Suitable Coordinates for Singular First Order Linear Partial Differential Equations in a Formal Gevrey Class

Hiroshi Yamazawa

Department of Language and Culture, Caritas Junior College, Azamino Aoba-ku, Yokohama 225-0011 Japan

1 Introduction

In this paper, we study a formal power series solution of th following linear partial differential equation;

Lu(χ) = ∑_{i=1}^{d} a_{i}(χ)∂_{χ_{i}}u(χ) + b(χ)u(χ) = f(χ),  (1.1)

where χ = (χ_{1}, …, χ_{d}) ∈ C^{d}, ∂_{χ_{i}} = ∂/∂χ_{i}, a_{i}(χ), b(χ) and f(χ) are holomorphic functions in a neighborhood of the origin with a_{i}(0) = 0 for i = 1, …, d, and the case when a_{i}(0) = 0 (∀i) the operator L is called a singular type.

There are many results for the equation (1.1). We are concerned with the following result. Oshima [O] characterized a solvability of holomorphic solutions of the equation (1.1) by the kernel and cokernel of the mapping

L : O → O

where O is the stalk at the origin of the sheaf of holomorphic functions over C^{d}. Kaplan [K] and Bengel-Gérard [BG] studied the existence of holomorphic solutions for non linear equations.

Yamazawa [Y] studied a uniqueness of a formal power series solution and an estimate of coefficients of the solution with the form ∑_{|α|≥0} u_{α}(x)y^{α} via an appropriate coordinates change under more relax conditions than those in [K] for nonlinear equations, where the coefficients {u_{α}(x)}_{|α|≥0}} are holomorphic functions in a common neighborhood of x = 0 and a dimension of x is a number of nonzero eigenvalues for Jacobian of (a_{1}(χ), …, a_{d}(χ)) at χ = 0. Hibino [H] studied divergent more general operators than those in [Y] for linear equations. But a formal power series solution in [H] is not the above form and diverges in all variables. It is our purpose to structure a solution with the above form for the equations that are studied in [H].

We show our motivation by the following essential example. In [H], for

(χ_{1}∂_{χ_{1}} + 1)u − χ_{3}∂_{χ_{2}}u − (χ_{1}^{2} + χ_{2}^{2})∂_{χ_{2}}u = χ_{2}

the formal solution u(χ) = ∑_{|l|≥0} u_{l}χ^{l} satisfies u(χ_{1}, 0, 0) ≫ ∑_{l_{1}≥1} 2^{−l_{1}}(l_{1}−1)!χ_{1}^{2l_{1}}, u(0, χ_{2}, 0) ≫ ∑_{l_{2}≥1}(l_{2} − 1)!χ_{2}^{l_{2}} and u(0, 0, χ_{3}) ≫ ∑_{l_{3}≥1} l_{3}!(l_{3} − 1)!χ_{3}^{l_{3}}. For this example, we change the coordinate

x = χ_{1}, y = χ_{2} + ϕ(χ_{1}) and z = χ_{3}

where the function ϕ satisfies the equation

χ_{1} \frac{dϕ}{dχ_{1}} (χ_{1}) = χ_{1}^{2} + ϕ(χ_{1})^{2}

that has a holomorphic solution in a neighborhood of the origin with ϕ(χ_{1}) ≡ O(χ_{1}^{2}). Then u(χ) = v(x(χ), y(χ), z(χ)) = ∑_{|l|≥0} v_{l}x^{l_{1}}y^{l_{2}}z^{l_{3}} satisfies

|v_{l}| ≤ c\frac{l_{2}!l_{3}!^{2}}{X^{|l|}} for some positive constant c and X.

Hence we understand that ∑ \frac{|u_{l}|}{l_{2}!l_{3}!^{2}} χ^{l} is not a convergent power series but

∑ \frac{|v_{l}|}{l_{2}!l_{3}!^{2}} x^{l_{1}}t^{l_{2}}y^{l_{3}} is a convergent power series in a neighborhood of the origin, and

show that formal power series solutions have better estimates via an appropriate coordinates change.

2 Notations and Result

In this section, for the operator L_{0} := ∑_{i=1}^{d} a_{i}(χ)∂_{χ_{i}}, Let us explain some signs and

define some conditions, and we assume the conditions.

N, C and R_{+} denote the set of all non negative integers, complex numbers and positive real numbers, respectively. C{·} and C[[·]] denote the set of all convergent and formal power series in a neighborhood of the origin of the variable {·}. e_{i}^{j} = {e = (e_{1}, …, e_{j}) ∈ N^{j}; e_{i} = 1 and e_{i′} = 0 for i′ ≠ i}.

