ|, maxs,k0 ,k1 ∈S |b|s,k0 ,k1 x0 , maxs,k0 ,k1 ∈S |b|
0 1
The function Yh,Sd ,E T, z, can be written as the Laplace transform of order 1 in the direction
d (in the sense of Definition 4.1) of a function Vh,Sd ,E τ, z, , which is holomorphic on the domain
Sd ∪ D0, τ0 × D0, x1 /2 × E and satisfies the following estimates.
There exists a constant u1 such that 0 < u1 < u0 (depending on u0 , S and b,σ) and a
constant CΩd,E > 0 (depending on max0≤j≤S−1 Wj,Sd ,E u0 (where Wj,Sd ,E are defined above), |λ|,
s,k ,k x0 , S, u0 , x0 , S, b) such that
maxs,k0 ,k1 ∈S |b|s,k0 ,k1 x0 , maxs,k0 ,k1 ∈S |b|
0 1
−1
h CΩd,E
2
σ
|τ|2
1 2
ζb|τ|
h!
exp
|Vh,Sd ,E τ, z, | ≤
1 − 2|z|/x1
u1
2||
||
4.62
for all τ, z, ∈ Sd ∪ D0, τ0 × D0, x1 /2 × E, all h ≥ 0.
Moreover, the function Vh,Sd ,E τ, z, is the analytic continuation of a function Vh τ, z, which is holomorphic on the punctured polydisc D0, τ0 × D0, x1 /2 × D0, 0 \ {0} and verifies
the following estimates.
There exists a constant CΩτ0 ,0 > 0 (depending on max0≤j≤S−1 Wj,τ0 ,0 u0 (where Wj,τ0 ,0 are
s,k ,k x0 , S, u0 , x0 , S, b) such that
defined above), |λ|, maxs,k ,k ∈S |b|s,k ,k x0 , maxs,k ,k ∈S |b|
0
1
0
CΩτ0 ,0
1
0
2
h!
|Vh τ, z, | ≤
1 − 2|z|/x1
u1
h 1
1
0
|τ|2
||2
1
−1
exp
σ
ζb|τ|
2||
4.63
for all τ ∈ D0, τ0 , all z ∈ D0, x1 /2, all ∈ D0, 0 \ {0}, and all h ≥ 0.
Proof. From the hypotheses of Proposition 4.12, we deduce from Proposition 4.11 applied
to the situation Ω D0, τ0 × D0, 0 \ {0} the existence of a holomorphic function
Vh τ, z, satisfying the estimates 4.63, which is the solution of the problem 4.20 with
j
initial conditions ∂z Vh τ, 0, vh,j τ, , 0 ≤ j ≤ S−1, on the domain D0, τ0 ×D0, x1 /2×
D0, 0 \{0}. Likewise, from Proposition 4.11 applied to the situation Ω Sd ∪D0, τ0 ×E,
we get the existence of a holomorphic function Vh,Sd ,E τ, z, satisfying 4.62, which is the
j
solution of the problem 4.20 with initial conditions ∂z Vh τ, 0, vh,j,Sd ,E τ, , 0 ≤ j ≤ S−1
on the domain Sd ∪ D0, τ0 × D0, x1 /2 × E.
With Proposition 4.3, we deduce that the formal solution Yh T, z, of the problem
4.12, 4.13 is 1-summable with respect to T in the direction d as series in the Banach space
Abstract and Applied Analysis
47
OD0, x1 /4, for all ∈ E. We denote by Yh,Sd ,E T, z, its 1-sum which is holomorphic with
respect to T on a domain Ud,θ,ι|| due to Definition 4.1 and the estimates 4.62. Moreover,
from the algebraic properties of the κ-summability procedure, see 27, Section 6.3, we deduce
that Yh,Sd ,E T, z, is a solution of the problem 4.55, 4.56.
4.6. Summability in a Complex Parameter
We recall the definition of a good covering.
Definition 4.13. Let ν ≥ 2 be an integer. For all 0 ≤ i ≤ ν − 1, we consider open sectors Ei
centered at 0, with radius 0 , bisecting direction κi ∈ 0, 2π and opening π δi , with δi > 0,
such that Ei ∩ Ei1 / ∅ for all 0 ≤ i ≤ ν − 1 with the convention that Eν E0 and such that
E
U
\
{0},
where
U is some neighborhood of 0 in C. Such a set of sectors {Ei }0≤i≤ν−1 is
∪ν−1
i0 i
called a good covering in C∗ .
Definition 4.14. Let {Ei }0≤i≤ν−1 be a good covering in C∗ . Let T be an open sector centered at 0
with radius rT and consider a family of open sectors
Udi ,θ,0 rT :
θ
,
t ∈ C : |t| < 0 rT , di − argt <
2
4.64
where di ∈ 0, 2π, for 0 ≤ i ≤ ν − 1, where θ > π, which satisfy the following properties:
arg−1 − λh.
1 For all 0 ≤ i ≤ ν − 1, all h ∈ N, argdi /
2 For all 0 ≤ i ≤ ν − 1, for all t ∈ T, and all ∈ Ei , we have that t ∈ Udi ,θ,0 rT .
3 3.1 We assume that d0 < argλ < d1 . We consider the two closed sectors
Md0
τ∈
!
"
C∗
∈ d0 , argλ ,
argτ
Md1
τ∈
!
"
C∗
∈ argλ, d1 .
argτ
4.65
We make the assumption that there exist two constants c , δ > 0 with
|τ 1 λh| ≥ c |τ 1| > δ
4.66
for all τ ∈ Md0 ∪ Md1 ∪ D0, τ0 and all h ≥ 0.
3.2 There exists 0 < δT < π/2 such that argλ/t ∈ −π/2 δT , π/2 − δT for all
∈ E0 ∩ E1 and all t ∈ T.
We say that the family {{Udi ,θ,0 rT }0≤i≤ν−1 , T, λ} is associated to the good covering
{Ei }0≤i≤ν−1 .
Now, we consider a set of functions ϕh,i,j T, for 0 ≤ i ≤ ν − 1, 0 ≤ j ≤ S − 1, h ≥ 0,
constructed as follows. For all 0 ≤ i ≤ ν − 1, let Sdi be an open sector of infinite radius centered
at 0, with bisecting direction di and with opening ni > θ − π. The numbers θ > π and ni > 0
are chosen in such a way that −1 − λh ∈
/ Sdi for all 0 ≤ i ≤ ν − 1 and all h ≥ 0. Now, we put
ϕh,i,j T, : ϕh,j,Sdi ,Ei T, 4.67
48
Abstract and Applied Analysis
for all T ∈ Udi ,θ,ι|| and all ∈ Ei , where ϕh,j,Sdi ,Ei T, is given by the formula 4.56. Recalling
how these functions are constructed, we consider a set of functions
4.68
vh,j τ, ∈ Ej,,σ,D0,τ0 ×D0,0 \{0}
for all ∈ D0, 0 \ {0} such that
Wj,τ0 ,0 u :
uh
vh,j τ, ∈ C{u},
j,,σ,D0,τ0 ×D0,0 \{0} h!
h≥0 ∈D0,0 \{0}
sup
4.69
0 ≤ j ≤ S − 1.
We also assume that for all h ≥ 0, all 0 ≤ j ≤ S − 1, vh,j τ, has an analytic continuation
denoted by vh,j,Sdi ,Ei τ, ∈ Ej,,σ,Sdi ∪D0,τ0 ×Ei for all ∈ Ei such that
Wj,Sdi ,Ei u :
sup
vh,j,Sdi ,Ei τ, h≥0 ∈Ei
uh
∈ C{u},
j,,σ,Sdi ∪D0,τ0 ×Ei h!
0 ≤ j ≤ S − 1.
4.70
Let
vh,j τ, ϕh,j,m m≥0
τm
m!2
4.71
be the convergent Taylor expansion of vh,j with respect to τ on D0, τ0 for all ∈ D0, 0 \{0}.
We consider the formal series
ϕh,j T, ϕh,j,m m≥0
Tm
m!
4.72
for all ∈ D0, 0 \{0}. We define ϕh,j,Sdi ,Ei T, as the 1-sum in the sense of Definition 4.1 of
ϕj,h T, in the direction di . We deduce that T → ϕh,j,Sdi ,Ei T, defines a holomorphic function
for all T ∈ Udi ,θ,ι|| and for all ∈ Ei , where
Udi ,θ,ι|| θ
T ∈ C∗ : |T | < ι||, di − argT <
2
4.73
for some θ > π and some constant ι > 0 independent of for all 0 ≤ j ≤ S − 1.
From Proposition 4.12, for all 0 ≤ i ≤ ν − 1, we consider the solution Yh,Sdi ,Ei T, z, of
the problem 4.55 with the initial conditions
j
∂z Yh,Sdi ,Ei T, 0, ϕh,i,j T, ,
0 ≤ j ≤ S − 1, h ≥ 0,
which defines a bounded and holomorphic function on Udi ,θ,ι|| × D0, x1 /4 × Ei .
4.74
Abstract and Applied Analysis
49
Proposition 4.15. The function defined by
Xh,i t, z, Yh,Sdi ,Ei t, z, 4.75
is holomorphic and bounded on T ∩ D0, ι × D0, x1 /4 × Ei , for all h ≥ 0, all 0 ≤ i ≤ ν − 1, and
for some 0 < ι < ι .
Moreover, the functions Gh,i : → Xh,i t, z, from Ei into the Banach space OT ∩
h ∈ OT ∩ D0, ι ×
D0, ι × D0, x1 /4 are the 1-sums on Ei of a formal series G
D0, x1 /4. In other words, for all h ≥ 0, there exists a function gh s, t, z which is holomorphic
on D0, sh × T ∩ D0, ι × D0, x1 /4 which admits for all 0 ≤ i ≤ ν − 1, an analytic continuation
gh,i s, t, z which is holomorphic on Gκi ∪ D0, sh × T ∩ D0, ι × D0, x1 /4, where Gκi is an
open sector centered at 0 with infinite radius and bisecting direction κi such that
Xh,i t, z, −1
gh,i s, t, ze−s/ ds
Lκi
4.76
along a half-line Lκi R eiκi ⊂ Gκi ∪ {0}.
Proof. The proof is based on a cohomological criterion for summability of formal series with
coefficients in a Banach space, see 27, page 121, which is known as the Ramis-Sibuya
theorem in the literature.
Theorem RS. Let E, || · ||E be a Banach space over C and {Ei }0≤i≤ν−1 a good covering in C∗ . For
all 0 ≤ i ≤ ν − 1, let Gi be a holomorphic function from Ei into the Banach space E, || · ||E and let the
cocycle Δi Gi1 − Gi be a holomorphic function from the sector Zi Ei1 ∩ Ei into E (with
the convention that Eν E0 and Gν G0 ). We make the following assumptions.
1 The functions Gi are bounded as ∈ Ei tends to the origin in C for all 0 ≤ i ≤ ν − 1.
2 The functions Δi are exponentially flat of order 1 on Zi for all 0 ≤ i ≤ ν − 1. This means
that there exist constants Ci , Ai > 0 such that
Δi E ≤ Ci e−Ai /||
4.77
for all ∈ Zi all 0 ≤ i ≤ ν − 1.
Then, for all 0 ≤ i ≤ ν − 1, the functions Gi are the 1 -sums on Ei of a 1 -summable formal
series G
in with coefficients in the Banach space E.
By Definition 4.14 and the construction of Yh,Sdi ,Ei T, z, in Proposition 4.12, we get
that the function Xh,i t, z, Yh,Sdi ,Ei t, z, defines a bounded and holomorphic function
on the domain T ∩ D0, ι × D0, x1 /4 × Ei for all h ≥ 0 all 0 ≤ i ≤ ν − 1, where 0 < x1 < ρ
depends on S, u0 > 0 which denotes a common radius of absolute convergence of the series
s,k ,k x0 , where x0 < ρ.
4.69, 4.70, S, b, σ, |λ|, maxs,k0 ,k1 ∈S |b|s,k0 ,k1 x0 , maxs,k0 ,k1 ∈S |b|
0 1
More precisely, we have the following.
Lemma 4.16. Consider the following:
1 There exist a constant 0 < ι < ι , a constant u1 such that 0 < u1 < u0 (depending
on u0 , S and b, σ), a constant x1 such that 0 < x1 < ρ (depending on S, u0 , S, b, σ,
s,k ,k x0 , where x0 < ρ), and a constant
|λ|, maxs,k0 ,k1 ∈S |b|s,k0 ,k1 x0 , maxs,k0 ,k1 ∈S |b|
0 1
50
Abstract and Applied Analysis
i > 0 (depending on max0≤j≤S−1 Wj,Sd ,Ei u0 (where Wj,Sd ,Ei are defined above), |λ|,
C
i
i
s,k ,k x0 , S, u0 , x0 , S, b) such that
maxs,k ,k ∈S |b|s,k ,k x0 , maxs,k ,k ∈S |b|
0
1
0
1
0
1
0
1
i h!
|Xh,i t, z, | ≤ 2C
sup
t∈T∩D0,ι ,z∈D0,x1 /4
2
u1
h
4.78
for all ∈ Ei , for all 0 ≤ i ≤ ν − 1, and all h ≥ 0.
