Using Ovcyannikov techniques and the method of majorants, a necessary and sufficient condition for existence and uniqueness of the solution of the problem under con- sideration is shown

http://jipam.vu.edu.au/

Volume 2, Issue 2, Article 24, 2001

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

HOLOMORPHIC FUCHSIAN OPERATORS

MEKKI TERBECHE

FLORIDAINSTITUTE OFTECHNOLOGY, DEPARTMENT OFMATHEMATICALSCIENCES,

MELBOURNE, FL 32901, USA terbeche@hotmail.com

Received 12 March, 2001; accepted 22 April, 2001.

Communicated by R.P. Agarwal

ABSTRACT. In this paper a Cauchy problem for holomorphic differential operators of Fuchsian 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 con- sideration is shown.

Key words and phrases: 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.

2000 Mathematics Subject Classification. 35A10, 58A99.

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 generalize the Goursat problem [8]. We also give a refinement of the method of successive approximations as in the Ovcyannikov Theorem given in [7]. Combining these two methods, we shall prove the theorem [6].

ISSN (electronic): 1443-5756

c 2001 Victoria University. All rights reserved.

022-01

2. NOTATIONS AND DEFINITIONS

Let us denote

x = (x₀, x₁, . . . , x_n)≡(x₀, x⁰)∈R×Rⁿ, wherex⁰ = (x₁, . . . , x_n)∈Rⁿ, ξ = (ξ₀, ξ₁, . . . , ξ_n)≡(ξ₀, ξ⁰)∈R×Rⁿ, whereξ⁰ = (ξ₁, . . . , ξ_n)∈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^α0D^α₁¹· · ·D^α_nⁿ, whereDj = ∂

∂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.

Definition 2.1. LetE be ann+ 1dimensional holomorphic differentiable manifold. Lethbe a holomorphic differentiable operator overE of orderm₀ ina, and of order≤m₀ neara. Let S be a holomorphic hypersurface of E containinga, letmbe an integer≥m0, and letϕbe a local equation ofS in some neighborhood ofa, that is, there exists an open neighborhoodΩof asuch 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 neighborhoodΩofa,

∀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.

(iii)

˜

τ_h,S(a) = inf

σ∈Z:∀Ω,∀Y,∀b∈Ω∩S, lim

x→b,x /∈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 operatorhis said to be a Fuchsian operator of weightτ inawith respect to Sif 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 polynomial C in λ of holomorphic coefficients iny∈S by

C(λ, y) = lim

x→y,x /∈Sϕ^{τh,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 which ϕ(x) = x₀ and a = (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. 3. MAJORANTS

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

Letαbe a multi-index ofNⁿ⁺¹andE be aC-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 in E[[x]]by

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

α∈Nⁿ⁺¹(uα+vα)^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

u_βvα−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]]

be two formal series. We say thatU majorizesu, written U(x) u(x), providedU_α ≥ ku_αk for 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), wherea_µ(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.

We sayP(x;D)majorizesP(x;D),writtenP(x;D)P(x;D), providedA_µ(x)a_µ(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. Let v ∈ E[[y]],(y = (y₁, ..., y_m)), V ∈ R[[y]] such that V(y) v(y). Let u^j(x) ∈ E[[x]]forj = 1, . . . , m, u^j₀ = 0,andU^j(x) ∈ R[[x]]forj = 1, . . . , m, U₀^j = 0 such 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 seriesU(x)converges for|x|< rthen u(x)converges for|x|< r. Let us construct a majorant series through an example.

Letr = (r₀, r₁, ..., r_n)and letu(x)be a bounded holomorphic function on the polydisc P_r ={x:x∈Cⁿ⁺¹,|x_j |< r_j, for allj = 0,1, . . . , n}.

LetM = sup

x∈Pr

ku(x)k, then it follows from Cauchy integral formula thatku_αk< ^M_rαα!.

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

Theorem 3.3. Leta_i,0 ≤ i ≤ m,be holomorphic functions near the origin inCⁿ that 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=0C_i(j) Pm

i=0C_i(j)a_i(x⁰) =

Pm

i=0C_i(j) Pm

i=0C_i(j)a_i(0) · 1

1− ^Pmi=0PCm^{i(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

a_m(0)

= 1.

Therefore, there exists a constantC ≥1such that

Pm

i=0Ci(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, . . . , mwithB(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 Theorem 3.1 that 1

1− ^Pmi=0PCm^{i(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 numbers M >0andr >0such that

1

C₁(j, x⁰) 1

(j+ 1)^m · M

1−rt(x⁰), ∀j ∈N, wheret(x⁰) =Pm

i=0x_i.

Proof. The proof is similar to the proof of the Theorem 3.3, 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.

Theorem 3.5. If A(x⁰)is holomorphic near the origin in Cⁿ, there existM > 0 andR > 0 such that:

A(x⁰) M R−t(x⁰). for allx⁰ ∈ {x⁰ :Pn

i=0 |x_i |< R}.

