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volume 2, issue 2, article 24, 2001.

Received —-;

accepted 5 March, 2001.

Communicated by:R.P. Agarwal

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Journal of Inequalities in Pure and Applied Mathematics

NECESSARY AND SUFFICIENT CONDITION FOR EXISTENCE AND UNIQUENESS OF THE SOLUTION OF CAUCHY PROBLEM FOR HOLOMORPHIC FUCHSIAN OPERATORS

MEKKI TERBECHE

Florida Institute of Technology, Department of Mathematical Sciences, Melbourne, FL 32901, USA

EMail:terbeche@hotmail.com

c

2000Victoria University ISSN (electronic): 1443-5756 022-01

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Necessary and Sufficient Condition for Existence and Uniqueness of the Solution of

Cauchy Problem for Holomorphic Fuchsian

Operators Mekki Terbeche

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Abstract

In this paper a Cauchy problem for holomorphic differential operators of Fuch- sian type is investigated. Using Ovcyannikov techniques and the method of majorants, a necessary and sufficient condition for existence and uniqueness of the solution of the problem under consideration is shown.

2000 Mathematics Subject Classification:35A10, 58A99.

Key words: Banach algebra, Cauchy problem, Fuchsian characteristic polynomial, Fuchsian differential operator, Fuchsian principal weight, holomorphic differentiable manifold, holomorphic hypersurface, Fuchsian principal weight, method of majorants, method of successive approximations, principal symbol, and reduced Fuchsian weight.

1. Introduction

We introduce the method of majorants [2], [5], and [8], which plays an important role for the Cauchy problem in proving the existence of a solution. This method has been applied by many mathematicians, in particular [1], [3], and [4]

to study Cauchy problems related to differential operators that are a “natural”

generalization of ordinary differential operators of Fuchsian type, and to gen- eralize the Goursat problem [8]. We also give a refinement of the method of successive approximations as in the Ovcyannikov Theorem given in [7]. Com- bining these two methods, we shall prove the theorem [6].

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2. Notations and Definitions

Let us denote

x = (x₀, x₁, . . . , x_n)≡(x₀, x⁰)∈R×Rⁿ, wherex⁰ = (x₁, . . . , x_n)∈Rⁿ, ξ = (ξ0, ξ1, . . . , ξn)≡(ξ0, ξ⁰)∈R×Rⁿ, whereξ⁰ = (ξ1, . . . , ξn)∈Rⁿ, α = (α₀, α₁, . . . , α_n)≡(α₀, α⁰)∈N×Nⁿ, whereα⁰ = (α₁, . . . , α_n)∈Nⁿ. We use Schwartz’s notations

x^α = x^α₀⁰x^α₁¹· · ·x^α_nⁿ ≡x^α₀⁰(x⁰)^α⁰,|x|^α =|x₀|^α⁰|x₁|^α¹· · · |x_n|^αⁿ α! = α₀!α₁!· · ·α_n!, |α|=α₀+α₁+· · ·+α_n,

β ≤ αmeansβ_j ≤α_j for allj = 0,1, . . . , n, D^α = ∂^|α|

∂x^α0⁰∂x^α1¹...∂_x^α_nⁿ ≡D₀^α⁰D₁^α¹· · ·D_n^αⁿ, whereD_j = ∂

∂x_j,0≤j ≤n.

Fork∈N,0≤k≤m,

max[0, α0+ 1−(m−k)]≡[α0+ 1−(m−k)]+, m

k

= m!

(m−k)!k!, Cq(j) =j(j−1)...(j−q+ 1),

by conventionC₀(j) = 1, and the gradient ofϕwith respect toxwill be denoted by

gradϕ(x) =

∂ϕ(x)

∂x₀ , . . . ,∂ϕ(x)

∂x_n

.

We denote a linear differential operator of orderm,P(x;D)byP

|α|≤ma_α(x)D^α.