(1) The matrix (∂_{χ}a(0)) denotes the Jacobian matrix of a(χ) = (a_{1}(χ), …, a_{d}(χ)) at the origin.

(A.1) Jordan normal form of (∂_{χ}a(0)) is

( \begin{matrix}A \\ & B_{1} \\ & & . & . & . \\ & & & & & B_{k} \\ & & & & & & O_{p} \end{matrix} )  (2.1)

where

A = ( \begin{matrix}λ_{1} & μ_{1} \\ & ... & ... \\ & & ... & μ_{m−1} \\ & & & λ_{m} \end{matrix} ) , B_{h} = ( \begin{matrix}0 & 1 \\ & ... & ... \\ & & ... & 1 \\ & & & 0 \end{matrix} ) ,

λ_{i} ≠ 0 for i = 1, 2, ⋯, m and μ_{i} = 0 or 1 for i = 1, 2, ⋯, m − 1 B_{h} is a matrix of order n_{h} for h = 1, …, k and O_{p} is a zero-matrix of order p where m, k, p ≥ 0, m + |n|_{k} + p = d and |n|_{h} = n_{1} + ⋯ + n_{h} for h = 1, 2, . . .. Without loss of generality, we may assume N = n_{1} ≥ n_{2} ≥ ⋯, ≥ n_{k} ≥ 2.

We put for the operator L_{0},

L_{0}^{0} := identity, L_{0}^{1} := L_{0}, L_{0}^{2} := L_{0} ∘ L_{0}, L_{0}^{3} := L_{0} ∘ L_{0} ∘ L_{0}, …,

and so on.

For the coefficients a(χ), let us consider L_{0}^{N−1}a_{1}(χ), …, L_{0}^{N−1}a_{d}(χ). By the condition (A.1), we can take m−functions L_{0}^{N−1}a_{i_{1}}(χ), …, L_{0}^{N−1}a_{i_{m}}(χ), that are functionally independent each other, from L_{0}^{N−1}a_{1}(χ), …, L_{0}^{N−1}a_{d}(χ). By rewriting number, we may put (a_{i_{1}}, …, a_{i_{m}}) ↦ (a_{1}, …, a_{m}). Then we define

I = I{L_{0}^{N−1}a_{1}(χ), …, L_{0}^{N−1}a_{m}(χ)} := ∑_{i=1}^{m} C{χ}L_{0}^{N−1}a_{i}(χ)  (2.2)

and let M be the maximal ideal of C[[χ]].

(2) For each i = 1, …, d, there exists a number J_{i} (J_{i} ≤ N − 1) such that

J_{i} = min{j_{i}; L_{0}^{j_{i}}a_{i}(χ) ∈ M^{2} mod I},

and we define

δ_{i} := sup{d_{i}; L_{0}^{J_{i}}a_{i}(χ) ∈ M^{d_{i}} mod I}.

Definition 2.1

δ = min{δ_{1}, …, δ_{d}}.

Remark 2.2 The number δ is said to be of multiplicity of the operator L in this paper, and multiplicity δ holds δ ≥ 2.

If we assume the condition (A.1) then the multiplicity δ is invariant under coordinates change and independent of a choice of m-independent functions from (a_{1} …, a_{d}), and a proof of these facts can be showed as Section 3 in [Y].

By the condition (A.1), by rewriting number, we can take (a_{1}, …, a_{m+|n|_{k}−k}), that are functionally independent each other, from (a_{1} …, a_{d}) where (a_{1} …, a_{m}) in (2.2).

(3) For (a_{m+1}, …, a_{m+|n|_{k}−k}), we define

S_{h} = {χ ∈ C^{n}; L_{0}^{n_{h}−2}a_{i}(χ) = 0 for i = m + 1, …, m + |n|_{k} − k}

for h = 1, …, k. Then we assume the following additional conditions; (A.C)

1 ≤ codimS_{h} ≤ |n|_{h−1} − (h − 1)n_{h} + h

for h = 1, …, k.

Let us define the function space that is studied in this space. Let x = (x_{1}, …, x_{m}) ∈ C^{m}, y = (y^{1}, …, y^{k}) ∈ C^{|n|_{k}} with y^{h} = (y_{1}^{h}, …, y_{n_{h}}^{h}) ∈ C^{n_{h}} and

z ∈ C^{p}. For u(x, y, z) ∈ C[[x, y, z]], we set u(x, y, z) = ∑_{|α|+|β|+|γ|≥0} u_{α,β,γ}x^{α}y^{β}z^{γ},

where u_{α,β,γ} ∈ C, α ∈ N^{m}, β ∈ N^{|n|_{k}} and γ ∈ N^{p}.