Proof. See [8].

4. STATEMENT OF THE MAINRESULT

Theorem 4.1 (Main Theorem). Lethbe a Fuchsian holomorphic differential operator of weight τ_h,S(a)inawith respect to a holomorphic hypersurfaceSpassing througha, of a holomorphic differential manifoldE of dimensionn+ 1, andϕa local equation ofSin some neighborhood ofa. Then the following assertions are equivalent:

i) for allλ≥τ_h,S(a),C(λ, x⁰)6= 0

ii) for all holomorphic functions f and v in a neighborhood of a, there exists a unique holomorphic functionusolving the Cauchy problem

h(u) = f (4.1)

u−v = O(ϕ^τh,S(a)).

If we choose a local card such thatϕ(x) =x₀ anda= (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 weightτ_h,S(a)inawith respect to a holomorphic hypersurface S passing through a, of a holomorphic differentiable manifoldEof dimension n+ 1, andϕa local equation ofSin some neighborhood ofa. If we define the operatorh₁by

Y →h₁(Y) = h(Y ϕ^τh,S(a)),

thenh₁ is a Fuchsian holomorphic differential operator of weight zero in arelative to a holo- morphic hypersurfaceS.

IfC(respectivelyC₁) denotes the Fuchsian polynomial characteristic ofh(respectivelyh₁), then C₁(λ, y) =C(λ+τ_h,S(a), y), ∀λ∈C, ∀y ∈S.

Proof. (1) We look for the Fuchsian weight ofh₁ inawith 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+τh,S(a))(x)

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

(2) We look for the principal Fuchsian weight ofh₁inarelative toS.

In the local card ϕ(x) = x₀ and a = (0, . . . ,0), it is the exponent of x₀ in the coefficient ofD^m₀ ofh₁. In this local card

P₁(u) = P(x^m₀^0−ku)

= x^k₀D₀^m(x^m₀^0−ku) +...

= x^k₀D₀^mu+...

(the points indicate the terms that have the order of differentiation with respect to x₀ less thanm0). Hence

τ_h,S^∗ (a) = 0.

(3) Ifm≥m₀, then

x→b,x /lim∈Sϕ^{τh1,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+τh,S(a))(x)−Y(x)h(ϕ^m+τh,S(a))(x)]

= lim

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

= 0, by hypothesis.

Finally, we have

ϕ^{τh1,S(a)−λ}(x)h₁(ϕ^λ)(x) = ϕ^−λ(x)h(ϕ^λ+τh,S(a))(x)

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

This concludes the proof of the Theorem.

5. FORMALPROBLEM

If we choose a local card for which ϕ(x) = x₀ and a = (0, . . . ,0)the Cauchy problem (4.1) becomes (5.1) below. We devote this section to formal calculations by looking for solutions as power series of the problem (5.1) below connected with a Fuchsian operatorP(x;D)of order mand weightm−kwith respect tox₀ atx₀ = 0.

We decompose this operator in the following form

P(x;D) =P_m(x;D₀)−Q(x;D), where

Pm(x;D0) =

k

X

p=0

am−p(x⁰)x^k−p₀ D^m−p₀ , Q(x;D) =− X

α0<m,|α|≤m

x^µ(α₀ ⁰⁾D^α₀⁰(a_α0,α⁰(x₀, x⁰)D^α_x0⁰), witham = 1andµ(α0) = [α0+ 1−(m−k)]+.

Theorem 5.1. If the coefficients ofP_m(x;D₀)andQ(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 functionfnear the origin inCⁿ⁺¹, there exists a unique holomorphic solutionunear the origin inCⁿ⁺¹ solving Cauchy problem

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

D^j₀u(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 determineu_j(x⁰)for allj ≥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)vp(x⁰)

#x^j₀ (5.2) j!

D^p₀u(x0, x⁰) =

∞

X

j=0

uj+p(x⁰)x^j₀ (5.3) j!

x^q₀D^p₀u(x₀, x⁰) =

∞

X

j=0

[C_q(j)u_j+p−q(x⁰)]x^j₀ (5.4) j!

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!, whereD(j, x⁰) =Pk

j=0am−q(x⁰)Ck−p(j)which can be written in terms ofC(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)uj+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!.

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)

C_m−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.

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

Proof. We have|P(λ;x⁰)| ≥ |λ|^m−

Pm−1

k=0 a_k(x⁰)λ^k

. LetM = 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

# . 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 that 0 ≤ 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 neighborhoodV of the origin such thatC (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)

×X_{j+α0−µ(α0)}

p=0

j +α₀−µ(α₀) p

a^p_α(x⁰)D_x^α0⁰u_{j+α0−µ(α0)−p}(x⁰) +f_j(x⁰)

yields for allx⁰ ∈V and allj ∈N.

Under the conditions of the Theorem 5.1, 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 allu_j(x⁰)are uniquely determined by (5.7).

