数理解析研究所講究録 1261 巻 2002 年 94-102

94

Maillet type theorem for first ordersingular nonlinear partial differential
equaitons of nilpotent type名古屋大学大学院多元数理科学研究科
Graduate School of Mathematics, Nagoya University
白井 朗 (Akira Shirai)

1

Introduction.

We consider the following first order nonlinear partial differential equation of generalform in the complex domain:
(1.1)

$\{$

$f(x, u(x)$ , $\partial_{x}u(x))=0$ ,$u(0)=0$

where $x=$,
, and $f(x, u, \xi)(\xi=(\xi_{1}, \ldots, \xi_{n})$is a holomorphic function in a neighborhood of the origin.
We assume that $f(x, u, \xi)$ is an entire function in variables when and are fixed.
As afundamental assumption, we always assume the existence of aformal solution of
the equation (1.1), that is,$(x_{1}, \ldots, x_{n})\in \mathrm{C}^{n}$$\partial_{x}u=(\partial_{x_{1}}u, \ldots, \partial_{x_{n}}u)$$\in \mathrm{C}^{n})$

$x$

$\xi$

$u$

Assumption 1The equation (1.1) has aformal solution of the form
(1.2)

where

$u(x)= \sum_{|\alpha|\geq 1}u_{\alpha}x^{\alpha}=\sum_{j=1}^{n}\xi_{j}^{0}x_{j}+\sum_{|\alpha|\geq 2}u_{\alpha}x^{\alpha}\in \mathrm{C}[[x]]$$\alpha=$$(\alpha_{1}, \ldots, \alpha_{n})\in \mathrm{N}^{n}(\mathrm{N}=\{0,1,2, \ldots\})$$\alpha_{1}+\cdots+\alpha_{n}$.

,denotes the multi-index and$|\alpha|=$Our interest in this note is to study the convergence or the divergence nature of suchformal solution in the case where the equation (1.1) is singular in the sense defined in
Miyake-Shirai [3] as follows:(1.3)$f(0,0, \xi)\equiv 0$

By (1.3), the coefficientssatisfy$\frac{\partial}{\partial x_{i}}f(x, u(x)$,,for all$\xi^{0}=(\xi_{1}^{0}, \ldots, \xi_{n}^{0})$$\xi\in \mathrm{C}^{n}$.

of linear part of the formal solution (1.2)$\partial_{x}u(x))|_{x=0}=\frac{\partial f}{\partial x_{i}}(0,0, \xi^{0})+\frac{\partial f}{\partial u}(0,0, \xi^{0})\xi^{0}.\cdot=0$

95
of such roots.
, . We take and fix one
Let $v(x)=u(x)- \sum_{j=1}^{n}\xi_{j}^{0}x_{j}$ be anew unknown function. By substituting this power
series into (1.1), we see that $v(x)$ satisfies the following equation:

for $i=1,2$ ,(1.4)$\xi^{0}$$n$$\ldots$$\partial_{x}v(x))$$P_{0}v(x)= \sum_{|\alpha|=2}c_{\alpha}x^{\alpha}+f_{3}(x, v(x),$$v(x)=O(|x|^{2})$ ,,where $f_{3}(x, v, \xi)$ is holomorphic in aneighborhood of the origin with Taylor expansion$f_{3}(x, v, \xi)=\sum_{|\alpha|+2r+|\kappa|\geq 3}f_{\alpha r\kappa}x^{\alpha}v^{r}\xi^{\kappa}$

and$P_{0}$,$\kappa=\{\kappa_{j}\}\in \mathrm{N}^{n}$,$| \kappa|=\sum_{j=1}^{n}\kappa_{j}$,denotes the operator of the form

(1.5)