2 There exist a constant 0 < ι ≤ ι , a constant u1 such that 0 < u1 < u0 (depending
on u0 , S and b, σ), a constant x1 such that 0 < x1 < ρ (depending on S, u0 , S, b,
s,k ,k x0 , where x0 < ρ), a constant
σ, |λ|, maxs,k0 ,k1 ∈S |b|s,k0 ,k1 x0 , maxs,k0 ,k1 ∈S |b|
0 1
Mi > 0, a constant Ki > 0 (depending on max0≤j≤S−1 Wj,Sdq ,Eq u0 , for q i, i 1
(where Wj,Sdq ,Eq are defined above), max0≤j≤S−1 Wj,τ0 ,0 u0 , |λ|, maxs,k0 ,k1 ∈S |b|s,k0 ,k1 x0 ,
s,k ,k x0 , S, u0 , x0 , S, b) such that
maxs,k ,k ∈S |b|
0
1
0
1
sup
|Xh,i1 t, z, − Xh,i t, z, | ≤ h!
t∈T∩D0,ι ,z∈D0,x1 /4
2
u1
h
2Ki e−Mi /||
4.79
for all ∈ Ei ∩ Ei1 , for all 0 ≤ i ≤ ν − 1, and all h ≥ 0 (where by convention Xh,ν Xh,0 ).
Proof. 1 Let i be an integer such that 0 ≤ i ≤ ν − 1. From Proposition 4.12, we can write
−1
Xh,i t, z, t
Lγi
Vh,Sdi ,Ei τ, z, e−τ/t dτ,
4.80
√
where Lγi R e −1γi ⊂ Sdi ∪ {0} and Vh,Sdi ,Ei is a holomorphic function on Sdi ∪ D0, τ0 ×
D0, x1 /4 × Ei for which the estimates 4.62 hold. By construction, the direction γi which
depends on t is chosen in such a way that cosγi − argt ≥ δ1 , for all ∈ Ei , all t ∈
T ∩ D0, ι , and for some fixed δ1 > 0. From the estimates 4.62, we get
−1
h CΩdi ,Ei r2
2
1 2
h!
eσζbr/2|| e−r/|||t| cosγi −argt dr
|Xh,i t, z, | ≤ |t|
1
−
2|z|/x
u
1
1
||
0
∞
h
CΩdi ,Ei 2
≤ |t|−1
h!
eσζb/2−δ1 /|t|r/|| dr
1
−
2|z|/x
u
1
1
0
h
h
CΩdi ,Ei CΩdi ,Ei 1
2
2
≤
h!
h!
1 − 2|z|/x1
u1
δ1 − σζb/2|t| δ2 1 − 2|z|/x1 u1
4.81
−1
∞
for all t ∈ T∩D0, ι , with |t| < 2δ1 −δ2 /σζb, for some 0 < δ2 < δ1 , and for all ∈ Ei1 ∩Ei .
Abstract and Applied Analysis
51
2 Let i an integer such that 0 ≤ i ≤ ν − 1. From Proposition 4.12, we can write again
−1
Xh,i t, z, t
Xi1 t, z, t−1
Lγi
Lγi1
√
Vh,Sdi ,Ei τ, z, e−τ/t dτ,
4.82
Vh,Sdi1 ,Ei1 τ, z, e−τ/t dτ,
√
where Lγi R e −1γi ⊂ Sdi ∪ {0}, Lγi1 R e −1γi1 ⊂ Sdi1 ∪ {0}, and Vh,Sdi ,Ei resp., Vh,Sdi1 ,Ei1 is a holomorphic function on Sdi ∪ D0, τ0 × D0, x1 /4 × Ei resp., on Sdi1 ∪ D0, τ0 ×
D0, x1 /4 × Ei1 for which the estimates 4.62 hold and which is moreover an analytic
continuation of a function Vh τ, z, which satisfies the estimates 4.63.
From the fact that τ → Vh τ, z, is holomorphic on D0, τ0 for all z, ∈ D0, x1 /4×
starting from
0
D0, 0 \√{0}, the integral of τ → Vh τ, z, along the union of a segment
√
√
to τ0 /2e −1γi1 , an arc of circle with√ radius τ0 /2 connecting τ0 /2e −1γi1 and τ0 /2e −1γi
and a segment starting from τ0 /2e −1γi to 0 is equal to zero. Therefore, we can rewrite the
difference Xh,i1 − Xh,i as a sum of three integrals:
Xh,i1 t, z, − Xh,i t, z, t
−1
Lτ0 /2,γi1
Vh,Sdi1 ,Ei1 τ, z, e−τ/t dτ
−
Lτ0 /2,γi
Vh,Sdi ,Ei τ, z, e−τ/t dτ
√
Cτ0 /2,γi ,γi1 4.83
Vh τ, z, e
−τ/t
dτ ,
√
where Lτ0 /2,γi τ0 /2, ∞e −1γi , Lτ0 /2,γi1 τ√0 /2, ∞e −1γi1 , and
Cτ0 /2, γi , γi1 is an
√
arc of circle with radius τ0 /2 connecting τ0 /2e −1γi with τ0 /2e −1γi1 with a well-chosen
orientation.
We give estimates for I1 |t−1 Lτ /2,γ Vh,Sdi1 ,Ei1 τ, z, e−τ/t dτ|. By construction,
0
i1
the direction γi1 which depends on t is chosen in such a way that cosγi1 − argt ≥ δ1 ,
for all ∈ Ei1 ∩ Ei , all t ∈ T ∩ D0, ι, and for some fixed δ1 > 0. From the estimates 4.62,
we get
I1 ≤
≤
≤
−1
h CΩdi1 ,Ei1 r2
2
1 2
h!
eσζbr/2|| e−r/|||t| cosγi1 −argt dr
|t|
1
−
2|z|/x
u
1
1
||
τ0 /2
∞
h
CΩdi1 ,Ei1 2
h!
eσζb/2−δ1 /|t|r/|| dr
|t|−1
1
−
2|z|/x
u
1
1
τ0 /2
h −δ1 /|t|−σζb/2τ0 /21/||
CΩdi1 ,Ei1 e
2
h!
1 − 2|z|/x1
u1
δ1 − σζb/2|t|
h
CΩdi1 ,Ei1 2
h!
e−δ2 τ0 /2/||ι
δ2 1 − 2|z|/x1 u1
−1
∞
4.84
for all t ∈ T∩D0, ι , with |t| < 2δ1 −δ2 /σζb, for some 0 < δ2 < δ1 , and for all ∈ Ei1 ∩Ei .
52
Abstract and Applied Analysis
We give estimates for I2 |t−1 Lτ /2,γ Vh,Sdi ,Ei τ, z, e−τ/t dτ|. By construction, the
0
i
direction γi which depends on t is chosen in such a way that there exists a fixed δ1 > 0 with
cosγi − argt ≥ δ1 , for all ∈ Ei1 ∩ Ei and all t ∈ T ∩ D0, ι . From the estimates 4.62, we
deduce as before that
h
CΩdi ,Ei 2
I2 ≤
h!
e−δ2 τ0 /2/||ι
δ2 1 − 2|z|/x1 u1
4.85
for all t ∈ T∩D0, ι , with |t| < 2δ1 −δ2 /σζb, for some 0 < δ2 < δ1 , and for all ∈ Ei1 ∩Ei .
Finally, we get estimates for I3 |t|−1 | Cτ0 /2,γi ,γi1 Vh τ, z, e−τ/t dτ|. From the
estimates 4.63, we have
−1
γi1
h 2
C
Ω
/2
τ
2
τ0
τ0 ,0
σζbτ0 /4|| −τ0 /2|||t| cosθ−argt 0
.
1
dθ
I3 ≤ |t|−1 h!
e
e
2
u1
2 ||
γi 1 − 2|z|/x1
4.86
By construction, the arc of circle Cτ0 /2, γi , γi1 is chosen in such a way that that cosθ −
argt ≥ δ1 for all θ ∈ γi , γi1 if γi < γi1 , θ ∈ γi1 , γi if γi1 < γi for all t ∈ T, all
∈ Ei ∩ Ei1 . From 4.86, we deduce that
I3 ≤ γi1 − γi ≤ γi1 − γi CΩτ0 ,0
1 − 2|z|/x1
CΩτ0 ,0
h!
2
u1
2
h!
1 − 2|z|/x1
u1
h
h
τ0 1 −δ1 /|t|−σζb/2τ0 /21/||
e
2 |t|
4.87
τ0 1 −δ2 τ0 /4/|t| −δ2 τ0 /4/||ι
e
e
2 |t|
for all t ∈ T∩D0, ι , with |t| < 2δ1 −δ2 /σζb, for some 0 < δ2 < δ1 , and for all ∈ Ei1 ∩Ei .
Using the inequality 4.87 and the estimates 2.41, we deduce that
I3 ≤ γi1 − γi CΩτ0 ,0
1 − 2|z|/x1
h!
2
u1
h
2e−1 −δ2 τ0 /4/||ι
e
δ2
4.88
for all t ∈ T ∩ D0, ι , with |t| < 2δ1 − δ2 /σζb, and for all ∈ Ei1 ∩ Ei .
Finally, collecting the inequalities 4.84, 4.85, and 4.88, we deduce from 4.83, that
|Xi1 t, z, − Xi t, z, |
2e−1 −δ2 τ0 /4/||ι
h!2/u1 h CΩdi1 ,Ei CΩdi ,Ei −δ2 τ0 /2/||ι ≤
e
γi1 − γi CΩτ0 ,0
e
1 − 2|z|/x1
δ2
δ2
4.89
for all t ∈ T ∩ D0, ι , with |t| < 2δ1 − δ2 /σζb, for some 0 < δ2 < δ1 , for all ∈ Ei1 ∩ Ei ,
and for all 0 ≤ i ≤ ν − 1. Hence, the estimates 4.79 hold.
Now, let us fix h ≥ 0. For all 0 ≤ i ≤ ν − 1, we define Gh,i : t, z → Xh,i t, z, ,
which is, by Lemma 4.16, a holomorphic and bounded function from Ei into the Banach space
Abstract and Applied Analysis
53
E OT ∩ D0, ι × D0, x1 /4 of holomorphic and bounded functions on the set
T ∩ D0, ι × D0, x1 /4 equipped with the supremum norm. Therefore, the property 1 of
Theorem RS is satisfied for the functions Gh,i , 0 ≤ i ≤ ν − 1. From the estimates 4.79, we
get that the cocycle Δi Gh,i1 − Gh,i is exponentially flat of order 1 on Zi Ei1 ∩ Ei for
all 0 ≤ i ≤ ν − 1. We deduce that the property 2 of Theorem RS is fulfilled for the functions
Gh,i , 0 ≤ i ≤ ν − 1. From Theorem RS, we get that Gh,i are the 1-sums of a formal series
h with coefficients in E. In particular, from Definition 4.1, we deduce the existence of the
G
functions gh,i s, t, z which satisfy the expression 4.76.
4.7. Analytic Transseries Solutions for a Singularly Perturbed
Cauchy Problem
We keep the notations of the previous section.
Proposition 4.17. The following singularly perturbed Cauchy problem
t2 ∂t ∂Sz Z0 t, z, t 1∂Sz Z0 t, z, s,k0 ,k1 ∈S
bs,k0 ,k1 z, ts ∂kt 0 ∂kz1 Z0 t, z, 4.90
for given initial data
exp−hλ/t
j
∂z Z0 t, 0, γ0,j t, ϕh,0,j t, ,
h!
h≥0
0 ≤ j ≤ S − 1,
4.91
which are holomorphic and bounded functions on T ∩ D0, ι × E0 ∩ E1 , has a solution
Z0 t, z, exp−hλ/t
h!
h≥0
Xh,0 t, z, ,
4.92
which defines a holomorphic and bounded function on T ∩ D0, ι × D0, δZ0 × E0 ∩ E1 , for some
ι , δZ0 > 0.
j
Proof. Let h ≥ 0 and 0 ≤ j ≤ S − 1. By construction, we have that ϕh,0,j t, ∂z Xh,0 t, 0, for all t ∈ T and all ∈ E0 . From Lemma 4.16, 1, we get that there exist a constant
ι > 0, a constant u1 such that 0 < u1 < u0 depending on u0 , S and b, σ, and a
constant Č0 > 0 depending on max0≤j≤S−1 Wj,Sd0 ,E0 u0 where Wj,Sd0 ,E0 are defined above,
s,k ,k x0 , S, u0 , x0 , S, b such that
|λ|, maxs,k ,k ∈S |b|s,k ,k x0 , maxs,k ,k ∈S |b|
0
1
0
1
0
sup
1
t∈T∩D0,ι 0
1
ϕh,0,j t, ≤ h!
2
u1
h
Č0
4.93
54
Abstract and Applied Analysis
for all ∈ E0 , all 0 ≤ j ≤ S − 1, all h ≥ 0. From 4.93 and from the property 3 of
Definition 4.14, we deduce the estimates
2 exp−|λ|/0 ι cosπ/2 − δT h
γ0,j t, ≤ Č0
,
u1
t∈T∩D0,ι h≥0
sup
4.94
for all ∈ E0 ∩ E1 . This latter sum converges provided that 0 is small enough. We deduce that
γ0,j t, defines a holomorphic and bounded function on T ∩ D0, ι × E0 ∩ E1 .