Let s_{j_{h}}^{h}, s_{q} ≥ 1, ϵ > 0, ξ = (ξ_{1}, …, ξ_{m}) ∈ R_{+}^{m} and η = (η^{1}, …, η^{k}) ∈ R_{+}^{|n|_{k}} for j_{h} = 1, …, n_{h}, h = 1, …, k and q = 1 …, p. Then we define a Banach space G_{ξ,η}^{{s_{y},s_{z}}}(ϵ)[x, y, z] ⊂ C[[x, y, z]] as follows for fixed ξ and η;

G_{ξ,η}^{{s_{y},s_{z}}}(ϵ)[x, y, z] := {u(x, y, z) ∈ C[[x, y, z]]; ||u||_{ξ,η}^{{s_{y},s_{z}}}(ϵ) < ∞} (2.3)

where

||u||_{ξ,η}^{\{s_{y},s_{z}\}}(ϵ) := sup_{α,β,γ} |u_{α,β,γ}|\frac{|α|!|β|!|γ|!ϵ^{|α|+|β|+|γ|}}{\{|α| + s_{y} · β + s_{z} · γ\}!ξ^{α}η^{β}} ,  (2.4)

s_{y} · β = ∑_{h=1}^{k} ∑_{j_{h}=1}^{n_{h}} s_{j_{h}}^{h}β_{j_{h}}^{h}, s_{z} · γ = ∑_{q=1}^{p} s_{q}γ_{q} and n! = Γ(n + 1) for n ≥ 0.

Here we define a formal Gevrey space. Definition 2.3

G^{{s_{y},s_{z}}}[x, y, z] := ∪_{ϵ>0}G_{ξ,η}^{{s_{y},s_{z}}}(ϵ)[x, y, z].

The Gevrey space G^{{s_{y},s_{z}}}[x, y, z] does not depend on the choice of ξ and η and u(x, y, z) belongs to G^{{s_{y},s_{z}}}[x, y, z] if and only if there exists positive constants ϵ′ and c such that

|u_{α,β,γ}| ≤ c\frac{∏_{h=1}^{k} ∏_{j_{h}=1}^{n_{h}} |β_{j_{h}}^{h}|!^{s_{j_{h}}^{h}−1} ∏_{q=1}^{p} |γ_{q}|!^{s_{q}−1}}{ϵ′^{|α|+|β|+|γ|}} .

Let us define

O(||x|| < R)[[y, z]] := { ∑_{|β|+|γ|≥0} u_{β,γ}(x)y^{β}z^{γ} ∈ C[[y, z]]}

where the coefficients {u_{β,γ}(x)}_{|β|+|γ|≥0} are holomorphic functions in |x_{1}| + ⋯ + |x_{m}| < R. We define

Definition 2.4

G^{∞}[x, y, z] := ∪_{R>0}O(||x|| < R)[[y, z]].

We obtain the following the main result.

Theorem 2.5 Assume that the operator L satisfies the conditions (A.1) and (A.C) and further assume that there exists a positive constant σ such that

|∑_{i=1}^{m} λ_{i}α_{i} + b(0)| ≥ σ(|α| + 1) for all α ∈ N^{m}.  (2.5)

Then there exist local coordinates (x(χ), y(χ), z(χ)) ∈ C^{m} × C^{|n|_{k}} × C^{p} in a neighborhood of the origin such that the mapping

L : G^{∞}[x, y, z]/G^{{s_{y},s_{z}}}[x, y, z] ↦ G^{∞}[x, y, z]/G^{{s_{y},s_{z}}}[x, y, z]

is bijective for s_{y} = (s_{y^{1}}, …, s_{y^{k}}) with s_{y^{h}} = (\frac{N}{δ−1} + 1, …, \frac{N}{δ−1} + n_{h}) and s_{z} = (\frac{N}{δ−1} + 1, …, \frac{N}{δ−1} + 1) for h=1, …,k.