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Definition 2.1. LetEbe ann+ 1dimensional holomorphic differentiable man- ifold. Let hbe a holomorphic differentiable operator overE of orderm0 ina, and of order≤m₀ neara. LetSbe a holomorphic hypersurface ofE contain- ing a, let m be an integer≥ m₀, and letϕ be a local equation of S in some neighborhood ofa, that is, there exists an open neighborhoodΩofasuch that:

∀x∈Ω, gradϕ(x)6= 0, x∈Ω∩S ⇐⇒ϕ(x) = 0.

Ifσ∈ZandY is a holomorphic function onΩ, forx∈Ω\S, we denote by h^σ_m(Y)(x) = ϕ^σ−m(x)h(Y ϕ^m)(x)

and byH_m^σ(x, ξ)the principal symbol of this differential operator.

(i)

τ_h,S(a) = inf

σ∈Z:∀Y holomorphic function in a neighborhood Ωofa, ∀x∈Ω∩S, lim

x→b,x /∈Sh^σ+1_m (Y)(x) = 0

denotes the Fuchsian weight ofhinawith respect toS.

(ii)

τ_h,S^∗ (a) = inf

σ∈Z: lim

x→b,x /∈Sϕ^−m⁰(x)H^σ+1(x;gradϕ(x)) = 0

is the Fuchsian principal weight ofhinawith respect toS.

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(iii)

˜

τ_h,S(a) = inf

σ∈Z:∀Ω,∀Y,∀b ∈Ω∩S,

x→b,x /lim∈S[h^σ+1_m (Y)(x)−Y(x)h^σ+1_m (1)(x)] = 0

denotes the reduced Fuchsian weight ofhinawith respect toS.

A differential operatorh is said to be a Fuchsian operator of weightτ ina with respect toSif the following assertions are valid:

(H-0) τ_h,S^∗ is finite and constant and equalτ neara ∈S, (H-1) τ_h,S(a) = τ,

(H-2) τ˜_h,S(a)≤τ −1.

A Fuchsian characteristic polynomial is defined to be a polynomialC inλof holomorphic coefficients iny∈Sby

C(λ, y) = lim

x→y,x /∈Sϕ^τ^h,S^(a)−λ(x)h(ϕ^λ)(x).

Set

C₁(λ, y) = C(λ+τ_h,S(a), y), ∀λ∈C, ∀y∈S.

Remark 2.1. If we choose a local card for whichϕ(x) =x0anda= (0, . . . ,0), we get Baouendi-Goulaouic’s definitions [1].

Remark 2.2. The numberτ_h,S(a)is independent onmwhich is greater or equal tom₀.

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3. Majorants

The majorants play an important role in the Cauchy method to prove the existence of the solution, where the problem consists of finding a majorant function which converges.

Letα be a multi-index of Nⁿ⁺¹ andE be a C-Banach algebra, we define a formal series inxby

u(x) = X

α∈Nⁿ⁺¹

u_αx^α α!, whereu_α ∈E.

We denote byE[[x]]the set of the formal series inxwith coefficients inE.

Definition 3.1. Let u(x), v(x) ∈ E[[x]], and λ ∈ C. We define the following operations inE[[x]]by

(a) u(x) +v(x) =u(x) = P

α∈Nⁿ⁺¹(u_α+v_α)^x_α!^α, (b) λu(x) =P

α∈Nⁿ⁺¹(λu_α)^x_α!^α, (c) u(x)v(x) = P

α∈Nⁿ⁺¹

P

0≤β≤α

α β

uβvα−βx^α α!. Definition 3.2. Let

u(x) = X

α∈Nⁿ⁺¹

u_αx^α

α! ∈E[[x]], and

U(x) = X

α∈Nⁿ⁺¹

U_αx^α

α! ∈R[[x]]

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be two formal series. We say that U majorizesu, written U(x) u(x), pro- videdUα ≥ kuαkfor all multi-indicesα.

Definition 3.3. Let

u(x) = X

α∈Nⁿ⁺¹

u_αx^α

α! ∈E[[x]].

We define the integro-differentiation ofu(x)by D^µu(x) = X

α≥[−µ]+

u_α+µx^α α!,

forµ∈Zⁿ⁺¹, and[−µ]+≡([−µ0]+,[−µ1]+, . . . ,[−µn]+).