Likewise, from 4.78 and from the property 3 of Definition 4.14, we deduce that
there exist a constant ι > 0, a constant u1 such that 0 < u1 < u0 depending on
u0 , S and b, σ, a constant x1 such that 0 < x1 < ρ depending on S, u0 , S, b,
s,k ,k x0 , where x0 < ρ and a constant
σ, |λ|, maxs,k0 ,k1 ∈S |b|s,k0 ,k1 x0 , maxs,k0 ,k1 ∈S |b|
0 1
0 > 0 depending on max0≤j≤S−1 Wj,Sd ,E0 u0 where Wj,Sd ,E0 are defined above, |λ|,
C
0
0
s,k ,k x0 , S, u0 , x0 , S, b such that
maxs,k ,k ∈S |b|s,k ,k x0 , maxs,k ,k ∈S |b|
0
1
0
sup
1
t∈T∩D0,ι ,z∈D0,δZ0
0
1
0
1
2 exp−|λ|/0 ι cosπ/2 − δT h
0
C
|Z0 t, z, | ≤
1 − 2δZ0 /x1 h≥0
u1
4.95
for all ∈ E0 ∩ E1 . Again, this latter sum converges if 0 is small enough and if 0 < δZ0 ≤
x1 /4. We get that Z0 t, z, defines a holomorphic and bounded function on T ∩ D0, ι ×
j
D0, δZ0 × E0 ∩ E1 . By construction, we have that ∂z Z0 t, 0, γ0,j t, , for 0 ≤ j ≤ S − 1.
Finally, from Proposition 4.3, we deduce that Z0 t, z, solves 4.90.
5. Parametric Stokes Relations and Analytic Continuation of
the Borel Transform in the Perturbation Parameter
5.1. Assumptions on the Initial Data
We keep the notations of the previous section. Now, we make the following additional
assumption that there exists a sequence of unbounded open sectors Sd0 ,ϑn such that
Sd0 ⊂ Sd0 ,ϑn ⊂ Md0 ∪ Sd0
5.1
for all n ≥ 0 and a sequence of real numbers ζn , n ≥ 0 such that
eiζn ∈ Sd0 ,ϑn ,
lim ζn argλ
n → ∞
5.2
with the property that argeiζn /t ∈ −π/2 δT , π/2 − δT for all ∈ E0 ∩ E1 , all t ∈ T, and for
all n ≥ 0 where T and δT were introduced in Definition 4.14. We also make the assumption
that for all n ≥ 0, the function vh,j,Sd0 ,E0 τ, can be analytically continued to a holomorphic
function τ → vh,j,Sd0 ,ϑn ,E0 τ, on Sd0 ,ϑn for all ∈ E0 such that
vh,j,Sd0 ,ϑn ,E0 τ, ∈ Ej,,σ,Sd0 ,ϑn ∪D0,τ0 ×E0
5.3
Abstract and Applied Analysis
55
with the property that
Wj,Sd0 ,ϑn ,E0 u :
sup
vh,j,Sd0 ,ϑn ,E0 τ, uh
∈ C{u},
j,,σ,Sd0 ,ϑn ∪D0,τ0 ×E0 h!
h≥0 ∈E0
0≤j ≤S−1
5.4
and has a common radius of absolute convergence denoted by uE0 > 0 for all n ≥ 0. From the
assumption 5.4, we get a constant u0,j > 0 depending on j ∈ {0, . . . , S − 1} and a constant
Cn,j > 0 depending on n and j ∈ {0, . . . , S − 1} such that
sup
vh,j,Sd0 ,ϑn ,E0 τ, j,,σ,Sd0 ,ϑn ∪D0,τ0 ×E0
∈E0
≤ Cn,j
1
u0,j
h
h!
5.5
for all h ≥ 0. We deduce that
vh,j,Sd0 ,ϑn ,E0 reiζn , ≤ Cn,j
1
u0,j
h
h! exp
σ rb j r
2||
5.6
for all r ≥ 0, all ∈ E0 , all 0 ≤ j ≤ S − 1, and all h ≥ 0. In particular, we have that
r → vh,j,Sd0 ,ϑn ,E0 reiζn , belongs to the space L0,σ /2, for σ > σrb S − 1. Moreover, from
and that
Proposition 2.7, we deduce that r → vh,j,Sd0 ,ϑn ,E0 reiζn , belongs to the space D0,
σ ,
there exists a universal constant C1 > 0 such that
vh,j,Sd0 ,ϑn ,E0 reiζn , ≤ C1 vh,j,Sd0 ,ϑn ,E0 reiζn , 0,
σ ,,d
0,
σ /2,
2||
≤
C1 Cn,j
σ − σrb S − 1
1
u0,j
h
5.7
h!
for all 0 ≤ j ≤ S − 1, all h ≥ 0, and all n ≥ 0, all ∈ E0 .
We make the crucial assumption that for all 0 ≤ j ≤ S − 1, there exists a sequence of
, for h ≥ 0, a constant uj > 0 and a sequence In,j > 0 with
distributions vh,j,Md0 ,E0 r, ∈ D0,
σ ,
limn → ∞ In,j 0 such that
sup
vh,j,Sd0 ,ϑn ,E0 reiζn , − vh,j,Md0 ,E0 r, ∈E0
0,
σ ,,d
≤ In,j h!
1
uj
h
5.8
for all n ≥ 0 and all h ≥ 0. From the estimates 5.7 and 5.8, we deduce that
sup
vh,j,Md0 ,E0 r, h≥0 ∈E0
uh
∈ C{u},
0,
σ ,,d h!
0 ≤ j ≤ S − 1.
5.9
56
Abstract and Applied Analysis
Lemma 5.1. Let σ > σrb S−1. One can write the initial data γ0,j t, in the form of a Laplace transform in direction argλ,
γ0,j t, Largλ Vj,argλ,Sd0 ,E0 r, t,
5.10
where Vj,argλ,Sd0 ,E0 r, ∈ D0,
and for all 0 ≤ j ≤ S − 1, and all ∈ E0 ∩ E1 , all t ∈ T ∩ D0, ι .
σ ,
Proof. For 0 ≤ j ≤ S − 1, from the definition of the initial data, we can write
exp−hλ/t 1 τ
dτ
vh,j,Sd0 ,ϑn ,E0 τ, exp −
γ0,j t, h!
t Lζn
t
h≥0
exp − h|λ|ei argλ /t eiζn ∞
eiζn
iζn
dr
vh,j,Sd0 ,ϑn ,E0 re , exp −r
h!
t 0
t
h≥0
5.11
for all ∈ E0 ∩ E1 , all t ∈ T ∩ D0, ι , and all n ≥ 0. Now, we can write
Lζn vh,j,Sd0 ,ϑn ,E0 reiζn , t Largλ vh,j,Sd0 ,ϑn ,E0 reiζn , teiargλ−ζn 5.12
for all ∈ E0 ∩ E1 , all t ∈ T ∩ D0, ι, all n ≥ 0. From the continuity estimates 3.3 for the
Laplace transform, we deduce that for given t ∈ T ∩ D0, ι, ∈ E0 ∩ E1 , there exists a constant
C,t depending on , t such that
teiargλ−ζn Largλ vh,j,Md0 ,E0 r, t − Largλ vh,j,Sd0 ,ϑn ,E0 reiζn , ≤ C,t vh,j,Md0 ,E0 r, − vh,j,Sd0 ,ϑn ,E0 reiζn , 0,
σ ,,d
5.13
Largλ vh,j,Md0 ,E0 r, teiargλ−ζn − Largλ vh,j,Md0 ,E0 r, t
for all n ≥ 0. By letting n tend to ∞ in this latter inequality and using the hypothesis 5.8,
we get that
Lζn vh,j,Sd0 ,ϑn ,E0 reiζn , t Largλ vh,j,Md0 ,E0 r, t
5.14
for all ∈ E0 ∩ E1 , all t ∈ T ∩ D0, ι , and all n ≥ 0.
On the other hand, from Corollary 2.10, we have that for all h ≥ 0, the distribution
v
and that there exists a universal constant C3 > 0 such
∂−h
h,j,Md0 ,E0 r, belongs to D0,
r
σ ,
that
−h
∂r vh,j,Md0 ,E0 r, 0,
σ ,,d
≤ C3
||
σ
h vh,j,Md0 ,E0 r, 0,
σ ,,d
5.15
Abstract and Applied Analysis
57
for all h ≥ 0, all 0 ≤ j ≤ S − 1. From 5.14 and using Propositions 3.3 and 3.7, we can write
exp −h|λ|ei argλ /t eiζn ∞
eiζn
iζn
vh,j,Sd0 ,ϑn ,E0 re , exp −r
dr
h!
t 0
t
h
exp −h|λ|ei argλ /t
ei argλ
Largλ ∂−h
vh,j,Md0 ,E0 r, t
r
t
h!
Largλ Vh,j,λ,Md0 ,E0 r, t,
5.16
where
Vh,j,λ,Md0 ,E0 r, fh,j,λ,Md0 ,E0 r − |λ|h, 1|λ|h,∞ r
h
h!
∈ D0,
σ ,
5.17
with fh,j,λ,Md0 ,E0 r, ∂−h
for all h ≥ 0 and all 0 ≤ j ≤ S − 1. From
r vh,j,Md0 ,E0 r, ∈ D0,
σ ,
Proposition 3.6, we have a universal constant A > 0 and a constant B
σ , b, depending on
σ , b, and , which tend to zero as → 0 such that
Vh,j,λ,Md0 ,E0 r, 0,
σ ,,d
≤A
σ , b, h B
.
fh,j,λ,Md0 ,E0 r, 0,
σ ,,d
h!
5.18
From the estimates 5.9 and using 5.15, 5.18, we deduce that the distribution
Vj,argλ,Sd0 ,E0 r, Vh,j,λ,Md0 ,E0 r, ∈ D0,
σ ,,d
h≥0
5.19
for all 0 ≤ j ≤ S − 1, if 0 > 0 is chosen small enough. Finally, by the continuity estimates
3.3 for the Laplace transform Largλ and the formula 5.11, 5.16, we get the expression
5.10.
On the other hand, we assume the existence of a sequence of unbounded open sectors
Sd1 ,δn with
Sd1 ⊂ Sd1 ,δn ⊂ Md1 ∪ Sd1
5.20
for all n ≥ 0 and a sequence of real numbers ξn , n ≥ 0 such that
eiξn ∈ Sd1 ,δn ,
lim ξn argλ
n → ∞
5.21
with the property that argeiξn /t ∈ −π/2 δT , π/2 − δT for all ∈ E0 ∩ E1 , all t ∈ T, and all
n ≥ 0 where T and δT are introduced in Definition 4.14. We make the assumption that for
58
Abstract and Applied Analysis
all n ≥ 0, the function vh,j,Sd1 ,E1 τ, can be analytically continued to a holomorphic function
τ → vh,j,Sd1 ,δn ,E1 τ, on Sd1 ,δn for all ∈ E1 such that
vh,j,Sd1 ,δn ,E1 τ, ∈ Ej,,σ,Sd1 ,δn ∪D0,τ0 ×E1
5.22
with the property that
Wj,Sd1 ,δn ,E1 u :
sup
vh,j,Sd1 ,δn ,E1 τ, uh
∈ C{u},
j,,σ,Sd1 ,δn ∪D0,τ0 ×E1 h!
h≥0 ∈E1
0≤j ≤S−1
5.23
and has a common radius of absolute convergence defined by uE1 > 0 for all n ≥ 0. From the
assumption 5.23, we get a constant u1,j > 0 depending on j ∈ {0, . . . , S − 1} and a constant
Cn,1,j > 0 depending on n and j ∈ {0, . . . , S − 1} such that
sup
vh,j,Sd1 ,δn ,E1 τ, j,,σ,Sd1 ,δn ∪D0,τ0 ×E1
∈E1
≤ Cn,1,j
1
u1,j
h
h!
5.24
for all h ≥ 0. We deduce that
vh,j,Sd1 ,δn ,E1 reiξn , ≤ Cn,1,j
1
u1,j
h
h! exp
σ rb j r
2||
5.25
for all r ≥ 0, all ∈ E1 , all 0 ≤ j ≤ S − 1, and all h ≥ 0. In particular, we have that
r → vh,j,Sd1 ,δn ,E1 reiξn , belongs to the space L0,σ /2, for σ > σrb S − 1. Moreover, from
and that
Proposition 2.7, we deduce that r → vh,j,Sd1 ,δn ,E1 reiξn , belongs to the space D0,
σ ,
there exists a universal constant C1 > 0 such that
vh,j,Sd1 ,δn ,E1 reiξn , 0,
σ ,,d
≤ C1 vh,j,Sd1 ,δn ,E1 reiξn , 0,
σ /2,
2||
C1 Cn,1,j
≤
σ − σrb S − 1
1
u1,j
h
5.26
h!
for all 0 ≤ j ≤ S − 1, all h ≥ 0, all n ≥ 0, and all ∈ E1 .
Now, we make the crucial assumption that for all 0 ≤ j ≤ S − 1, there exists a sequence
Jn,j > 0 with limn → ∞ Jn,j 0 such that
sup v0,j,Sd1 ,δn ,E1 reiξn , − Vj,argλ,Sd0 ,E0 r, ∈E0 ∩E1
0,
σ ,,d
≤ Jn,j
for all n ≥ 0, where Vj,argλ,Sd0 ,E0 r, are the distributions defined in Lemma 5.1.
5.27
Abstract and Applied Analysis
59
5.2. The Stokes Relation and the Main Result
In the next proposition, we establish a connection formula for the two holomorphic solutions
X0,0 t, z, and X0,1 t, z, of 4.90 constructed in Proposition 4.15.
Proposition 5.2. Let the assumptions 5.1, 5.4, 5.8, 5.20, 5.23, and 5.27 hold for the initial
data. Then, there exists 0 < δD0,1 < δZ0 such that one can write the following connection formula:
X0,1 t, z, Z0 t, z, X0,0 t, z, exp−hλ/t
h!
h≥1
Xh,0 t, z, 5.28
for all ∈ E0 ∩ E1 , all t ∈ T ∩ D0, ι , and all z ∈ D0, δD0,1 .