Remark 2.6 The equation (x∂_{x}+1)u−y_{2}∂_{y_{1}}u−(xy_{1}+y_{1}^{2})∂_{y_{2}}u = \frac{1}{1−x} + \frac{1}{1−y_{1}} + \frac{1}{1−y_{2}} does not satisfy (A.C). Then we have

u_{l} ≥ \frac{(l_{1} + l_{2})!(l_{2} + l_{3})!l_{1}!}{(l_{1} + 1)!l_{2}!l_{3}!} (\frac{l_{2} + l_{3}}{2} )! for l_{1} ≤ l_{2}

for the solution u = ∑_{|l|≥0} u_{l}x^{l_{1}}y_{1}^{l_{2}}y_{2}^{l_{3}}. Therefore we need the condition (A.C) to obtain this result.

3 Preliminary lemmas

In this section, we show some lemmas that are needed to prove Theorem 2.5.

3.1 Existence of a holomorphic solution of system equations

Let x = (x_{1}, …, x_{N}) ∈ C^{N} and the functions a_{I}(x, Φ), b_{J}(x, Φ) are convergent power series in a neighborhood of the origin in C^{N} ×C^{M} where Φ = (ϕ_{1}, ⋯, ϕ_{M}) for I = 1, …, N and J = 1, …, M. We assume that a_{I}(x, Φ), b_{J}(x, Φ) have the following expansion;

\begin{aligned}a_{I}(x, Φ) & = ∑_{|p|+|q|≥2} a_{I,p,q}\{x\}^{p}\{Φ\}^{q} \\ b_{J}(x, Φ) & = ∑_{|p|+|q|≥2} b_{J,p,q}\{x\}^{p}\{Φ\}^{q}. \end{aligned}  (3.1)

We consider the following system equation;

∑_{I=1}^{N} {λ_{I}x_{I} + μ_{I}x_{I+1} + a_{I}(x, Φ)} ∂_{x_{I}}ϕ_{J} = A^{J+1}ϕ_{J+1} + b_{J}(x, Φ)  (3.2)

for J = 1, …, M, where μ_{N} = 0, A^{J} = 0 or 1 for J = 1, …, M and A^{M+1} = 0. Then we have the following lemma.

Lemma 3.1 Assume that there exists a positive constant σ such that

|∑_{I=1}^{N} λ_{I}K_{I} | ≥ σ|K_{1} + ⋯ + K_{N}| for ∀K = (K_{1}, K_{2}, ⋯, K_{N}) ∈ N^{N}. (3.3)

Then we obtain that (3.2) has a tuple of unique holomorphic solution (ϕ_{1}(x), …, ϕ_{M}(x)) in a neighborhood of the origin with ϕ_{J}(x) = O(|x|^{2}) for J = 1, 2, …, M.

3.2 Estimates of Gamma function

Let δ ≥ 2, α′ + α′′ = α ∈ N^{m}, β′ + β′′ = β ∈ N^{|n|_{k}} and γ′ + γ′′ = γ ∈ N^{p}, and we put for j_{h} = 1, …, n_{h}, h = 1, …, k and q = 1, …, p,

s_{j_{h},y}^{h} = \frac{N}{δ − 1} + j_{h} and s_{q,z} = \frac{N}{δ − 1} + 1.

Here we assume the following conditions (C.δ); (C.δ) Assume |α′′| + |β′′| + |γ′′| ≥ 2, further if |α′′| = 0, assume |β′′| + |γ′′| ≥ δ. Then we have the following lemmas. Lemma 3.2 Assume the condition (C.δ). Then we have

\frac{α′_{i} + 1}{|α| + 1} \frac{(|α′| + s_{y} · β′ + s_{z} · γ′)!}{(|α′| + 1)!|β′|!|γ′|!} ≤ \frac{(|α| + s_{y} · β + s_{z} · γ)!}{|α|!|β|!|γ|!}  (3.4)

for i = 1, …, m. Lemma 3.3 Assume the condition (C.δ) and further assume

∑_{k=1}^{h} (β_{n_{h}}^{′′k} + ⋯ + β_{n_{k}}^{′′k}) ≥ 1,  (3.5)

then we have

\frac{β_{n_{h}}^{′h} + 1}{|α| + 1} \frac{(|α′| + s_{y} · (β′ + e_{n_{h}}^{n_{h}}) + s_{z} · γ′)!}{|α′|!(|β′| + 1)!|γ′|!} ≤ \frac{(|α| + s_{y} · β + s_{z} · γ)!}{|α|!|β|!|γ|!} (3.6)

for h = 1, …, k. Lemma 3.4 Assume the condition (C.δ) and further assume |β′′| + |γ′′| ≥ 1, then we have