6. PROOF OF THE MAIN THEOREM

Lethbe a differential operator of Fuchsian type inawith respect toSof weightτ_h,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^τh,S(a)), whereg =f −h(v).

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

U verifies

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

h1(U) =g,

whereh₁ is a Fuchsian operator of weight zero inarelative 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, where

P˜ =

m

X

p=0

am−p(x⁰)x^m−p₀ D₀^m−p, (am = 1),

Q = −

m−1

X

α0=0

x^α₀⁰⁺¹D₀^α0Bm−α₀, and

Bm−α₀ = X

|α⁰|≤m−α0

a_α(x)D^α_x0⁰.

Let us denote byC(j+m−k, x⁰), (respectivelyC₁(λ, x⁰)) the Fuchsian polynomial characteristic ofh, (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 all j ∈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

u_j(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).

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 V_p+1 =

P˜⁻¹◦Q (V_p).

LetD =

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

V_p(x) kV₀k 1

1−x₀ξ₀ · 1 1−st(x⁰), where t(x⁰) = Pn

i=0x_i, ξ₀ ≥ _η^R

0, s ≥ _η^r

0, and η₀ 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),

and we want to study the action of the operatorsBm−α0,x^α₀⁰⁺¹D^α₀⁰, andP˜⁻¹onV_p. 1) We have

Bm−α0(x;Dx⁰) = X

|α⁰|≤m−α0

aα0,α⁰(x0, x⁰)D_x^α00.

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

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 Definition 3.3, Bm−α₀(x;D_x⁰) Cm−α₀(x;D_x⁰). Let σ be in the open interval (s, s⁰), then

Bm−α0(V_p)(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_x^α0⁰

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. 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(V_p)(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

∼

C_m−α0(a, b)such that B_m−α0(V_p)(x) kV₀k K^p

(σ−s)^mp · 1 1−η₀R_α0

∼

C_m−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|. 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^α₀⁰B_m−α0(V_p)(x) kV₀k K^p

(σ−s)^mp · R_α0 (1−η₀)² ·

∼

Cm−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⁺¹D₀^α0Bm−α₀(V_p)(x) =

∞

X

j=0

ξ₀^j−p−1w_p,j(x⁰)x^j₀, 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

F_p(x) = ˜P⁻¹(w_p)(x) =

∞

X

j=0

F_p,j(x⁰)x^j₀, then

F_p,j(x⁰) = wp,j(x⁰) C₁(j, x⁰). It follows from Corollary 3.4 that

F_p,j(x⁰) kV₀k K^p

(σ−s)^mp · Rα0

(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 · ξ^j−p−1₀

(p+ 1)^m−α0 · 1 1−s⁰t(x⁰), for allj ≥p+ 1, whereR˜_α0 = _(1−η^Rα0

0)³

∼

Cm−α0(a, b), hence F_p(x) kV₀k K^p

(σ−s)^mp · R˜_α0

[(p+ 1)(s⁰−σ)]^m−α0

" _∞ X

j=p+1

ξ^j−p−1₀ x^j₀

# 1 1−s⁰t(x⁰) kV₀k K^p

(σ−s)^mp · R˜_α0

[(p+ 1)(s⁰−σ)]^m−α0 · x^p+1₀

1−ξ₀x₀ · 1 1−s⁰t(x⁰).

If we chooseσ such thats < σ < s⁰ ands⁰ −σ = ^s_p+1^0−s then the following majoration holds

1

(σ−s)^mp · 1

[(p+ 1)(s⁰−σ)]^m−α0 ≤e^m (b−a)^α0 (s⁰−s)^m(p+1) and consequently

F_p(x) kV₀k K^p

(s⁰−s)^m(p+1) ·R˜_α0e^m(b−a)^α0 · x^p+1₀

1−ξ₀x₀ · 1 1−s⁰t(x⁰).

Finally

V_p+1(x) kV₀k · K^p

(s⁰−s)^m(p+1)e^m ·

"_m−1 X

α0=0

R˜_α0(b−a)^α0

# x^p+1₀

1−ξ₀x₀ · 1 1−s⁰t(x⁰). Choosing then

K =e^m

m−1

X

α0=0

R˜_α0(b−a)^α0 and yields easily the lemma.

If we impose|ξ₀x₀| ≤ρ₀ <1andb(|x₁|+· · ·+|x_n|)≤ρ₀ <1then

|V_p(x)| ≤ kV₀k

K|x₀| (s⁰−s)^m

p

1 (1−ρ₀)² for allp∈Nand alls⁰ > s.

If |x₀| ≤ ^(s0−s)_K0 ^m, where K⁰ > K, then the series of general term V_p converges normally and the sequence of general termU_p converges uniformly to a holomorphic functionU on some suitable choice of polydiscs centered at the origin inCⁿ⁺¹.

Since

P˜(U_p+1) =Q(U_p) +g, then the limitU satisfies the equation

P˜(U) = Q(U) +g thereforeh₁(U) = gas desired.

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