The proof of this proposition will need two long steps and will be the consequence of
the formula 5.79 and 5.124 from Lemmas 5.5 and 5.7.
Step 1. In this step, we show that the function Z0 t, z, can be expressed as a Laplace
transform of some staircase distribution in direction argλ satisfying the problem 5.80,
5.81.
From the assumption 5.4, we deduce from Proposition 4.12 that the function
Vh,Sd0 ,E0 τ, z, constructed in 4.80 has an analytic continuation denoted by Vh,Sd0 ,ϑn ,E0 τ, z, on the domain Sd0 ,ϑn ∪ D0, τ0 × D0, δE0 × E0 which satisfies estimates of the form 4.62 for
all n ≥ 0, where δE0 > 0 depends on S, uE0 which denotes a common radius of convergence
s,k ,k x0 , where x0 < ρ.
of the series 5.4, S, b, σ, |λ|, maxs,k0 ,k1 ∈S |b|s,k0 ,k1 x0 , maxs,k0 ,k1 ∈S |b|
0 1
This constant δE0 is, therefore, independent of n and h. Now, one defines the functions
Vh,Sd0 ,ϑn ,E0 r, z, : Vh,Sd0 ,ϑn ,E0 reiζn , z, 5.29
for all r ≥ 0, all z ∈ D0, δE0 , all ∈ E0 , and all n ≥ 0.
Lemma 5.3. Let σ̌ > σ > σrb S − 1. Then, there exists 0 < δD < δE0 (depending on S, b, σ̌, |λ|, uj ,
0 ≤ j ≤ S − 1 (introduced in 5.8), S, uE0 , ρ, μ, A, B (introduced in Lemma 5.4)), there exist M1 > 0
(depending on S, S, σ̌, |λ|, uj , for 0 ≤ j ≤ S − 1, ρ, μ, A, B), M1 > 0 (depending on S, S, σ̌, |λ|, ρ, μ,
ρ , μ (introduced in Lemma 5.4), A, B, uj for 0 ≤ j ≤ S − 1) and a constant U1 (depending on S, S,
σ̌, |λ|, ρ, μ, A, B, uE0 , uj for 0 ≤ j ≤ S − 1) such that for all h ≥ 0 all n ≥ 0, there exists a staircase
distribution Vh,Md0 ,E0 r, z, ∈ D σ̌, , δD with
sup
Vh,Sd0 ,ϑn ,E0 r, z, − Vh,Md0 ,E0 r, z, ∈E0
σ̌,,d,δD ≤
M1 max In,j 0≤j≤S−1
M1 Dn
h
2
h!
,
U1
5.30
60
Abstract and Applied Analysis
where In,j is a positive sequence (converging to 0 as n tends to ∞) introduced in the assumption 5.8
and Dn is the positive sequence (tending to 0 as n → ∞) introduced in Lemma 5.4. Moreover, one
has
sup
Vh,Md0 ,E0 r, z, h≥0 ∈E0
uh
∈ C{u}.
σ̌,,d,δD h!
5.31
Proof. From the estimates 4.54, we can write
Vh,Sd0 ,ϑn ,E0 τ, z, zβ
Vh,β,Sd0 ,ϑn ,E0 τ, ,
β!
β≥0
5.32
where Vh,β,Sd0 ,ϑn ,E0 τ, are holomorphic functions such that there exists a constant u1 such that
0 < u1 < uE0 depending on uE0 , S, and b, σ, a constant x1 such that 0 < x1 < ρ depending
s,k ,k x0 , where x0 < ρ, and a
on S, uE0 , S, b, σ, |λ|, maxs,k0 ,k1 ∈S |b|s,k0 ,k1 x0 , maxs,k0 ,k1 ∈S |b|
0 1
constant CΩd0 ,E0 ,n > 0 depending on max0≤j≤S−1 Wj,Sd0 ,ϑn ,E0 uE0 where Wj,Sd0 ,ϑn ,E0 are defined
s,k ,k x0 , S, uE , x0 , S, b with
in 5.4, |λ|, maxs,k ,k ∈S |b|s,k ,k x0 , maxs,k ,k ∈S |b|
0
1
0
1
0
1
0
1
0
−1
h β 2
σ 2
|τ|2
1 2
rb β |τ|
exp
Vh,β,Sd0 ,ϑn ,E0 τ, ≤ CΩd0 ,E0 ,n h!β!
u1
x1
2||
||
5.33
for all τ ∈ Sd0 ,ϑn ∪ D0, τ0 , ∈ E0 , all h ≥ 0, all β ≥ 0, and all n ≥ 0. We deduce that
h β
σ 2
2
iζn
r
re
,
≤
C
h!β!
exp
β
r
Vh,β,Sd0 ,ϑn ,E0
Ωd0 ,E0 ,n
b
u1
x1
2||
5.34
for all r ≥ 0, all ∈ E0 , all β ≥ 0, all h ≥ 0, and all n ≥ 0. In particular, r → Vh,β,Sd0 ,ϑn ,E0 reiζn , belongs to Lβ,σ̌/2, . From Proposition 2.7, we deduce that r → Vh,β,Sd0 ,ϑn ,E0 reiζn , belongs to
. From Proposition 2.7 and 5.34, we get a universal constant C1 > 0 such that
Dβ,
σ̌,
Vh,β,Sd0 ,ϑn ,E0 reiζn , β,σ̌,,d
≤ C1 Vh,β,Sd0 ,ϑn ,E0 reiζn , β,σ̌/2,
2||
≤ C1 CΩd0 ,E0 ,n
σ̌ − σ
2
u1
h 2
x1
5.35
β
h!β!
for all β ≥ 0, all h ≥ 0, and all n ≥ 0. From 5.35, we deduce that the distribution
Vh,Sd0 ,ϑn ,E0 r, z, zβ
∈ D σ̌, , δ̌
Vh,β,Sd0 ,ϑn ,E0 reiζn , β!
β≥0
for all ∈ E0 , all δ̌ < x1 /2, all h ≥ 0, and all n ≥ 0.
5.36
Abstract and Applied Analysis
61
One gets from 4.20, 4.21 and the assumption 4.44 that the following problem
holds:
reiζn 1 λh ∂Sz Vh,Sd0 ,ϑn ,E0 r, z, ⎛
⎜
k0 bs,k0 ,k1 z, k0 −s eis−k0 ζn ⎝
s,k0 ,k1 ∈S
⎞
1
m,p∈Os−k
0
−p
⎟
α1m,p r m ∂r ∂kz1 Vh,Sd0 ,ϑn ,E0 r, z, ⎠
5.37
1
k0
k0 ! k01
k0 c
b
z,
hλq k0 −s eis−k0 −qζn
s,k
,k
q
0
1
1 2
k
!k
!
k1 k2 k0 ,k1 ≥1 0 0 q1
0
0
⎛
0
⎞
⎜
×⎝
2
m,p∈Os−k
0 −q
2,q
−p
⎟
αm,p r m ∂r ∂kz1 Vh,Sd0 ,ϑn ,E0 r, z, ⎠
with initial data
j
∂z Vh,Sd0 ,ϑn ,E0 r, 0, vh,j,Sd0 ,ϑn ,E0 reiζn , ,
0 ≤ j ≤ S − 1.
5.38
On the other hand, we consider the problem
rei argλ 1 λh ∂Sz Vh,Md0 ,E0 r, z, ⎛
⎞
−p
⎜
⎟
k0 bs,k0 ,k1 z, k0 −s eis−k0 argλ ⎝
α1m,p r m ∂r ∂kz1 Vh,Md0 ,E0 r, z, ⎠
1
s,k0 ,k1 ∈S
m,p∈Os−k
0
1
k0
k0 ! k1
k0 bs,k0 ,k1 z, cq 0 hλq k0 −s eis−k0 −q argλ
1 2
k !k !
k1 k2 k0 ,k1 ≥1 0 0 q1
0
⎛
⎜
×⎝
0
0
⎞
2
m,p∈Os−k
5.39
2,q
−p
⎟
αm,p r m ∂r ∂kz1 Vh,Md0 ,E0 r, z, ⎠
0 −q
with initial data
j
∂z Vh,Md0 ,E0 r, 0, vh,j,Md0 ,E0 r, ,
0 ≤ j ≤ S − 1.
5.40
In the next lemma, we give estimates for the coefficients of 5.37 and 5.39.
Lemma 5.4. Let
bs,k0 ,k1 z, zβ
bs,k0 ,k1 ,β β!
β≥0
5.41
62
Abstract and Applied Analysis
the convergent Taylor expansion of bs,k0 ,k1 with respect to z near 0. Let α ∈ R be a real number. Then,
there exist positive constants A, B, ρ, ρ , μ, μ and a sequence Dn > 0 such that limn → ∞ Dn 0 with
iα argλ
β!q!
q bs,k0 ,k1 ,β e
∂r
≤ AB−β q1 ,
i
argλ
re
1 λh
ρ rμ
5.42
iαζn
β!q!
q bs,k0 ,k1 ,β e
∂r
≤ AB−β q1 ,
iζ
n
re 1 λh
ρ rμ
iα argλ
bs,k0 ,k1 ,β eiαζn β!q!
q
q bs,k0 ,k1 ,β e
− ∂r
∂r
≤ Dn B−β q1
iζ
i
argλ
n
re 1 λh
re
1 λh
ρ r μ
5.43
for all q ≥ 0, all β ≥ 0, all n ≥ 0, all h ≥ 0, all r ≥ 0, and all ∈ E0 .
Proof. We first show 5.42. From the fact that bs,k0 ,k1 z, is holomorphic near z 0, we get
from the Cauchy formula that there exist A, B > 0 such that
bs,k ,k ,β ≤ AB−β β!
0
5.44
1
for all β ≥ 0, and all ∈ E0 . On the other hand, from Definition 4.143.1, there exist ρ, μ > 0
such that |reiζn 1 λh| ≥ ρr μ for all r ≥ 0, all h ≥ 0, and all n ≥ 0. Hence,
q!
q!
eiαζn
q
≤ ∂r
≤ q1
reiζn 1 λh reiζn 1 λhq1
ρ rμ
5.45
for all r ≥ 0, and all h ≥ 0, all q ≥ 0, all n ≥ 0. We deduce 5.42 from 5.44 and 5.45.
Now, we show 5.43. Using the classical identities ab − cd a − cb cb − d and
q
bq1 − aq1 b − a × s0 as bq−s , we get the estimates
eiα argλ
eiαζn
q
q
−
∂
∂r
r
reiζn 1 λh rei argλ 1 λh
eiα argλ eiq argλ
eiαζn eiqζn
≤ q! −
rei argλ 1 λhq1 reiζn 1 λhq1 ≤ q! eiζn − ei argλ ×
q1
r
5.46
q2−s reiζn 1 λhs
1 λh
iα argλ
e
− eiαζn ei argλ − eiζn q 1
.
rei argλ 1 λhq1
s1 re
i argλ
On the other hand, again from Definition 4.14 3.1, there exist ρ1 , μ1 > 0 such that
i argλ
1 λh ≥ ρ1 r μ1 ,
re
iζn
re 1 λh ≥ ρ1 r μ1
5.47
Abstract and Applied Analysis
63
for all r ≥ 0, all h ≥ 0, and all n ≥ 0. Using 5.46, 5.47 and the fact that q 1 ≤ 2q1 for all
q ≥ 0, we deduce the estimates 5.43.
In the first part of the proof of Lemma 5.3, we show the existence of a staircase
distribution solution of the problem 5.39, 5.40, which satisfies the estimates 5.31. As
a starting point, it is easy to check that the problem 5.39, 5.40 has a formal solution of the
form
Vh,Md0 ,E0 r, z, zβ
Vh,β,Md0 ,E0 r, ,
β!
β≥0
5.48
where r → Vh,β,Md0 ,E0 r, are distributions on R , for which the next recursion holds:
Vh,βS,Md0 ,E0 r, β!
s,k0 ,k1 ∈S β1 β2 β
k0 bs,k0 ,k1 ,β1 k0 −s
β1 !
⎛
×
is−k0 argλ
e
⎜
⎝
rei argλ 1 λh
1
m,p∈Os−k
0
⎞
V
r,
h,β
k
,M
,E
2
1
d0 0
−p
⎟
α1m,p r m ∂r
⎠
β2 !
k0
k0 bs,k0 ,k1 ,β1 k01
k0 ! cq hλq
β!
1 2
β
!
k
!k
!
1
1
2
1
q1
β
β
β
1
2
k k k0 ,k ≥1 0 0
1
0
0
× k0 −s
0
⎛
is−k0 −q argλ
e
⎜
⎝
rei argλ 1 λh
2
m,p∈Os−k
0 −q
⎞
V
r,
2,q
−p h,β2 k1 ,Md0 ,E0
⎟
αm,p r m ∂r
⎠
β2 !
5.49
for all β ≥ 0, h ≥ 0, with initial conditions
Vh,j,Md0 ,E0 r, vh,j,Md0 ,E0 r, ,
0 ≤ j ≤ S − 1, h ≥ 0.
5.50
Using Corollary 2.10, Propositions 2.11 and 2.12, the estimates 5.9, and Remark 2.4, we
for all h, β ≥ 0 and that the following inequalities hold
deduce that Vh,β,Md0 ,E0 r, ∈ Dβ,
σ̌,
1
2
, C23.0
for the real numbers Vh,β,Md0 : ||Vh,β,Md0 ,E0 r, ||β,σ̌,,d : there exist constants C23.0
depending on S, σ̌, S, ρ, μ with
Vh,βS,Md0 ≤
s,k0 ,k1 ∈S β1 β2
s−k0 b Vh,β2 k1 ,Md0 1
C23.0
β!AB−β1 β S 1
β2 !