\frac{γ′_{q} + 1}{|α| + 1} \frac{(|α′| + s_{y} · β′ + s_{z} · (γ′ + e_{q}^{p}))!}{|α′|!|β′|!(|γ′| + 1)!} ≤ \frac{(|α| + s_{y} · β + s_{z} · γ)!}{|α|!|β|!|γ|!}  (3.7)

for q = 1, …, p. Lemma 3.5 Assume |α′′| + |β′′| + |γ′′| ≥ 1, then we have

\frac{1}{|α| + 1} \frac{(|α′| + s_{y} · β′ + s_{z} · γ′)!}{|α′|!|β′|!|γ′|!} ≤ \frac{(|α| + s_{y} · β + s_{z} · γ)!}{|α|!|β|!|γ|!}.  (3.8)

4 Formal Gevrey estimates of formal solution

for special case

In this section, we consider the following operator P that satisfies the assumptions of Theorem 2.5;

\begin{aligned}P & = ∑_{i=1}^{m} \{λ_{i}x_{i} + μ_{i}x_{i+1} + a_{i}(x, y, z)\} ∂_{x_{i}} \\ & + ∑_{h=1}^{k} \{ ∑_{j_{h}=1}^{n_{h}−1} y_{j_{h}+1}^{h}∂_{y_{j_{h}}^{h}} + b_{h}(x, y, z)∂_{y_{n_{h}}^{h}} \} + ∑_{q=1}^{p} c_{q}(x, y, z)∂_{z_{q}} + d(x, y, z). \end{aligned}

The coefficients of the operator P satisfy

\begin{matrix}a_{i}(0, y, z) ≡ b_{h}(0, y, z) ≡ c_{q}(0, y, z) ≡ O ((|y| + |z|)^{δ}) , \\ b_{h} (x, y_{(n_{h}−1)}^{1}, o, …, y_{(n_{h}−1)}^{h}, o, y^{h+1}, …, y^{k}, z) ≡ c_{q}(x, 0, 0) ≡ 0, \\ a_{i}(x, y, z) ≡ O ((|x| + |y| + |z|)^{2}) \end{matrix} (4.1)

where (y_{(n_{h}−1)}^{h′}, o) = (y_{1}^{h′}, …, y_{n_{h}−1}^{h′}, 0, …, 0) ∈ C^{n_{h′}} for h′ = 1, …, h, i = 1, …, m, h = 1, …, k and q = 1, …, p, and we put δ ≥ 2 and μ_{m} := 0.

Theorem 4.1 Assume that for the operator P, there exists a positive constant σ such that

P (α) := |∑_{i=1}^{m} λ_{i}α_{i} + d(0, 0, 0)| ≥ σ(|α| + 1) for α ∈ N^{m}.  (4.2)

Then the mapping P : G^{{s_{y},s_{z}}}[x, y, z] → G^{{s_{y},s_{z}}}[x, y, z] is bijective, where {s_{y}, s_{z}} in Theorem 2.5.

We show some lemmas to prove Theorem 4.1.

We put

P_{0} = ∑_{i=1}^{m} {λ_{i}x_{i}} ∂_{x_{i}} + d(0, 0, 0).

Here we study an estimate of the mapping

(P − P_{0})P_{0}^{−1} : G_{ξ,η}^{{s_{y},s_{z}}}(ϵ)[x, y, z] → G_{ξ,η}^{{s_{y},s_{z}}}(ϵ)[x, y, z],  (4.3)

where

P_{0}^{−1} : ∑_{α,β,γ} u_{α,β,γ}x^{α}y^{β}z^{γ} ↦ ∑_{α,β,γ} P (α)^{−1}u_{α,β,γ}x^{α}y^{β}z^{γ}.

Lemma 4.2 Suppose the assumption of Theorem 4.1, then the mapping (4.3) is a contraction mapping on G_{ξ,η}^{{s_{y},s_{z}}}(ϵ)[x, y, z] for sufficiently small ϵ > 0.

Proof. We have

a_{i}(x, y, z) = ∑_{|α′′|+|β′′|+|γ′′|≥2} a_{i,α′′,β′′,γ′′}x^{α′′}y^{β′′}z^{γ′′},

b_{h}(x, y, z) = ∑_{|α′′|+|β′′|+|γ′′|≥2} b_{h,α′′,β′′,γ′′}x^{α′′}y^{β′′}z^{γ′′},

c_{q}(x, y, z) = ∑_{|α′′|+|β′′|+|γ′′|≥2} c_{q,α′′,β′′,γ′′}x^{α′′}y^{β′′}z^{γ′′},

where

a_{i,0,β′′,γ′′} = b_{h,0,β′′,γ′′} = c_{q,0,β′′,γ′′} = 0 for |β′′| + |γ′′| < δ,
b_{h,α′′,β′′,γ′′} = 0 for β′′ = (β′′_{(n_{h}−1)}^{1}, o, …, β′′_{(n_{h}−1)}^{h}, o, β′′^{h+1}, …, β′′^{k}) ,
c_{q,α′′,0,0} = 0