(a) Let A be a finite subset of Zⁿ⁺¹, P(x;D)is said to be a formal integro- differential operator overE[[x]]if foru(x)∈E[[x]],

P(x;D) = X

µ∈A

a_µ(x)D^µu(x),

whereaµ(x)∈E[[x]].

(b) Let

P(x;D) = X

µ∈Zⁿ⁺¹

A_µ(x)D^µ

and

P(x;D) = X

µ∈Zⁿ⁺¹

aµ(x)D^µ

be formal integro-differential operators overR[[x]]andE[[x]]respectively.

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We sayP(x;D)majorizesP(x;D), writtenP(x;D)P(x;D), provided Aµ(x)aµ(x)for all multi-indicesµ∈Zⁿ⁺¹.

Definition 3.4. Consider a family {u^j}_j∈J, u^j ∈ E[[x]]. The family{u^j}_j∈J is said to be summable if for anyα∈Nⁿ⁺¹,J_α ={j ∈J :u^j_α 6= 0}is finite.

Theorem 3.1. Letv ∈E[[y]],(y= (y₁, ..., y_m)), V ∈R[[y]]such thatV(y) v(y). Let u^j(x) ∈ E[[x]]for j = 1, . . . , m, u^j₀ = 0, and U^j(x) ∈ R[[x]]for j = 1, . . . , m, U₀^j = 0such thatU(x)u(x)for allj = 1, . . . , m.Then

V U¹(x), . . . , U^m(x)

v u¹(x), . . . , u^m(x) . Proof. See [7].

Definition 3.5. Ifu(x)∈E[[x]], we denote the domain of convergence ofuby

d(u) = (

x:x∈Cⁿ⁺¹, u(x) =X

α≥0

ku_αk|x|^α α! <∞

) .

Theorem 3.2. (majorants): IfU(x)u(x), then d(U)⊂d(u).

The above theorem is practical because, if the majorant series U(x) converges for|x|< rthenu(x)converges for|x|< r. Let us construct a majorant series through an example.

Letr = (r₀, r₁, ..., r_n)and let u(x) be a bounded holomorphic function on the polydisc

Pr ={x:x∈Cⁿ⁺¹,|xj |< rj, for allj = 0,1, . . . , n}.

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LetM = sup

x∈Pr

ku(x)k, then it follows from Cauchy integral formula thatku_αk<

M r^αα!.

If we letUα = _r^Mαα!, thenU(x)majorizesu(x).

Theorem 3.3. Let a_i,0 ≤ i ≤ m,be holomorphic functions near the origin in Cⁿthat satisfy the following condition

m

X

i=0

C_i(j)a_i(0)6= 0, ∀j ∈N, (a_m = 1),

then there exists a holomorphic functionAnear the origin inCⁿsuch that 1

Pm

i=0C_i(j)a_i(x⁰) A(x⁰) Pm

i=0C_i(j), ∀j ∈N Proof. If we write

Pm

i=0Ci(j) Pm

i=0C_i(j)a_i(x⁰) =

Pm

i=0Ci(j) Pm

i=0C_i(j)a_i(0) · 1

1−^P^mⁱ⁼⁰^P^Cmⁱ^(j)(aⁱ^(0)−aⁱ^(x⁰⁾⁾ i=0Ci(j)ai(0)

then we have

j→∞lim

Pm

i=0C_i(j) Pm

i=0C_i(j)a_i(0) = lim

j→∞

C_m(j) C_m(j)a_m(0)

= 1

am(0)

= 1.

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Theorem 3.5.

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Therefore, there exists a constantC ≥1such that

Pm

i=0C_i(j) Pm

i=0C_i(j)a_i(0)

≤C, ∀j ∈N.

LetB(x⁰)be a common majorant toa_i(0)−a_i(x⁰)for alli = 0,1, . . . , mwith B(0) = 0, that is

Pm

i=0C_i(j)(a_i(0)−a_i(x⁰)) Pm

i=0C_i(j)a_i(0) B(x⁰)

Pm

i=0C_i(j) Pm

i=0C_i(j)a_i(0) CB(x⁰).