β
k01
1 q q
k0 ! 2
−β1 k0 C
β!AB
23.0
cq |λ| h
k1 !k2 !
k1 k2 k0 ,k1 ≥1 0 0 q1 β1 β2 β
0
0
0
s−k0 −qb Vh,β2 k1 ,Md0 × βS1
β2 !
5.51
64
Abstract and Applied Analysis
for all β, h ≥ 0, where A, B > 0 are defined in Lemma 5.4. We define the following Cauchy
problem:
∂Sx WMd0 u, x s,k0 ,k1 ∈S
1
C23.0
x∂x S 1bs−k0 A
∂k1 WMd0 u, x
1 − x/B x
k01
1 q
k0 ! bs−k0 −q
2 k0 C
23.0
cq |λ| x∂x S 1
1 2
k
!k
!
1
2
1
q1
0
0
k k k0 ,k ≥1
0
×
0
0
A
u∂u q ∂kx1 WMd0 u, x
1 − x/B
5.52
for given initial data
j
∂x WMd0 u, 0 WMd0 ,j u sup
vh,j,Md0 ,E0 r, h≥0 ∈E0
uh
∈ C{u},
j,σ̌,,d h!
0 ≤ j ≤ S − 1.
5.53
From the assumption 4.42 and the fact that b > 1, we deduce that
S > b s − k0 − q q k 1
5.54
for all s, k0 , k1 ∈ S and all 0 ≤ q ≤ k0 . Hence, the assumption 2.108 is satisfied in
Proposition 2.22 for the Cauchy problem 5.52, 5.53. Since the initial data WMd0 ,j u is
an analytic function on a disc containing some closed disc D0, U0 , for 0 ≤ j ≤ S − 1 and
since the coefficients of 5.52 are analytic on C × D0, B, we deduce that all the hypotheses
of Proposition 2.22 are fulfilled for the problem 5.52, 5.53. We deduce the existence of a
formal solution WMd0 u, x ∈ GUMd0 , XMd0 , where 0 < UMd0 < U0 depending on S and
0 < XMd0 ≤ B/2 depending on S, σ̌, |λ|, ρ, μ, U0 , S, A, B.
h
β
Now, let WMd0 u, x h,β≥0 wh,β,Md0 u /h!x /β! be its Taylor expansion at the
origin. Then, the sequence wh,β,Md0 satisfies the next equalities:
wh,βS,Md0 s,k0 ,k1 ∈S β1 β2
s−k0 b wh,β2 k1 ,Md0
1
C23.0
β!AB−β1 β S 1
β2 !
β
k01
1 q q
k0 ! 2
−β1 k0 C
β!AB
23.0
cq |λ| h
k1 !k2 !
k1 k2 k0 ,k1 ≥1 0 0 q1 β1 β2 β
0
0
5.55
0
s−k0 −qb wh,β2 k1 ,Md0
× βS1
β2 !
for all β, h ≥ 0, with
wh,j,Md0 sup
vh,j,Md0 ,E0 r, ∈E0
j,σ̌,,d
,
h ≥ 0, 0 ≤ j ≤ S − 1.
5.56
Abstract and Applied Analysis
65
Gathering the inequalities 5.51, the equalities 5.55 with the initial conditions 5.56, one
gets
supVh,β,Md0 ≤ wh,β,Md0
5.57
∈E0
for all h, β ≥ 0. From 5.57 and the fact that WMd0 u, x ∈ GUMd0 , XM0 , we get a constant
CMd0 > 0 such that
sup
Vh,β,Md0 ,E0 r, ∈E0
≤ CMd0
hβ !
β,σ̌,,d
h 1
UMd0
1
β
≤ CMd0 h!β!
XMd0
h 2
UMd0
2
β
5.58
XMd0
for all h, β ≥ 0. From this last estimates 5.58, we deduce that for all h ≥ 0, Vh,Md0 ,E0 r, z, belongs to D σ̌, , δMd0 for 0 < δMd0 ≤ XMd0 /4 and moreover that
sup
Vh,Md0 ,E0 r, z, h≥0 ∈E0
uh
∈ C{u}
σ̌,,d,δMd h!
0
5.59
holds. This yields the property 5.31.
In the second part of the proof, we show 5.30. One defines the distribution
VΔ
h,Sd ,ϑ
,E
0 n 0
r, z, : Vh,Md0 ,E0 r, z, − Vh,Sd0 ,ϑn ,E0 r, z, 5.60
for all r ≥ 0, all z ∈ D0, δMd0 ∩ D0, δ̌, with 0 < δ̌ < x1 /2, all ∈ E0 . If one writes the Taylor
expansion
VΔ
h,Sd ,ϑ
0
r, z, n ,E0
VΔ
h,β,Sd ,ϑ
β≥0
0
r, n ,E0
zβ
β!
5.61
66
Abstract and Applied Analysis
r, satisfy the following recurfor z ∈ D0, δMd0 ∩ D0, δ̌, then the coefficients VΔ
h,β,Sd0 ,ϑn ,E0
sion:
VΔ
h,βS,Sd0 ,ϑn ,E0 r, k0 bs,k0 ,k1 ,β1 k0 −s
β!
β1 !
β β
s,k0 ,k1 ∈S β1
⎛
2
1
m,p∈Os−k
eis−k0 ζn
reiζn 1 λh
⎞
Δ
V
r,
−p h,β2 k1 ,Sd0 ,ϑn ,E0
⎟
α1m,p r m ∂r
⎠
β2 !
⎜
×⎝
0
k0
k0 bs,k0 ,k1 ,β1 k01
k0 ! cq hλq
β!
1 2
β
!
k
!k
!
1
k1 k2 k0 ,k1 ≥1 0 0 q1 β1 β2 β
1
0
0
0
⎛
× k0 −s
is−k0 −qζn
e
⎜
×⎝
reiζn 1 λh
2
m,p∈Os−k
0 −q
⎞
Δ
V
r,
2,q
−p h,β2 k1 ,Sd0 ,ϑn ,E0
⎟
αm,p r m ∂r
⎠
β2 !
Bh,β,n r, ,
5.62
where
Bh,β,n r, k0 bs,k0 ,k1 ,β1 k0 −s
β!
β1 !
β β
s,k0 ,k1 ∈S β1
⎛
2
1
m,p∈Os−k
eis−k0 argλ
eis−k0 ζn
−
rei argλ 1 λh reiζn 1 λh
0
k0
k0 bs,k0 ,k1 ,β1 k01
k0 ! cq hλq
β!
1 2
β
!
k
!k
!
1
1
2
1
k k k0 ,k ≥1 0 0 q1 β1 β2 β
1
0
0
0
×
k0 −s
⎛
⎜
×⎝
⎞
V
r,
h,β
k
,M
,E
2
1
d0 0
−p
⎟
α1m,p r m ∂r
⎠
β2 !
⎜
×⎝
eis−k0 −q argλ
eis−k0 −qζn
−
rei argλ 1 λh reiζn 1 λh
2
m,p∈Os−k
0 −q
⎞
V
r,
2,q
−p h,β2 k1 ,Md0 ,E0
⎟
αm,p r m ∂r
⎠
β2 !
5.63
Abstract and Applied Analysis
67
||VΔ
r, ||β,σ̌,,d . Using
for all h ≥ 0 and all β ≥ 0. Now, we put VΔ
h,β,n
h,β,Sd0 ,ϑn ,E0
Corollary 2.10, Propositions 2.11 and 2.12, and Lemma 5.4, we get that there exist constants
1
2
C23.1
, C23.1
depending on S, σ̌, S, ρ, μ such that the following inequalities:
VΔ
h,βS,n ≤
s,k0 ,k1 ∈S β1 β2 β
1
C23.1
β!AB−β1
Δ
s−k0 b Vh,β2 k1 ,n βS1
β2 !
1
k0
k0 ! 2
C23.1
β!AB−β1
1 2
k
!k
!
k1 k2 k0 ,k1 ≥1 0 0 q1 β1 β2 β
0
0
5.64
0
Δ
k1 s−k0 −qb Vh,β2 k1 ,n Bh,β,n × cq 0 |λ|q hq β S 1
β2 !
hold for all h, β ≥ 0, where A, B > 0 are defined in Lemma 5.4 and Bh,β,n is a sequence which
3
4
, C23.1
> 0 depending on S,σ̌,S,ρ ,μ satisfies the next estimates: there exist constants C23.1
with
Bh,β,n ≤
s,k0 ,k1 ∈S β1 β2 β
3
C23.1
β!Dn B−β1
s−k0 b
× βS1
Vh,β2 k1 ,Md0 ,E0 r, β2 k1 ,σ̌,,d
β2 !
5.65
1
k0
k0 ! 4
C23.1
β!Dn B−β1
1 2
k
!k
!
1
2
1
k k k0 ,k ≥1 0 0 q1 β1 β2 β
0
0
0
r,
V
h,β
k
,M
,E
2
1
d0 0
k01 q q s−k0 −qb
β2 k1 ,σ̌,,d
× cq |λ| h β S 1
β2 !
for all h, β, n ≥ 0, where Dn , n ≥ 0 is the sequence defined in Lemma 5.4.
We consider the following sequence of Cauchy problem:
∂Sx WΔ
n u, x
s,k0 ,k1 ∈S
1
C23.1
x∂x
S 1
bs−k0 A
∂k1 WΔ u, x
1 − x/B x n
k01
1 q
k0 ! bs−k0 −q
2 k0 C
23.1 cq |λ| x∂x S 1
1 2
k
!k
!
1
2
1
k k k0 ,k ≥1 0 0 q1
0
×
0
0
A
q k1
Δ
u∂u ∂x Wn u, x Dn u, x,
1 − x/B
5.66
68
Abstract and Applied Analysis
where
Dn u, x s,k0 ,k1 ∈S
3
C23.1
x∂x S 1bs−k0 Dn
∂k1 WMd0 u, x
1 − x/B x
k01
1 q
k0 ! bs−k0 −q
4 k0 C
23.1 cq |λ| x∂x S 1
1 2
k
!k
!
1
2
1
k k k0 ,k ≥1 0 0 q1
0
×
0
0
Dn
u∂u q ∂kx1 WMd0 u, x
1 − x/B
5.67
and WMd0 u, x is already defined as the solution of the problem 5.52, 5.53, for given initial
data
j
Δ
∂x WΔ
n u, 0 Wj,n u
sup
vh,j,Md0 ,E0 r, − vh,j,Sd0 ,ϑn ,E0 reiζn , uh
∈ C{u},
j,σ̌,,d h!
h≥0 ∈E0
5.68
0 ≤ j ≤ S − 1,
which are convergent near the origin with respect to u due to the assumption 5.8 and
Remark 2.4. Moreover, the initial data satisfy the estimates
Δ
Wj,n u ≤
In,j
1 − |u|/uj
5.69
for all |u| < uj , 0 ≤ j ≤ S − 1, all n ≥ 0.
From the assumption 4.42 and the fact that b > 1, we deduce that
S > b s − k0 − q q k1
5.70
for all s, k0 , k1 ∈ S and all 0 ≤ q ≤ k0 . Therefore, the assumption 2.108 is satisfied in
Proposition 2.22 for the problem 5.66, 5.68.
On the other hand, from Lemmas 2.20 and 2.21, there exist a constant DMd0 > 0
depending on S, σ̌, S, ρ , μ , |λ|, B, UMd0 , XMd0 , a constant 0 < U1,Md0 < UMd0 , and a constant
0 < X1,Md0 < XMd0 such that
Dn u, xU1,M
d0
,X1,Md 0
≤ Dn DMd0 WMd0 u, x
UMd ,XMd 0
≤ Dn DMd0 CMd0
5.71
0
for all n ≥ 0, where the constant CMd0 is introduced in 5.58.
Since the initial data WΔ
j,n u is an analytic function on some disc containing the closed
disc D0, uj /2, for 0 ≤ j ≤ S − 1 and the coefficients of 5.66 are analytic on C × D0, B, we
deduce that all the hypotheses of Proposition 2.22 for the problem 5.66, 5.68 are fulfilled.
Abstract and Applied Analysis
69
We deduce the existence of a formal solution WΔ
n u, x ∈ GU1 , X1 of 5.66, 5.68, where
0 < U1 < minU1,Md0 , min0≤j≤S−1 uj /2 depending on S and 0 < X1 ≤ minB/2, X1,Md0 depending on S, σ̌, |λ|, uj , for 0 ≤ j ≤ S − 1, S, A, B, ρ, μ.
Moreover, from 2.111 and 5.71, there exist constants M1 > 0 depending on S, σ̌, λ,
uj , for 0 ≤ j ≤ S − 1, S, A, B, ρ, μ and M2 > 0 depending on S, uj for 0 ≤ j ≤ S − 1, B, S such
that
Δ
Wn u, x
U1 ,X1 for all n ≥ 0. Now, let WΔ
n u, x ≤ M1 max In,j Dn M2 DMd0 CMd0
h,β≥0
0≤j≤S−1
5.72
Δ
wh,β,n
uh /h!xβ /β! be its Taylor expansion at the
Δ
satisfies the following equalities:
origin. Then, the sequence wh,β,n
Δ
wh,βS,n
Δ
s,k0 ,k1 ∈S β1
s−k0 b wh,β2 k1 ,n
1
C23.1
β!AB−β1 β S 1
β2 !