by (4.1), where (β_{(n_{h}−1)}^{′′h′}, o) = (β_{1}^{′′h′}, …, β_{n_{h}−1}^{′′h′}, 0, …, 0) for h′ = 1, …, h. Then

for u = ∑_{|α|+|β|+|γ|≥0} u_{α,β,γ}x^{α}y^{β}z^{γ}, we have

(P − P_{0})P_{0}^{−1}u = ∑_{|α|+|β|+|γ|≥0} v_{α,β,γ}x^{α}y^{β}z^{γ},

where

v_{α,β,γ} = P (α)^{−1}{ ∑_{i=1}^{m−1} μ_{i}(α_{i} + 1)u_{α+e_{i}^{m}−e_{i+1}^{m},β,γ} + ∑_{h=1}^{k} ∑_{j_{h}=1}^{n_{h}−1}(β_{j_{h}}^{h} + 1)u_{α,β+e_{j_{h}}^{n_{h}}−e_{j_{h}+1}^{n_{h}},γ}

+ ∑_{\begin{matrix}α′+α′′=α \\ β′+β′′=β \\ γ′+γ′′=γ \end{matrix}} \{ ∑_{i=1}^{m}(α′_{i} + 1)a_{i,α′′,β′′,γ′′}u_{α′+e_{i}^{m},β′,γ′}

+ ∑_{h=1}^{k}(β′_{n_{h}}^{h} + 1)b_{h,α′′,β′′,γ′′}u_{α′,β′+e_{n_{h}}^{n_{h}},γ′}

+ ∑_{q=1}^{p}(γ′_{q} + 1)c_{q,α′′,β′′,γ′′}u_{α′,β′,γ′+e_{q}^{p}} + d_{α′′,β′′,γ′′}u_{α′,β′,γ′}}}.

Then by Lemma 3.2, 3.3, 3.4 and 3.5, we have

\frac{|α|!|β|!|γ|!ϵ^{|α|+|β|+|γ|}}{\{|α| + s_{y} · β + s_{z} · γ\}!ξ^{α}η^{β}} |v_{α,β,γ}|

≤ ( ∑_{i=1}^{m−1} μ_{i} \frac{ξ_{i}}{ξ_{i+1}} ) ||u||_{ξ,η}^{\{s_{y},s_{z}\}}(ϵ) + (∑_{h=1}^{k} ∑_{j_{h}=1}^{n_{h}−1} \frac{η_{j_{h}}^{h}}{η_{j_{h}+1}^{h}} ) ||u||_{ξ,η}^{\{s_{y},s_{z}\}}(ϵ)

+ ∑_{|α′′|+|β′′|+|γ′′|≥2} \{ ∑_{i=1}^{m} \frac{|a_{i,α′′,β′′,γ′′}|}{ξ^{α′′−e_{i}^{m}}η^{β′′}} + ∑_{h=1}^{k} \frac{|b_{h,α′′,β′′,γ′′}|}{ξ^{α′′}η^{β′′−e_{n_{h}}^{n_{h}}}} + ∑_{q=1}^{p} \frac{|c_{q,α′′,β′′,γ′′}|}{ξ^{α′′}η^{β′′}}\}

× ϵ^{|α′′|+|β′′|+|γ′′|−1}||u||_{ξ,η}^{{s_{y},s_{z}}}(ϵ)

+ ∑_{|α′′|+|β′′|+|γ′′|≥1} \frac{|d_{α′′,β′′,γ′′}|}{ξ^{α′′}η^{β′′}}ϵ^{|α′′|+|β′′|+|γ′′|}||u||_{ξ,η}^{\{s_{y},s_{z}\}}(ϵ).

For any ϵ > 0, we can take ξ and η such that

∑_{i=1}^{m−1} μ_{i} \frac{ξ_{i}}{ξ_{i+1}} < ϵ and ∑_{h=1}^{k} ∑_{j_{h}=1}^{n_{h}−1} \frac{η_{j_{h}}^{h}}{η_{j_{h}+1}^{h}} < ϵ.

Therefore we have

\frac{|α|!|β|!|γ|!ϵ^{|α|+|β|+|γ|}}{\{|α| + s_{y} · β + s_{z} · γ\}!ξ^{α}η^{β}} |v_{α,β,γ}| = O(ϵ).