It follows from Theorem3.1that 1

1− ^P^mⁱ⁼⁰^P^Cmⁱ^(j)(aⁱ^(0)−aⁱ^(x⁰⁾⁾ i=0Ci(j)ai(0)

1

1−CB(x⁰).

Choosing thenA(x⁰) = _1−CB(x¹ 0),the desired conclusion easily yields.

Corollary 3.4. Under the conditions of Theorem 3.3, there exist two positive real numbersM > 0andr >0such that

1

C₁(j, x⁰) 1

(j + 1)^m · M

1−rt(x⁰), ∀j ∈N, wheret(x⁰) = Pm

i=0xi.

Proof. The proof is similar to the proof of the Theorem3.3, it suffices to observe that

j→∞lim

Pm

i=0C_i(j) Pm

i=0C_i(j)a_i(0) = 1, then apply the following theorem.

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IfA(x⁰)is holomorphic near the origin inCⁿ, there existM >0andR > 0 such that:

A(x⁰) M R−t(x⁰). for allx⁰ ∈ {x⁰ :Pn

i=0 |xi |< R}.

Proof. See [8].

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4. Statement of the Main Result

Theorem 4.1 (Main Theorem). Let hbe a Fuchsian holomorphic differential operator of weight τ_h,S(a) ina with respect to a holomorphic hypersurface S passing througha, of a holomorphic differential manifoldEof dimensionn+ 1, and ϕ a local equation of S in some neighborhood of a. Then the following assertions are equivalent:

i) for allλ≥τ_h,S(a),C(λ, x⁰)6= 0

ii) for all holomorphic functionsf andvin a neighborhood ofa, there exists a unique holomorphic functionusolving the Cauchy problem

h(u) = f (4.1)

u−v = O(ϕ^τ^h,S^(a)).

If we choose a local card such that ϕ(x) = x0 and a = (0, ...,0), then we obtain the Baouendi-Goulaouic’s Theorem [1], if k = 0, we obtain Cauchy- Kovalevskaya Theorem, and ifk = 1, we obtain Hasegawa’s Theorem [3].

The following theorem gives a relationship between a Fuchsian operator of arbitrary weight and a Fuchsian operator of weight zero.

Theorem 4.2. Lethbe a Fuchsian holomorphic differential operator of weight τ_h,S(a)in a with respect to a holomorphic hypersurfaceS passing through a, of a holomorphic differentiable manifoldE of dimensionn+ 1, andϕa local equation ofSin some neighborhood ofa. If we define the operatorh1by

Y →h1(Y) = h(Y ϕ^τ^h,S^(a)),

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then h₁ is a Fuchsian holomorphic differential operator of weight zero in a relative to a holomorphic hypersurfaceS.

If C (respectively C₁) denotes the Fuchsian polynomial characteristic of h (respectivelyh₁), then

C₁(λ, y) = C(λ+τ_h,S(a), y), ∀λ∈C, ∀y∈S.

Proof. 1. We look for the Fuchsian weight ofh₁inawith respect toS.

Letm≥m₀, wherem₀ is the order ofh, then

ϕ^σ+1−m(x)h1(Y ϕ^m)(x) =ϕ^σ+1−m(x)h(Y ϕ^m+τ^h,S^(a))(x)

=ϕ^(σ+τ^h,S(a)+1)−(m+τ_h,S(a))(x)h(Y ϕ^m+τ^h,S^(a))(x), consequentlyτ_h,S(a) = 0.

2. We look for the principal Fuchsian weight ofh₁ inarelative toS.

In the local cardϕ(x) =x₀ anda= (0, . . . ,0), it is the exponent ofx₀in the coefficient ofD^m₀ ofh₁. In this local card

P₁(u) = P(x^m₀⁰^−ku)

= x^k₀D^m₀ (x^m₀⁰^−ku) +...

= x^k₀D^m₀ u+...

(the points indicate the terms that have the order of differentiation with respect tox₀ less thanm₀). Hence

τ_h,S^∗ (a) = 0.