β β
2
Δ
k01
1 q q
s−k0 −qb wh,β2 k1 ,n
k0 ! 2
−β1 k0 c
C
β!AB
h
β
S
1
|λ|
q
23.1
β2 !
k1 !k2 !
k1 k2 k0 ,k1 ≥1 0 0 q1 β1 β2 β
0
0
0
Dh,β,n ,
5.73
where
Dh,β,n s,k0 ,k1 ∈S β1 β2
s−k0 b wh,β2 k1 ,Md0
3
C23.1
β!Dn B−β1 β S 1
β2 !
β
k01
1
s−k0 −qb wh,β2 k1 ,Md0
k0 ! 4
−β1 k0 C23.1 β!Dn B cq |λ|q hq β S 1
1 2
β2 !
k !k !
k1 k2 k0 ,k1 ≥1 0 0 q1 β1 β2 β
0
0
0
5.74
for all h, β, n ≥ 0, with
Δ
wh,j,n
sup
vh,j,Md0 ,E0 r, − vh,j,Sd0 ,ϑn ,E0 reiζn , j,σ̌,,d
∈E0
,
∀h ≥ 0, ∀0 ≤ j ≤ S − 1 .
5.75
Gathering the inequalities 5.64, 5.65 and the equalities 5.73, with the initial conditions
5.75, one gets that
Δ
supVΔ
≤ wh,β,n
h,β,n
∈E0
for all h, β ≥ 0 and all n ≥ 0.
5.76
70
Abstract and Applied Analysis
From 5.76 and the estimates 5.72, we deduce that
Δ
sup
Vh,β,S
d ,ϑ
0
∈E0
r, n ,E0
β,σ̌,,d
≤
M1 max In,j
0≤j≤S−1
≤
M1 max In,j
0≤j≤S−1
1 h 1 β
Dn M2 DMd0 CMd0 h β !
U1
X1
h β
2
2
Dn M2 DMd0 CMd0 h!β!
U1
X1
5.77
for all h, β ≥ 0, all n ≥ 0. From 5.77, we get that
sup
Vh,Sd0 ,ϑn ,E0 r, z, − Vh,Md0 ,E0 r, z, σ̌,,d,δD ∈E0
≤
M1 max In,j
0≤j≤S−1
h
2
Dn M2 DMd0 CMd0 h!
U1
5.78
for all h ≥ 0 and all 0 < δD ≤ X1 /4. This yields the estimates 5.30.
In the next lemma, we express Z0 t, z, as a Laplace transform of a staircase
distribution.
Lemma 5.5. Let σ̌ > σ > σrb S − 1. Then, one can write the solution Z0 t, z, of 4.90, 4.91 in
the form of a Laplace transform in direction argλ
Z0 t, z, Largλ Vargλ,Sd0 ,E0 r, z, t
5.79
for all t, z, ∈ T ∩ D0, ι × D0, δD,Z0 × E0 ∩ E1 , where Vargλ,Sd0 ,E0 r, z, ∈ D σ̌, , δD,Z0 (with δD,Z0 minδD , δZ0 ) solves the following Cauchy problem:
rei argλ 1 ∂Sz Vargλ,Sd0 ,E0 r, z, ⎛
⎜
k0 −s bs,k0 ,k1 z, ⎝eis−k0 argλ
s,k0 ,k1 ∈S
1
m,p∈Os−k
0
⎞
−p
⎟
α1m,p r m ∂r ∂kz1 Vargλ,Sd0 ,E0 r, z, ⎠,
5.80
1
where the sets Os−k
and the integers α1m,p are introduced in 4.20, with initial data
0
j
∂z Vargλ,Sd0 ,E0 r, 0, Vj,argλ,Sd0 ,E0 r, ,
0 ≤ j ≤ S − 1.
5.81
Abstract and Applied Analysis
71
Proof. From Proposition 4.17, we can write the solution Z0 t, z, of 4.90, 4.91 in the form
Z0 t, z, exp−hλ/t 1 τ
dτ
Vh,Sd0 ,ϑn ,E0 τ, z, exp −
h!
t Lζn
t
h≥0
exp −h|λ|ei argλ /t eiζn ∞
eiζn
iζn
dr
Vh,Sd0 ,ϑn ,E0 re , z, exp −r
h!
t 0
t
h≥0
5.82
for all t, z, ∈ T ∩ D0, ι × D0, δZ0 × E0 ∩ E1 and all n ≥ 0. Now, we write
Lζn Vh,Sd0 ,ϑn ,E0 reiζn , z, t Largλ Vh,Sd0 ,ϑn ,E0 reiζn , z, teiargλ−ζn 5.83
for all t, z, ∈ T ∩ D0, ι × D0, δZ0 × E0 ∩ E1 and all n ≥ 0. Now, we define δD,Z0 minδD , δZ0 . From the continuity estimates 3.5 for the Laplace transform, we deduce that
for given ∈ E0 ∩ E1 , t ∈ T ∩ D0, ι , there exists a constant C,t depending on , t such that
teiargλ−ζn Largλ Vh,Md0 ,E0 r, z, t − Largλ Vh,Sd0 ,ϑn ,E0 reiζn , z, ≤ C,t Vh,Md0 ,E0 r, z, − Vh,Sd0 ,ϑn ,E0 reiζn , z, σ̌,,d,δD,Z0 5.84
Largλ Vh,Md0 ,E0 r, z, teiargλ−ζn − Largλ Vh,Md0 ,E0 r, z, t
for all z ∈ D0, δD,Z0 , all n ≥ 0. By letting n tend to ∞ in this latter inequality and using the
estimates 5.30, we obtain
Lζn Vh,Sd0 ,ϑn ,E0 reiζn , z, t Largλ Vh,Md0 ,E0 r, z, t
5.85
for all t, z, ∈ T ∩ D0, ι × D0, δD,Z0 × E0 ∩ E1 and all n ≥ 0.
On the other hand, from Corollary 2.10, we have that for all h ≥ 0, the distribution
∂−h
r Vh,Md0 ,E0 r, z, belongs to D σ̌, , δD,Z0 and that there exists a universal constant C3 > 0
such that
−h
∂r Vh,Md0 ,E0 r, z, for all h ≥ 0.
σ̌,,d,δD,Z0
≤ C3
||
σ̌
h Vh,Md0 ,E0 r, z, σ̌,,d,δD,Z0
5.86
72
Abstract and Applied Analysis
From 5.85 and using Propositions 3.3 and 3.7, we can write
exp −h|λ|ei argλ /t eiζn ∞
eiζn
iζn
Vh,Sd0 ,ϑn ,E0 re , z, exp −r
dr
h!
t 0
t
ei argλ
t
h
exp −h|λ|ei argλ /t
V
Largλ ∂−h
z,
r,
t
h,M
,E
r
d0 0
h!
5.87
Largλ Vh,λ,Md0 ,E0 r, z, t,
where
Vh,λ,Md0 ,E0 r, fh,λ,Md0 ,E0 r − |λ|h, z, 1|λ|h,∞ r
h!
h
∈ D σ̌, , δD,Z0 5.88
with fh,λ,Md0 ,E0 r, z, ∂−h
r Vh,Md0 ,E0 r, z, ∈ D σ̌, , δD,Z0 , for all h ≥ 0, all 0 ≤ j ≤
S − 1. From Proposition 3.6, we have a universal constant A > 0 and a constant Bσ̌, b, depending on σ̌, b, and , which tend to zero as → 0 such that
Vh,λ,Md0 ,E0 r, z, σ̌,,d,δD,Z0
≤A
Bσ̌, b, h .
fh,λ,Md0 ,E0 r, z, σ̌,,d,δD,Z0
h!
5.89
From the convergence of the series 5.31 near the origin and using 5.86, 5.89, we deduce
the distribution
Vargλ,Sd0 ,E0 r, z, Vh,λ,Md0 ,E0 r, z, ∈ D σ̌, , δD,Z0 ,
h≥0
5.90
if 0 > 0 is chosen small enough. Finally, by the continuity estimates 3.5 of the Laplace
transform Largλ and the formula 5.82, 5.87, we get the expression 5.79. Moreover, from
the formulas in Proposition 3.3, as Z0 t, z, solves the problem 4.90, 4.91, we deduce that
the distribution Vargλ,Sd0 ,E0 r, z, solves the Cauchy problem 5.80, 5.81.
Step 2. In this step, we show that the function X0,1 t, z, can be expressed as a Laplace
transform of some staircase distribution in direction argλ, satisfying the problem 5.80,
5.81.
From the assumption 5.23, we deduce from Proposition 4.12, that the function
V0,Sd1 ,E1 τ, z, constructed in 4.80 has an analytic continuation denoted by V0,Sd1 ,δn ,E1 τ, z, on Sd1 ,δn ∪ D0, τ0 × D0, δE1 × E0 ∩ E1 and satisfies estimates 4.62 for all n ≥
0, where δE1 > 0 depends on S,uE1 which denotes a common radius of absolute
s,k ,k x0 ,
convergence of the series 5.23, S, b, σ, |λ|, maxs,k0 ,k1 ∈S |b|s,k0 ,k1 x0 , maxs,k0 ,k1 ∈S |b|
0 1
Abstract and Applied Analysis
73
where x0 < ρ. This constant δE1 is, therefore, independent of n. Now, one defines the
functions
V0,Sd1 ,δn ,E1 r, z, V0,Sd1 ,δn ,E1 reiξn , z, 5.91
for all r ≥ 0, all z ∈ D0, δE1 , and all n ≥ 0.
Lemma 5.6. Let σ̌ > σ > σrb S − 1 as in Lemma 5.3. Then, there exists 0 < δD0,1 < minδE1 , δD,Z0 #1
B,
ρ,
#1 , M
(depending on S, S, σ̌, |λ|, A, B, ρ, μ and A,
μ
introduced in Lemma 5.7), there exist M
(depending on S, S, σ̌, |λ|, A, B, ρ, μ, A, B, ρ,
μ
and ρ , μ
introduced in Lemma 5.7) such that
sup V0,Sd1 ,δn ,E1 r, z, − Vargλ,Sd0 ,E0 r, z, ∈E0 ∩E1
σ̌,,d,δD0,1 ≤
#1 max Jn,j M
#1 D
n
M
0≤j≤S−1
5.92
for all n ≥ 0, where Vargλ,Sd0 ,E0 r, z, is defined in Lemma 5.5 and solves the problem 5.80, 5.81
n is the sequence (which tends to zero as n → ∞) defined in Lemma 5.7.
and D
Proof. From the estimates 4.54, we can write
V0,Sd1 ,δn ,E1 τ, z, zβ
V0,β,Sd1 ,δn ,E1 τ, ,
β!
β≥0
5.93
where V0,β,Sd1 ,δn ,E1 τ, are holomorphic functions such that there exist a constant u1 with 0 <
u1 < uE1 depending on uE1 , S, and b, σ, a constant x1 such that 0 < x1 < ρ depending
s,k ,k x0 , where x0 < ρ, and a
on S, uE1 , S, b, σ, |λ|, maxs,k0 ,k1 ∈S |b|s,k0 ,k1 x0 , maxs,k0 ,k1 ∈S |b|
0 1
constant CΩd1 ,E1 ,n > 0 depending on max0≤j≤S−1 Wj,Sd1 ,δn ,E1 uE1 where Wj,Sd1 ,δn ,E1 are defined
s,k ,k x0 , S, uE , x0 , S, b with
in 5.23, |λ|, maxs,k ,k ∈S |b|s,k ,k x0 , maxs,k ,k ∈S |b|
0
1
0
1
0
1
0
1
1
−1
β 2
σ |τ|2
1 2
rb β |τ|
exp
V0,β,Sd1 ,δn ,E1 τ, ≤ CΩd1 ,E1 ,n β!
x1
2||
||
5.94
for all τ ∈ Sd1 ,δn ∪ D0, τ0 , ∈ E1 , all β ≥ 0, and all n ≥ 0. We deduce that
β
σ 2
rb β r
β! exp
V0,β,Sd1 ,δn ,E1 reiξn , ≤ CΩd1 ,E1 ,n
x1
2||
5.95
for all r ≥ 0, all ∈ E1 , all β ≥ 0, and all n ≥ 0. In particular, r → V0,β,Sd1 ,δn ,E1 reiξn , belongs to
Lβ,σ̌/2, . From Proposition 2.7, we deduce that r → V0,β,Sd1 ,δn ,E1 reiξn , belongs to Dβ,
. From
σ̌,
Proposition 2.7 and 5.95, we get a universal constant C1 > 0 such that
V0,β,Sd1 ,δn ,E1 reiξn , β,σ̌,,d
≤ C1 V0,β,Sd1 ,δn ,E1 reiξn , β,σ̌/2,
2||
≤ C1 CΩd1 ,E1 ,n
σ̌ − σ
2
x1
5.96
β
β!
74
Abstract and Applied Analysis
for all β ≥ 0 and all n ≥ 0. From 5.96, we deduce that the distribution
V0,Sd1 ,δn ,E1 r, z, zβ
∈ D σ̌, , δ̌
V0,β,Sd1 ,δn ,E1 reiξn , β!
β≥0
5.97
for all ∈ E0 ∩ E1 , all δ̌ < x1 /2, and all n ≥ 0.