Hence we have the desired result for sufficiently small ϵ > 0. Q.E.D.

Proof of Theorem 4.1. By Lemma 4.2, the operator P P_{0}^{−1} is bijective on G_{ξ,η}^{{s_{y},s_{z}}}(ϵ)[x, y, z], and since P_{0}P_{0}^{−1} = P_{0}^{−1}P_{0} = I (identity) holds on G_{ξ,η}^{{s_{y},s_{z}}}(ϵ)[x, y, z], the operator P is bijective on G_{ξ,η}^{{s_{y},s_{z}}}(ϵ)[x, y, z]. Hence we have the desired result. Q.E.D.

5 Proof of Theorem 2.5

In this section, we prove Theorem 2.5 to show that the operator L becomes the form of the operator P in Section 4 via some coordinates change.

For the operator L we put

\begin{aligned}L_{0} & = ∑_{i=1}^{m} \{λ_{i}x_{i} + μ_{i}x_{i+1} + a_{i}(x, y, z)\} ∂_{x_{i}} \\ & + ∑_{h=1}^{k} ∑_{j_{h}=1}^{n_{h}} \{A_{j_{h}+1}^{h}y_{j_{h}+1}^{h} + b_{j_{h}}^{h}(x, y, z)\} ∂_{y_{j_{h}}^{h}} + ∑_{q=1}^{p} c_{q}(x, y, z)∂_{z_{q}}, \end{aligned} (5.1)

where μ_{i} = 0 or 1 for i = 1, …, m − 1, μ_{m} = 0, A_{j_{h}}^{h} = 1 for j_{h} = 2, …, n_{h}, A_{n_{h}+1}^{h} = 0 and

a_{i} ≡ b_{j_{h}}^{h} ≡ c_{q} ≡ O ((|x| + |y| + |z|)^{2}) .

Remark 5.1 By the condition (A.1), we may assume that a form of L_{0} is (5.1). In fact, this can be proved via linear transformations.

For the operator (5.1), we consider the following system equation;

\begin{aligned}∑_{i=1}^{m} \{λ_{i}x_{i} + μ_{i}x_{i+1} + a_{i}(x, ϕ^{1}, …, ϕ^{k}, ϕ^{z})\} ∂_{x_{i}}ϕ_{j_{h}}^{h}(x) & = A_{j_{h}+1}^{h}ϕ_{j_{h}+1}^{h}(x) \\ & + b_{j_{h}}^{h}(x, ϕ^{1}, …, ϕ^{k}, ϕ^{z}), \\ ∑_{i=1}^{m} \{λ_{i}x_{i} + μ_{i}x_{i+1} + a_{i}(x, ϕ^{1}, …, ϕ^{k}, ϕ^{z})\} ∂_{x_{i}}ϕ_{q}^{z}(x) & = c_{q}^{h}(x, ϕ^{1}, …, ϕ^{k}, ϕ^{z}) \end{aligned}(5.2)

for j_{h} = 1, …, n_{h}, h = 1, …, k and q = 1, …, p, where

ϕ^{h} = (ϕ_{1}^{h}, …, ϕ_{n_{h}}^{h}) and ϕ^{z} = (ϕ_{1}^{z}, …, ϕ_{p}^{z}) for h = 1, …, k.

Then we have the following lemma.

Lemma 5.2 The system equation (5.2) has a holomorphic solution Φ = (ϕ^{1}, …, ϕ^{k}, ϕ^{z}) with ϕ_{j_{h}}^{h}(x) ≡ ϕ_{q}(x)^{z} ≡ O(|x|^{2}) for j_{h} = 1, …, n_{h}, h = 1, …, k and q = 1, …, p.

Proof. We put

m ↦ N, (x_{1}, …, x_{m}) ↦ (x_{1}, …, x_{N}), (ϕ^{1}, …, ϕ^{k}, ϕ^{z}) ↦ (Φ),
(a_{1}, …, a_{m}) ↦ (a_{1}, …, a_{N}), (b^{1}, …, b^{k}, c) ↦ (b_{1}, …, b_{M}) and
(A^{1}, …, A^{k}, 0, …, 0) ↦ (A^{1}, …, A^{M}, 0),

where N, (x_{1}, …, x_{N}), (Φ), (a_{1}, …, a_{N}), (b_{1}, …, b_{M}) and (A^{1}, …, A^{M}, 0) in Lemma 3.1. Hence we obtain the desired result by Lemma 3.1. Q.E.D.