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3. Ifm ≥m₀, then lim

x→b,x /∈Sϕ^τ^h¹^,S^(a)−m(x)[h₁(Y ϕ^m)(x)−Y(x)h₁(ϕ^m)(x)]

= lim

x→b,x /∈Sϕ^−m(x)[h(Y ϕ^m+τ^h,S^(a))(x)−Y(x)h(ϕ^m+τ^h,S^(a))(x)]

= lim

x→b,x /∈Sϕ^τ^h,S^(a)−(m+τ^h,S^(a))(x)

×[h(Y ϕ^m+τ^h,S^(a))(x)−Y(x)h(ϕ^m+τ^h,S^(a))(x)]

= 0, by hypothesis.

Finally, we have

ϕ^τ^h¹^,S^(a)−λ(x)h₁(ϕ^λ)(x) = ϕ^−λ(x)h(ϕ^λ+τ^h,S^(a))(x)

= ϕ^τ^h,S^(a)−(λ+τ^h,S^(a))(x)h(ϕ^λ+τ^h,S^(a))(x) which tends toC(λ+τ_h,S(a), y)asxtends toyandx /∈S.

This concludes the proof of the Theorem.

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5. Formal Problem

If we choose a local card for which ϕ(x) = x₀ anda = (0, . . . ,0)the Cauchy problem (4.1) becomes (5.1) below. We devote this section to formal calcula- tions by looking for solutions as power series of the problem (5.1) below con- nected with a Fuchsian operator P(x;D) of orderm and weightm −k with respect tox₀ atx₀ = 0.

We decompose this operator in the following form P(x;D) =P_m(x;D₀)−Q(x;D), where

P_m(x;D₀) =

k

X

p=0

am−p(x⁰)x^k−p₀ D₀^m−p,

Q(x;D) =− X

α0<m,|α|≤m

x^µ(α₀ ⁰⁾D^α₀⁰(a_α₀_,α⁰(x₀, x⁰)D^α_x0⁰),

witha_m = 1andµ(α₀) = [α₀+ 1−(m−k)]₊.

Theorem 5.1. If the coefficients of P_m(x;D₀) and Q(x;D) are holomorphic functions near the origin inCⁿ⁺¹, then the following conditions are equivalent

i) For all integersλ≥m−k 6= 0,

ii) For any holomorphic Cauchy datau_j,0≤j ≤m−k−1, near the origin inCⁿand for each holomorphic functionf near the origin inCⁿ⁺¹, there

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exists a unique holomorphic solution u near the origin in Cⁿ⁺¹ solving Cauchy problem

P(x;D)u(x) = f(x) (5.1)

D₀^ju(0, x⁰) = u_j(x⁰), 0≤j ≤m−k−1.

Suppose that the solutionu(x0, x⁰)has the formP∞

j=0uj(x⁰)^x

j 0

j!.

The problem is to determine u_j(x⁰) for all j ≥ 0. It is easy to check the following statements:

If

u(x0, x⁰) =

∞

X

j=0

uj(x⁰)x^j₀ j!

and

v(x₀, x⁰) =

∞

X

µ=0

v_µ(x⁰)x^µ₀ µ!,

then

u(x₀, x⁰)v(x₀, x⁰) =

∞

X

j=0

" _j X

p=0

j p

uj−p(x⁰)v_p(x⁰)

#x^j₀ (5.2) j!

D₀^pu(x₀, x⁰) =

∞

X

j=0

u_j+p(x⁰)x^j₀ (5.3) j!

x^q₀D₀^pu(x₀, x⁰) =

∞

X

j=0

[C_q(j)uj+p−q(x⁰)]x^j₀ (5.4) j!

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and by conventionu_k = 0fork < 0.

By using (5.2), (5.3), and (5.4), one can check easily that P_m(x;D₀)u(x₀, x⁰) =

∞

X

j=0

[D(j, x⁰)uj+m−k(x⁰)]x^j₀ j!,

where D(j, x⁰) = Pk

j=0am−q(x⁰)Ck−p(j) which can be written in terms of C(j, x⁰)as

D(j, x⁰) = C(j+m−k, x⁰) Cm−k(j+m−k), and we have

(5.5) P_m(x;D₀)u(x₀, x⁰) =

∞

X

j=0

C(j +m−k, x⁰)

Cm−k(j+m−k)u_j+m−k(x⁰) x^j₀

j!. Similarly, ifa_α(x₀, x⁰) = P∞

ν=0a^ν_α(x⁰)^x_ν!^ν⁰, then (5.6) Q(x;D)u(x₀, x⁰) = −

∞

X

j=0





X

α0<m,|α|≤m

C_µ(α₀₎(j)

×

j+α0−µ(α0)

X

p=0

j+α₀−µ(α₀) p

a^p_α(x⁰)D^α_x0⁰u_j+α₀_−µ(α₀_)−p(x⁰)



 x^j₀

j!.