From 4.20, 4.21, we have that the distribution V0,Sd1 ,δn ,E1 r, z, solves the following
problem:
reiξn 1 ∂Sz V0,Sd1 ,δn ,E1 r, z, ⎛
⎜
k0 −s bs,k0 ,k1 z, ⎝eis−k0 ξn
s,k0 ,k1 ∈S
1
m,p∈Os−k
0
⎞
−p
⎟
α1m,p r m ∂r ∂kz1 V0,Sd1 ,δn ,E1 r, z, ⎠,
5.98
1
is the set and α1m,p are the integers from 5.80, with initial data
where Os−k
0
j
∂z V0,Sd1 ,δn ,E1 r, 0, v0,j,Sd1 ,δn ,E1 reiξn , ,
0 ≤ j ≤ S − 1.
5.99
In the next lemma, we give estimates for the coefficients of 5.98 and 5.80. The proof is
exactly the same as the one described for Lemma 5.4.
Lemma 5.7. Let
bs,k0 ,k1 z, zβ
bs,k0 ,k1 ,β β!
β≥0
5.100
be the convergent Taylor expansion of bs,k0 ,k1 with respect to z near 0. Then, there exist positive
n > 0 such that limn → ∞ D
B,
ρ,
n 0 with
, μ
and a sequence D
constants A,
ρ , μ
is−k0 argλ
β!q!
q bs,k0 ,k1 ,β e
−β
∂r
≤ AB q1 ,
i
argλ
re
1
ρ r μ
is−k0 ξn
β!q!
q bs,k0 ,k1 ,β e
−β
∂r
≤ AB q1 ,
reiξn 1
ρ r μ
is−k0 argλ
bs,k0 ,k1 ,β eis−k0 ξn q
q bs,k0 ,k1 ,β e
n B−β β!q! − ∂r
∂r
≤D
q1
iξ
i
argλ
re n 1
re
1
ρ r μ
for all q ≥ 0, all β ≥ 0, all n ≥ 0, all r ≥ 0 and all ∈ E0 ∩ E1 .
5.101
Abstract and Applied Analysis
75
Now, we consider the distribution
VΔ
0,Sd ,δn ,E1 r, z, : Vargλ,Sd0 ,E0 r, z, − V0,Sd1 ,δn ,E1 r, z, 5.102
1
for all r ≥ 0, all z ∈ D0, δD,Z0 ∩ D0, δ̌, with 0 < δ̌ < x1 /2 and δD,Z0 defined in Lemma 5.5,
for all ∈ E0 ∩ E1 . One writes the Taylor expansions as follows:
VΔ
0,Sd ,δn ,E1 r, z, 1
VΔ
0,β,Sd ,δ
β≥0
1
r, n ,E1
zβ
,
β!
5.103
zβ
Vargλ,Sd0 ,E0 r, z, Vβ,argλ,Sd0 ,E0 r, ,
β!
β≥0
for z ∈ D0, δD,Z0 ∩ D0, δ̌; then the coefficients VΔ
0,β,Sd ,δ
,E
1 n 1
VΔ
0,βS,Sd ,δ
,E
1 n 1
r, β!
s,k0 ,k1 ∈S β1 β2 β
⎛
1
m,p∈Os−k
0
s,k0 ,k1 ∈S β1
⎛
bs,k0 ,k1 ,β1 k0 −s
β!
β1 !
β β
1
m,p∈Os−k
2
eis−k0 argλ eis−k0 ξn
−
rei argλ 1 reiξn 1
5.104
⎞
V
r,
−p β2 k1 ,argλ,Sd0 ,E0
⎟
α1m,p r m ∂r
⎠
β2 !
⎜
×⎝
bs,k0 ,k1 ,β1 k0 −s eis−k0 ξn
β1 !
reiξn 1
⎞
Δ
V
r,
−p 0,β2 k1 ,Sd1 ,δn ,E1
⎟
α1m,p r m ∂r
⎠
β2 !
⎜
×⎝
r, satisfy the next recursion:
0
for all h ≥ 0, all β ≥ 0. We put VΔ
||VΔ
0,β,n,E1
0,β,Sd ,δ
,E
1 n 1
r, ||β,σ̌,,d . Using Corollary 2.10,
1
> 0 depending on
Propositions 2.11 and 2.12 and Lemma 5.7, we get a constant C23.2
2
such that the next inequalities:
S, σ̌, S, ρ,
μ
and C23.2 > 0 depending on S, σ̌, S, ρ , μ
VΔ
0,βS,n,E1 ≤
s,k0 ,k1 ∈S β1 β2
s,k0 ,k1 ∈S β1 β2 β
×
Δ
V
1
B−β1 β S 1 bs−k0 0,β2 k1 ,n,E1
C23.2
β!A
β2 !
β
2
n B−β1 β S 1 bs−k0 C23.2
β!D
Vβ2 k1 ,argλ,Sd0 ,E0 r, 5.105
β2 k1 ,σ̌,,d
β2 !
B > 0 and the sequence D
n , n ≥ 0 are defined in Lemma 5.7.
hold for all β ≥ 0, where A,
76
Abstract and Applied Analysis
We consider the following sequence of Cauchy problems:
∂Sx WΔ
n,E1 x
s,k0 ,k1 ∈S
1
C23.2
x∂x
S 1
bs−k0 A
1 − x/B
∂kx1 WΔ
n,E1 x
n x,
D
5.106
where
n x D
s,k0 ,k1 ∈S
2
C23.2
x∂x
S 1
bs−k0 n
D
1 − x/B
∂kx1 Wargλ,E0 x
5.107
with
Wargλ,E0 x sup Vβ,argλ,Sd0 ,E0 x
xβ
β,σ̌,,d β!
β≥0 ∈E0 ∩E1
5.108
for given initial data
j
∂x WΔ
n,E1 0
iξn
WΔ
sup
r,
−
v
re
,
V
j,argλ,S
,E
0,j,S
,E
d0 0
d1 ,δn 1
j,n,E1
j,σ̌,,d
∈E0 ∩E1
,
0 ≤ j ≤ S − 1,
5.109
which are finite positive numbers due to the assumption 5.27 and Remark 2.4. Moreover,
the initial data satisfy the estimates
Δ Wj,n,E1 ≤ Jn,j
5.110
for all 0 ≤ j ≤ S − 1 and all n ≥ 0.
On the other hand, we have that Wargλ,E0 x is convergent for all |x| ≤ XMd0 /4 where
XMd0 is chosen in 5.58. Indeed, we know, from 5.90, that
Vh,λ,Md0 ,E0 r, z, zβ
Vh,β,λ,Md0 ,E0 r, β!
β≥0
5.111
is convergent for all |z| < δD,Z0 , all r > 0, and all h ≥ 0. From 5.86 and 5.89, we know that
Vh,β,λ,Md0 ,E0 r, β,σ̌,,d
≤ C3 A
||Bσ̌, b, /σ̌h Vh,β,Md0 ,E0 r, β,σ̆,,d
h!
5.112
Abstract and Applied Analysis
77
for all h ≥ 0 and all β ≥ 0. From 5.58 and 5.112, we deduce that
Vβ,argλ,Sd0 ,E0 r, β,σ̌,,d
Vh,β,λ,Md0 ,E0 r, h≥0
≤ C3 ACMd0 β!
2
β,σ̌,,d
β
XMd0
2||Bσ̌, b, h
5.113
σ̌UMd0
h≥0
and this last sum is convergent provided that 0 is small enough. We deduce that Wargλ,E0 x
Md : ||Wargλ,E0 x||U,XM /4 .
belongs to GU, XMd0 /4, for any U > 0. Let C
0
d0
Md
From Lemmas 2.20 and 2.21, we get constants D
>
0 depending on
0
, B, U, XMd , 0 < U1,Md < U, and 0 < X1,Md < XMd /4 such that
S, σ̌, S, ρ , μ
0
0
0
Dn x
1,M ,X
1,M U
d
d
0
0
nD
Md C
Md
≤D
0
0
5.114
0
for all n ≥ 0.
From the assumption 4.42 and the fact that b > 1, we deduce that
S > bs − k0 k1
5.115
for all s, k0 , k1 ∈ S. Hence, the assumption 2.108 is satisfied in Proposition 2.22 for the
problem 5.106, 5.109. Moreover, the initial data WΔ
j,n can be seen as constant functions
therefore analytic with respect to a variable u on the closed disc D0, U for any given
and constant
U > 0 and the coefficients of 5.106 are analytic with respect to x on D0, B/2
therefore analytic with respect to u on D0, U. We deduce that all the hypotheses of
Proposition 2.22 for the problem 5.106, 5.109 are fulfilled. A direct computation shows
Δ
x β≥0 wβ,n,E
xβ /β!,
that the problem 5.106, 5.109 has a unique formal solution WΔ
n,E1
1
1, X
1 where
∈ C. From Proposition 2.22, we deduce that WΔ x ∈ GU
with wΔ
n,E1
β,n,E1
1,Md depending on S and 0 < X
1 < minB/2,
1,Md depending
1 < U
X
0 < U
0
0
B,
ρ,
#1 > 0
on S, S, σ̌, A,
μ
. Moreover, from 2.111 and 5.114, there exist constants M
#
depending on S, S, σ̌, A, B, ρ,
μ
and M2 > 0 depending on S, B, S such that
Δ
Wn,E1 x
1 ,X
1
U
for all n ≥ 0.
#1 max Jn,j D
nM
#2 D
Md C
Md
≤M
0
0
0≤j≤S−1
5.116
78
Abstract and Applied Analysis
Δ
satisfy the following equalities:
Now, the coefficients wβ,n,E
1
Δ
wβS,n,E
1
s,k0 ,k1 ∈S β1 β2 β
1
B−β1
C23.2
β!A
s,k0 ,k1 ∈S β1 β2 β
×
Δ
bs−k0 wβ2 k1 ,n,E1
βS1
β2 !
2
n B−β1 β S 1 bs−k0 C23.2
β!D
5.117
sup∈E0 ∩E1 Vβ2 k1 ,argλ,Sd0 ,E0 r, β2 k1 ,σ̆,,d
β2 !
for all β ≥ 0 and all n ≥ 0, with
Δ
WΔ
wj,n,E
j,n,E1 ,
1
0 ≤ j ≤ S − 1.
5.118
Gathering the inequalities 5.105 and the equalities 5.117, with the initial data 5.118, one
gets that
Δ
sup VΔ
≤ wβ,n,E
0,β,n,E1
1
5.119
∈E0 ∩E1
for all β, n ≥ 0.
From 5.119 and the estimates 5.116, we deduce that
sup VΔ
r,
0,β,Sd ,δn ,E1
∈E0 ∩E1
β,σ̌,,d
1
β
1
#
#
≤ M1 max Jn,j Dn M2 DMd0 CMd0 β!
0≤j≤S−1
1
X
5.120
for all β, n ≥ 0. From 5.120, we get that
sup V0,Sd1 ,δn ,E1 r, z, − Vargλ,Sd0 ,E0 r, z, ∈E0 ∩E1
σ̌,,d,δD0,1 #
#
≤ 2 M1 max Jn,j Dn M2 DMd0 CMd0
5.121
0≤j≤S−1
1 /2. This implies the estimates 5.92.
for all n ≥ 0 and for all 0 < δD0,1 < X
In the following lemma, we express the function X0,1 t, z, as Laplace transform of a
staircase distribution.
Lemma 5.7. Let σ̌ > σ > σrb S − 1 as in Lemma 5.3. Then, one can write the function X0,1 t, z, ,
which by construction of Proposition 4.12, solves the singularly perturbed Cauchy problem
t2 ∂t ∂Sz X0,1 t, z, t 1∂Sz X0,1 t, z, s,k0 ,k1 ∈S
bs,k0 ,k1 z, ts ∂kt 0 ∂kz1 X0,1 t, z, 5.122
Abstract and Applied Analysis
79
for given initial data
j
∂z X0,1 t, 0, ϕ0,1,j t, ,
0≤j ≤S−1
5.123
X0,1 t, z, Largλ Vargλ,Sd0 ,E0 r, z, t
5.124
in the form of a Laplace transform in direction argλ
for all t, z, ∈ T ∩ D0, ι × D0, δD0,1 × E0 ∩ E1 , where Vargλ,Sd0 ,E0 r, z, ∈ D σ̌, , δD0,1 solves the Cauchy problem 5.80, 5.81.
Proof. From Proposition 4.12 and the assumption 5.23, we get that the function X0,1 t, z, can be expressed as a Laplace transform in the direction ξn ,
τ
dτ
V0,Sd1 ,δn ,E1 τ, z, exp −
t
Lξn
eiξn ∞
reiξn
iξn
dr
V0,Sd1 ,δn ,E1 re , z, exp −
t 0
t
X0,1 t, z, 1
t
5.125
for all t, z, ∈ T ∩ D0, ι × D0, δE1 × E0 ∩ E1 , all n ≥ 0. Now, let t ∈ T ∩ D0, ι ,
∈ E0 ∩ E1 . For all n ≥ 0, we can rewite X0,1 t, z, as a Laplace transform in the direction
argλ as follows:
X0,1 t, z, Largλ V0,Sd1 ,δn ,E1 r, z, teiargλ−ξn 5.126
for all z ∈ D0, δE1 . Using the expression 5.126, we deduce that from the estimates 3.5,
there exists a constant Ct, > 0 such that
X0,1 t, z, − Largλ Vargλ,Sd0 ,E0 r, z, t
≤ Ct, V0,Sd1 ,δn ,E1 r, z, − Vargλ,Sd0 ,E0 r, z, σ̌,,d,δD0,1 Largλ Vargλ,Sd0 ,E0 r, z, teiargλ−ξn − Largλ Vargλ,Sd0 ,E0 r, z, t
5.127
for all n ≥ 0 and all z ∈ D0, δD0,1 . By letting n tend to ∞ and using the estimates 5.92, we
get the formula 5.124.