We change coordinates as follows;

X_{i} = x_{i}, Y_{j_{h}}^{h} = y_{j_{h}}^{h} − ϕ_{j_{h}}^{h}(x) and Z_{q} = z_{q} − ϕ_{q}^{z}(x)  (5.3)

for i = 1, …, m, j_{h} = 1, …, n_{h}, h = 1, …, k and q = 1, …, p. Then by the coordinate change (5.3), the operator L_{0} becomes the following form as Section 6 in [Y];

\begin{aligned}L_{0} & = ∑_{i=1}^{m} \{λ_{i}X_{i} + μ_{i}X_{i+1} + A_{i}(X, Y, Z)\} ∂_{X_{i}} \\ & + ∑_{h=1}^{k} ∑_{j_{h}=1}^{n_{h}} \{A_{j_{h}+1}^{h}Y_{j_{h}+1}^{h} + B_{j_{h}}^{h}(X, Y, Z)\} ∂_{Y_{j_{h}}^{h}} + ∑_{q=1}^{p} C_{q}(X, Y, Z)∂_{Z_{q}}, \end{aligned} (5.4)

where μ_{i} = 0 or 1 for i = 1, …, m − 1, μ_{m} = 0, A_{j_{h}}^{h} = 1 for j_{h} = 2, …, n_{h}, h = 1, …, k, A_{n_{h}+1}^{h} = 0 and

A_{i} ≡ B_{j_{h}}^{h} ≡ C_{q} ≡ O ((|X| + |Y | + |Z|)^{2}) and B_{j_{h}}^{h}(X, 0, 0) ≡ C_{q}(X, 0, 0) ≡ 0

for j_{h} = 2, …, n_{h}, h = 1, …, k and q = 1, …, p.

For economizing signs, let us put

X_{i} ↦ x_{i}, Y_{j_{h}}^{h} ↦ y_{j_{h}}^{h}, Z_{q} ↦ z_{q}, A_{i} ↦ a_{i}, B_{j_{h}}^{h} ↦ b_{j_{h}}^{h} and C_{q} ↦ c_{q}.

Further we change the following coordinate

\begin{aligned}& X_{i} = L_{0}^{N−1}\{λ_{i} + μ_{i}x_{i+1} + a_{i}(x, y, z)\}, Y_{j_{h}}^{h} = y_{j_{h}}^{h} + ∑_{j_{0}=1}^{j_{h}−2} L^{j_{0}}b_{j_{h}−1−j_{0}}^{h}(x, y, z), \\ and & Z_{q} = z_{q} \end{aligned} (5.5)

for i = 1, …, m, j_{h} = 2, …, n_{h}, h = 1, …, k and q = 1, …, p. Then by the coordinate change (5.5), the operator L_{0} becomes the following form;

\begin{aligned}L_{0} & = ∑_{i=1}^{m} \{λ_{i}X_{i} + μ_{i}X_{i+1} + A_{i}(X, Y, Z)\} ∂_{X_{i}} \\ & + ∑_{h=1}^{k} \{ ∑_{j_{h}=1}^{n_{h}−1} Y_{j_{h}+1}^{h}∂_{Y_{j_{h}}^{h}} + B_{h}(X, Y, Z)∂_{Y_{n_{h}}^{h}} \} + ∑_{q=1}^{p} C_{q}(X, Y, Z)∂_{Z_{q}}, \end{aligned} (5.6)

and in particular B_{h} = L^{n_{h}−2}Y_{2}^{h}. Then the coefficients of the operator L_{0} satisfy

A_{i}(0, Y, Z) ≡ B_{h}(0, Y, Z) ≡ C_{q}(0, Y, Z) ≡ O ((|Y | + |Z|)^{δ}) ,

B_{h}(X, Y_{(n_{h}−1)}^{1}, 0, Y_{(n_{h}−1)}^{2}, 0, …, Y_{(n_{h}−1)}^{h}, 0, Y^{h+1}, …, Y^{k}, Z) ≡ C_{q}(X, 0, 0) ≡ 0,

A_{i}(X, Y, Z) ≡ O ((|X| + |Y | + |Z|)^{2})

by Definition 2.1, the condition (A.C) and the form of L_{0} for i = 1, …, m, h = 1, …, k and q = 1, …, p, as Section 6 in [Y] and we may omit the details.

Therefore we find the form of L_{0} satisfies the form of P − d(x, y, z) in Section 4. Hence this completes the proof of Theorem 2.5 by Theorem 4.1. Q.E.D.

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