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Finally, iff(x₀, x⁰) =

∞

P

j=0

f_j(x⁰)^x

j 0

j!, then by using (5.5), (5.6), and by identifying the coefficients ofP(x;D)u(x) =f(x)we get the following expression

C(j+m−k, x⁰)

Cm−k(j +m−k)uj+m−k(x⁰)

=− X

α0<m,|α|≤m

C_µ(α₀₎(j)

j+α0−µ(α₀)

X

p=0

j+α0−µ(α0) p

a^p_α(x⁰)

×D^α_x⁰⁰u_j+α₀−µ(α0)−p(x⁰) +f_j(x⁰) for allj ∈N.

Lemma 5.2. Let P(λ;x⁰) = Pm

k=0a_k(x⁰)λ^k, (a_m = 1), be a polynomial inλ with continuous coefficients on some neighborhood

∼

V of the origin inCⁿ. If P(j; 0) 6= 0for allj ∈ N, there exists a neighborhoodV of the origin such thatP(j;x⁰)6= 0for allx⁰ ∈V and allj ∈N.

Proof. We have

|P(λ;x⁰)| ≥ |λ|^m−

m−1

X

k=0

ak(x⁰)λ^k . Let

M = max

0≤k≤m−1,x⁰∈

∼

V

|a_k(x⁰)|

then

|P(λ;x⁰)| ≥ |λ|^m

"

1− M

|λ|

m−1

X

k=0

1

|λ|^m−k−1

# .

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If|λ|>1, then

|P(λ;x⁰)|>1− M

|λ| −1. Ifx⁰ ∈V˜ and|λ| ≥2M + 1, then

|P(λ;x⁰)|> 1 2.

In other words, ifj is an integer such thatj ≥2M+ 1andx⁰ ∈ ^∼V then P(j;x⁰)6= 0.

Now letj ∈Nsuch that0≤j <2M+1. SinceP(j; 0) 6= 0,then by continuity, there is a neighborhood of the originV_j such thatP(j;x⁰)6= 0for allx⁰ ∈V_j. In conclusion we chooseV = (∩0≤j<2M+1V_j)∩V˜, and we have

P(j;x⁰)6= 0 for allx⁰ ∈V and allj ∈N, as required.

Corollary 5.3. There exists a neighborhood V of the origin such that C (j + m−k, x⁰)6= 0for allx⁰ ∈V and allj ∈N, and the induction formula

(5.7) uj+m−k(x⁰) = −Cm−k(j+m−k) C(j+m−k, x⁰)





X

α0<m,|α|≤m

C_µ(α₀₎(j)

×

j+α0−µ(α₀)

X

p=0

j+α₀−µ(α₀) p

a^p_α(x⁰)D^α_x0⁰u_j+α₀−µ(α0)−p(x⁰) +f_j(x⁰)





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yields for allx⁰ ∈V and allj ∈N.

Under the conditions of the Theorem5.1, there exists a unique formal series u(x₀, x⁰) =

∞

X

j=0

u_j(x⁰)x^j₀ j!

solution of the problem (5.1) since all u_j(x⁰) are uniquely determined by (5.7).

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6. Proof of the Main Theorem

Lethbe a differential operator of Fuchsian type inawith respect toSof weight τh,S(a)and of orderm. We want to solve (4.1) in some neighborhood ofa.

Setu−v =wand the problem (4.1) becomes h(w) = g,

(6.1)

w = O(ϕ^τ^h,S^(a)), whereg =f −h(v).