Now, we are in the position to state the main result of our work.
Theorem 5.8. Let the assumptions 4.42, 4.44, 4.67, 4.69, 4.70, 5.1, 5.4, 5.8, 5.20,
5.23, and 5.27 hold. Then, if one denote by op¿Gκ0 (resp. opGκ1 ) the opening of the sector
Gκ0 (resp. Gκ1 ), one has that for all t ∈ T ∩ D0, ι , z ∈ D0, δD0,1 , the function s → g0 s, t, z
80
Abstract and Applied Analysis
(constructed in Proposition 4.15) can be analytically continued along any path Γ in the punctured
sector
Ṡκ0 ,κ1 ,t,λ $
∞
opGκ1 C∗ opGκ0 λk
< args < κ1 s∈
−
\
,
κ0
2
2
t
k1
5.128
as a function denoted by g0Γ,t,z s. Moreover, for all k ≥ 1, and any path Γ0,k ⊂ Ṡκ0 ,κ1 ,t,λ from 0 to a
neighborhood of λk/t, there exists a constant Ck > 0 such that
λk Γ0,k ,t,z log
s
−
≤
C
s
g0
k
t 5.129
as s tends to λk/t in a sector centered at λk/t.
Proof. The proof is based on the following version of a result on analytic continuation of Borel
transforms obtained by Fruchard and Schäfke in 3. This result extends a former statement
obtained by the same authors in 28.
Theorem FS. Let r > 0 and let g : D0, r → C be a holomorphic function that can be analytically
continued as a function g (resp., g − ) with exponential growth of order 1 on an unbounded sector
Sκ ,δ (resp. Sκ− ,δ− ) centered at 0, with bisecting direction κ (resp. κ− ) and opening δ (resp. δ− ).
Let C > r be a real number and let m ≥ 1 be an integer. Let {ak ∈ C∗ , 1 ≤ k ≤ m} ⊂ D0, C be a
set of aligned points and let α > 0 with argak α ∈ κ− , κ , for all 1 ≤ k ≤ m. For all integers
1 ≤ k ≤ m, let Sk be an unbounded open sector centered at ak , with bisecting direction which is
parallel to κ− , and opening μ > 0 such that the Sk ∩ D0, C do not intersect for all 1 ≤ k ≤ m.
Now, for all 1 ≤ k ≤ m, let gk be a holomorphic and bounded function on a small neighborhood
of 0 and with exponential growth of order 1 on the sector Sk −ak {s ∈ C/sak ∈ Sk } with bisecting
direction κ− . We consider the Laplace transforms
f g se
−s/
ds,
−
−
f Lκ
g se
Lκ−
−s/
ds,
fk− Lκ−
gk se−s/ ds
5.130
for all k ≥ 1, where Lκ is the half-line starting from 0 in the direction κ and Lκ− is the half-line
starting from 0 in the direction κ− . The function f (resp. f − ) defines a holomorphic and bounded
function on an open sector E (resp. E− ) with finite radius, with bisecting direction κ (resp. κ− ) and
opening π δ (resp. π δ− ). The sectors E , E− are chosen in such a way that E ∩ E− is contained
in a sector with direction α and with opening less than π. Assume that the following Stokes relation
f f − m −ak /
e
k1
k!
iα
fk− O e−Ce /
5.131
Abstract and Applied Analysis
81
holds for all ∈ E ∩ E− , where Oe−Ce
exists a constant H > 0 with
iα
/
is a holomorphic function R on E ∩ E− such that there
iα/ |R| ≤ H e−Ce He−C/|| cosα−arg
5.132
for all ∈ E ∩ E− .
Then, the function g : D0, r → C can be analytically continued along any path Γ in the
punctured sector
Ṡκ− ,κ ,C
$
m
C∗
δ−
δ
−
< args < κ < C, κ −
s∈
\ {ak }.
2
2
|s|
k1
5.133
Moreover, for all 1 ≤ k ≤ m, and any path Γ0,k ⊂ Ṡκ− ,κ ,C from 0 to a neighborhood of ak , if we denote
by g Γ0,k s the analytic continuation of g along Γ0,k , then there exists a constant Ck > 0 such that
Γ0,k g s ≤ Ck logs − ak 5.134
as s tends to ak in a sector centered at ak .
Proof. For the sake of completeness, we give a sketch of proof of this theorem. In the first step,
let us consider the following sums of Cauchy integrals
ht m
1
1 2iπ k1 k!
Lak ,κ− ,C
gk τ − ak dτ,
τ −t
5.135
where Lak ,κ− ,C is the segment starting from ak in the direction κ− with length C. The
multivalued function ht can be analytically continued along any path Γ in C \ {a1 , . . . , am }
by deforming the path of integration Lak ,κ− ,C in the sector Sk and keeping the endpoints of
the segment Lak ,κ− ,C fixed for all 1 ≤ k ≤ m. Moreover, let 1 ≤ k ≤ m and t ∈ Lak ,κ− \ {ak },
where Lak ,κ− denotes the half-line starting from ak in the direction κ− . We denote by hΓak ,t,ρ t
the analytic continuation of ht along a loop Γak ,t,ρ around ak constructed as follows: the loop
follows a segment starting from t in the direction ak then turns around ak along a circle Γak ,ρ
of small radius ρ > 0 positively oriented and then goes back to t following the same segment.
We have that
ht − hΓak ,t,ρ t gk t − ak .
k!
5.136
Indeed, by the Cauchy theorem, one can write ht − hΓak ,t,ρ t as a Cauchy integral
1
Ik 2iπk!
Cak ,C
gk τ − ak dτ,
τ −t
5.137
82
Abstract and Applied Analysis
where Cak ,C is a positively oriented closed curve enclosing t starting from ak and containing
−
the point ak Ceiκ . By the residue theorem, one gets that Ik gk t − ak /k!. From the relation
5.136, we also deduce the existence of a holomorphic function bt near ak such that
ht −
gk t − ak logt − ak bt
2iπk!
5.138
for all t near ak , for a well-chosen determination of the logarithm logx.
In the second step, let us define the truncated Laplace transforms and Laplace transforms
HC
hse−s/ ds,
−
HC
Lκ ,C
H hse
−s/
−
hse−s/ ds,
Lκ− ,C
H ds,
Lκ
hse
5.139
−s/
ds,
Lκ−
where Lκ ,C is the segment starting from 0 to C eiκ and Lκ− ,C is the segment starting from 0
−
to C eiκ , for any fixed C > C. By the Cauchy formula, one can write the difference HC −
HC− as the sum
HC − HC− m −
hse−s/ ds k1
Γak ,ρ
iα
hs − hΓak ,s,ρ s e−s/ ds O e−Ce / ,
Lak ,ρ,C ,κ−
5.140
−
−
where Lak ,ρ,C ,κ− is the segment starting from ak ρeiκ to ak C eiκ for any ρ > 0 small
enough. Due to the decomposition 5.138, hs is integrable at ak . By letting ρ tending to 0
and C tending to infinity, using the relation 5.136 in 5.140, ones gets that
−
H − H m
1
k1
k!
iα
gk s − ak e−s/ ds O e−Ce /
Lak ,κ−
m −ak / e
k1
k!
gk se
−s/
iα
ds O e−Ce / ,
5.141
Lκ−
where Lak ,κ− is the half-line starting from ak in the direction κ− .
Now, one considers the differences D f − H and D− f − − H − .
From the Stokes relations 5.131 and 5.141, one deduces that
iα
D − D− O e−Ce /
5.142
for all ∈ E ∩ E− . Using a similar Borel transform integral representation as in the proof of
Theorem 1 in 28, one can show that the difference gs − hs, which is by construction
Abstract and Applied Analysis
83
analytic near the origin in C, can be analytically continued to a function Gs, which is
holomorphic on the sector Sκ− ,κ ,C {s ∈ C∗ /|s| < C, κ− < args < κ }. Since h can be
analytically continued along any path in C \ {a1 , . . . , an }, one gets that the function g can
be analytically continued along any path in Ṡκ− ,κ ,C and from the decomposition 5.138 one
deduces the estimates 5.134.
Now, we return to the proof of Theorem 5.8. From the formula 4.76 and
Proposition 5.2, the following equality
g0,1 s, t, ze−s/ ds Lκ1
g0,0 s, t, ze−s/ ds
Lκ0
5.143
exp−hλ/t h!
h≥1
gh,0 s, t, ze
−s/
ds
Lκ0
holds for all ∈ E0 ∩ E1 , and all t ∈ T ∩ D0, ι , all z ∈ D0, δD0,1 . Let t ∈ T ∩ D0, ι and
z ∈ D0, δD0,1 fixed. Let m ≥ 1 be an integer. From the estimates 4.78, we get that
exp−hλ/t gh,0 s, t, ze−s/ ds
h!
Lκ0
h≥m1
0
≤ 2C
h
exp −h λ 2
t u
0
≤ 2C
2
u1
5.144
1
h≥m1
m1 1
exp −m 1 λ t 1 − 2exp−λ/t/u
1
for all ∈ E0 ∩ E1 . From 5.143 and 5.144, we deduce that the following Stokes relation
g0,1 s, t, ze
Lκ1
−s/
g0,0 s, t, ze−s/ ds
ds Lκ0
m
exp−hλ/t
h1
h!
gh,0 s, t, ze
−s/
O e−m1λ/t .
5.145
Lκ0
Holds, where Oe−m1λ/t is a holomorphic function R on E0 ∩ E1 such that there exists
a constant H > 0 with
|R| ≤ H e−m1λ/t 5.146
84
Abstract and Applied Analysis
for all ∈ E0 ∩ E1 . We can apply Theorem FS with ak kλ/t, for 1 ≤ k ≤ m, C |λ|m 1/|t|
to get that the function s → g0 s, t, z constructed in Proposition 4.15 can be analytically
continued along any path in the punctured sector
Ṡκ0 ,κ1 ,t,λ,m
$
m opGκ0 opGκ1 C∗ |λ|m 1
λk
< args < κ1 s∈
<
, κ0 −
\
2
2
t
|s|
|t|
k1
5.147
as a function denoted by g0Γ,t,z s. Moreover, for all 1 ≤ k ≤ m, and any path Γ0,k ⊂ Ṡκ0 ,κ1 ,t,λ,m
Γ ,t,z
from 0 to a neighborhood of λk/t, there exists a constant Ck > 0 such that |g0 0,k s| ≤
Ck | logs − λk/t| as s tends to λk/t in a sector centered at λk/t. Since this result is true for all
m ≥ 1, Theorem 5.8 follows.
In the next result, Onee show that under the additional hypothesis that the coefficients
of 4.90 are polynomials in the parameter , the function g0 s, t, z solves a singular linear
partial differential equation in C3 .
Corollary 5.9. Let the assumptions of Theorem 5.8 hold. We assume moreover that, for all tuple
s, k0 , k1 chosen in the set S, the coefficients bs,k0 ,k1 z, belong to C{z} with the following
expansion in :
bs,k0 ,k1 z, ds,k0 ,k1
mk0
m
bs,k
zm
0 ,k1
5.148
for some ds,k0 ,k1 ≥ k0 . Then, for all K ∈ N with K ≥ 1 and K ≥ max{ds,k0 ,k1 ∈ N/s, k0 , k1 ∈ S},
the function g0 u, t, z (constructed in Proposition 4.15) satisfies the following singular linear partial
differential equation
S
K S
t2 ∂t ∂K−1
u ∂z g0 u, t, z ∂u ∂z g0 u, t, z
S
−t∂K−1
u ∂z g0 u, t, z
ds,k0 ,k1
s,k0 ,k1 ∈S mk0
m
s
K−m k0 k1
∂
bs,k
∂
∂
g
zt
u, t, z
0
z
u
t
,k
0 1
5.149
for all u, t, z ∈ D0, s0 × T ∩ D0, ι × D0, δD0,1 . From Theorem 5.8, for all t, z ∈ T ∩
D0, ι × D0, δD0,1 , this solution g0 u, t, z can be analytically continued with respect to u along
any path in the punctured sector Ṡκ0 ,κ1 ,t,λ with logarithmic estimates 5.129 near the singular points
λk/t for all k ≥ 1.
Proof. From Proposition 4.15, we have that the function
X0,0 t, z, −1
g0,0 s, t, ze−s/ ds
Lκ0
5.150
Abstract and Applied Analysis
85
solves 5.122 on T ∩ D0, ι × D0, δD0,1 × E0 . From the formulas in Proposition 4.2, we
deduce that the function g0,0 u, t, z solves the singular integrodifferential equation
S
S
t2 ∂t ∂−1
u ∂z g0,0 u, t, z ∂z g0,0 u, t, z
S
−t∂−1
u ∂z g0,0 u, t, z ds,k0 ,k1
s,k0 ,k1 ∈S mk0
m
s
−m k0 k1
∂
bs,k
∂
∂
g
zt
u, t, z
0,0
z
u
t
,k
0 1
5.151
for all u, t, z ∈ Gκ0 ∪ D0, s0 × T ∩ D0, ι × D0, δD0,1 . Since g0 u, t, z is holomorphic
on D0, s0 × T ∩ D0, ι × D0, δD0,1 and has g0,0 u, t, z as analytic continuation on Gκ0 ∪
D0, s0 ×T∩D0, ι ×D0, δD0,1 , we get that g0 u, t, z also solves 5.151. By differentiating
K times of each hand side of the equation with respect to u, one gets that g0 u, t, z solves the
partial differential equation 5.149.
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