It follows from the second condition of (6.1) that there is a unique holomorphic function U in some neighborhood ofasuch thatw= O(ϕ^τ^h,S^(a))and findingU is equivalent to findingw.

U verifies

h(U ϕ^τ^h,S^(a)) = g, i.e. U satisfies the equation

h1(U) =g,

whereh₁ is a Fuchsian operator of weight zero ina relative toS (by Theorem 4.2).

If we choose a local card such thatϕ(x) =x₀anda= (0, ...,0); in this local card the equation becomes

P˜(U) = Q(U) +g,

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where

P˜ =

m

X

p=0

am−p(x⁰)x^m−p₀ D^m−p₀ , (a_m = 1),

Q = −

m−1

X

α0=0

x^α₀⁰⁺¹D₀^α⁰Bm−α0, and

Bm−α₀ = X

|α⁰|≤m−α₀

a_α(x)D_x^α0⁰.

Let us denote by C(j + m − k, x⁰), (respectively C₁(λ, x⁰)) the Fuchsian polynomial characteristic of h, (respectivelyh₁). It follows from Lemma 5.2 that ifC(j,0)6= 0for allj ∈ N, j ≥ m−k, then there is a neighborhoodV of the origin inCⁿfor whichC(j, x⁰)6= 0for allx⁰ ∈V and allj ∈N,j ≥m−k .HenceP˜is one to one on the set of holomorphic functions at the origin.

Ifu(x₀, x⁰) = P∞

j=0u_j(x⁰)^x

j 0

j! then P˜⁻¹u(x) =

∞

X

j=0

uj(x⁰) C₁(j, x⁰)

x^j₀ j!,

and the problem (6.1) is equivalent to the following problem

(6.2) U =

P˜⁻¹ ◦Q

(U) +U₀, where

U0 = ˜P⁻¹(g).

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As in [1], [4], [5], [7], and [8], a successive approximation method will be used in the following sense.

LetUp+1 =

P˜⁻¹◦Q

(Up) +U0forp≥0, and setVp =Up+1−Up. Then Vp+1 =

P˜⁻¹◦Q

(Vp).

Let D =

(x₀, x⁰) : (x₀, x⁰)∈Cⁿ⁺¹,|x₀| ≤ _R¹ and |x₁|+· · ·+|x_n| ≤ ¹_r , thenV₀is holomorphic in some open neighborhood inD, and there is a constant denotedkV0ksuch that

Vp(x) kV0k 1

1−x₀ξ₀ · 1 1−st(x⁰), where t(x⁰) = Pn

i=0x_i, ξ₀ ≥ _η^R

0, s ≥ _η^r

0, andη₀ is some given number in the open interval(0,1).

Lemma 6.1. There exists a constantK such that (6.3) V_p(x) kV₀k K^p

(s⁰−s)^mp · x^p₀ 1−x0ξ0

· 1 1−s⁰t(x⁰) for alls⁰ > s.

Proof. Clearly (6.3) holds forp= 0. Suppose (6.3) holds forp, and let us prove it forp+ 1.

We have

P˜⁻¹ ◦Q=

m−1

X

α0=0

P˜⁻¹◦(x^α₀⁰⁺¹D₀^α⁰Bm−α₀),

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and we want to study the action of the operatorsB_m−α₀,x^α₀⁰⁺¹D^α₀⁰, andP˜⁻¹on V_p.

1) We have

Bm−α0(x;D_x⁰) = X

|α⁰|≤m−α0

a_α₀_,α⁰(x₀, x⁰)D^α_x⁰⁰.

For allαsuch that|α| ≤mthere isM_αfor whicha_α(x) _1−x^M^α

0R·_1−rt(x¹ 0). Set

Cm−α0(x;D_x⁰) = 1

1−x₀R · 1 1−rt(x⁰)

X

|α⁰|≤m−α0

M_α₀_,α⁰D^α_x⁰⁰.

By Definition3.3,Bm−α₀(x;D_x⁰) Cm−α₀(x;D_x⁰). Let σbe in the open interval(s, s⁰), then

Bm−α0(Vp)(x) kV0k K^p

(σ−s)^mp · 1

1−Rx₀ · x^p₀

1−ξ₀x₀ · 1 1−rt(x⁰)

× X

|α⁰|≤m−α0

M_α₀_,α⁰D^α_x0⁰

1 1−σt(x⁰)

kV0k K^p

(σ−s)^mp · 1

1−Rx₀ · x^p₀

1−ξ₀x₀ · 1 1−rt(x⁰)

× X

|α⁰|≤m−α0

M_α₀_,α⁰ σ^|α⁰^||α⁰|!

[1−σt(x⁰)]^|α⁰^|+1.

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One can check easily that 1

1−Rx₀ · 1

1−ξ₀x₀ 1 1− _ξ^R

0

· 1 1−ξ₀x₀

1

1−η₀ · 1 1−ξ₀x₀. By using [8], we obtain the following majoration

|α⁰|!

[1−σt(x⁰)]^|α⁰^|+1 (m−α₀)!

[1−σt(x⁰)]^m−α⁰⁺¹ hence

Bm−α0(Vp)(x) kV0k K^p

(σ−s)^mp · 1

1−η₀ · x^p₀ 1−ξ₀x₀

× 1

1−rt(x⁰)· (m−α₀)!

[1−σt(x⁰)]^m−α⁰⁺¹

X

|α⁰|≤m−α₀

M_α₀_,α⁰σ^|α⁰^|.

Again by [8], there exists

∼

Cm−α0(a, b)such that Bm−α0(V_p)(x) kV₀k K^p

(σ−s)^mp · 1 1−η₀R_α₀

∼

Cm−α0(a, b) x^p₀ 1−ξ₀x₀

× 1

1−s⁰t(x⁰)· 1

1−rt(x⁰) · (m−α₀)!

(s⁰−σ)^m−α⁰ whereR_α₀ =P

|α⁰|≤m−α0M_α₀_,α⁰b^|α⁰^|.

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If we leta > _η^r

0, then 1

1−s⁰t(x⁰) · 1

1−rt(x⁰) 1

1−η₀ · 1 1−s⁰t(x⁰). Finally,

Bm−α₀(V_p)(x) kV₀k K^p

(σ−s)^mp · R_α₀ (1−η₀)²

×

∼

Cm−α₀(a, b)

(s⁰−σ)^m−α⁰ · 1

1−s⁰t(x⁰)· x^p₀ 1−ξ₀x₀. 2) A straightforward computation leads to the following majoration

x^α₀⁰⁺¹D^α₀⁰Bm−α0(Vp)(x) kV0k K^p

(σ−s)^mp · R_α₀ (1−η₀)² ·

∼

C_m−α₀(a, b) (s⁰−σ)^m−α⁰

×

" _∞ X

j=p+1

ξ₀^j−p−1C_α₀(j −1)x^j₀

# 1 1−s⁰t(x⁰).

Set

w_p(x) = x^α₀⁰⁺¹D^α₀⁰Bm−α₀(V_p)(x) =

∞

X

j=0

ξ₀^j−p−1w_p,j(x⁰)x^j₀,

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hence

w_p,j(x⁰) kV₀k K^p

(σ−s)^mp · R_α₀ (1−η₀)²

×

∼

Cm−α₀(a, b)

(s⁰−σ)^m−α⁰ ·ξ₀^j−p−1C_α₀(j−1)· 1 1−s⁰t(x⁰). If

Fp(x) = ˜P⁻¹(wp)(x) =

∞

X

j=0

Fp,j(x⁰)x^j₀,

then

F_p,j(x⁰) = w_p,j(x⁰) C₁(j, x⁰). It follows from Corollary3.4that

F_p,j(x⁰) kV₀k K^p

(σ−s)^mp · R_α₀ (1−η₀)² ·

∼

Cm−α0(a, b) (s⁰−σ)^m−α⁰

× ξ₀^j−p−1

(j+ 1)^m−α⁰ · 1

1−rt(x⁰) · 1 1−s⁰t(x⁰) kV₀k K^p

(σ−s)^mp · R˜_α₀

(s⁰ −σ)^m−α⁰ · ξ₀^j−p−1

(p+ 1)^m−α⁰ · 1 1−s⁰t(x⁰),