$P_{0}--(x_{1}, \ldots, x_{n})A$$(\begin{array}{l}\partial_{x_{1}}\vdots\partial_{x_{h}}\end{array})$$+f_{u}(0,0, \xi^{0})$by an $n\cross n$ matrix $A=(a_{ij})_{i,j=1,2,\ldots,n}=(f_{x:\epsilon_{j}}(0,0, \xi^{0})+f_{u\xi_{j}}(0,0, \xi^{0})\xi_{i}^{0})_{i,j=1,2,\ldots n}’$.
Let the Jordan canonical form of be given by$A$$A\sim(\begin{array}{lllll}A_{m} B_{1} \ddots B_{p} O_{q}\end{array})$where$A_{m}=(\begin{array}{llll}\lambda_{1} \delta_{1} \lambda_{2} \ddots \ddots \delta_{m-1} \lambda_{m}\end{array})$,$B_{j}=$$(\begin{array}{llll}0 1 0 \ddots \ddots 1 0\end{array})$and$O_{q}$

are the block of nonzero eigenvalues of size , nilpotent block of size and zero matrix
block of size , respectively. It is obvious that $m+n_{1}+\cdots+n_{p}+q=n$ .
$nj$

$m$

$q$

Under the above situation, Miyake-Shirai [3] proved the following results:

Theorem 1(Miyake-Shirai) (i) Let $m=n$ andtion which is called the Poincare’ condition:(1.6)$\{\lambda_{j}\}_{j=1}^{n}$$\mathrm{C}\mathrm{h}(\lambda_{1}, \ldots, \lambda_{n})\geq 0$,

satisfy the following condi-

96
where

denotes the convex hull of. Then the for$mal$ solution
converges in a neighborhood of the origin.
(ii) If $q=n$ and $f_{u}(0,0, \xi^{0})\neq 0$ , then the fomal solution $u(x)$ belongs to the Gevrey
class of order at most 2, that is, power series
which is a formal 2-B0rel
converges
inaneighborhoodtransfom oforigin.
$\mathrm{C}\mathrm{h}(\lambda_{1}, \ldots, \lambda_{n})$$\{\lambda_{1}, \ldots, \lambda_{n}\}$$u(x)$$\sum_{|\alpha|>1}u_{\alpha}x^{\alpha}/|\alpha|!_{f}$$u(x)_{f}$$of^{-}the$

Our purpose in this note is to determine the Gevrey order in the case where thematrix is nilpotent, that is, the case where $m=0$ and $p\geq 1$ , which is not studied in$A$

Miyake-Shirai [3].

Theorem 2 If $m=0$ , $p\geq 1$ and $f_{u}(0,0,\xi^{0})\neq 0$ , then the fomal solution $u(x)$ of (1.1)
belongs to the Gevrey class of order at most $2N$ with $N= \max\{n_{1}, \ldots, n_{p}\}$ , that is,
the power series, which is a formal $2N$ -Borel transform of $u(x)$ ,converges in a neighborhood of the origin.
$\sum_{|\alpha|\geq 1}u_{\alpha}x^{\alpha}/|\alpha|!^{2N-1}$

In the case of first order linear singular equations, Hibino [2] and Yamazawa [6], [7]studied the same problem and they determined the Gevrey order of the formal solutions
which deeply depends on the Jordan canonical form of . Theorem 2is anonlinear
version of their results in the case where the matrix is nilpotent.
$A$

$A$

At the end of this introduction we give amention about the study by G\’erard-Tahara
on singular partial differential equations which can be seen in their book [1] and thereferences therein. Their research goes to many kinds of problems for singular (nonlinear)partial differential equations such as the convergence of formal solutions, the Maillet typetheorem for divergent formal solutions, the existence of singular solutions, etc. However,their study is somewhat restricted to the equation of reduced form such as(1.7)$\{\begin{array}{l}\dot{.}\sum_{\dot{v}=1}^{n}a_{j}\dot{.}x_{\dot{l}}\partial_{x_{\mathrm{j}}}u+cu=\sum_{j=1}^{n}a_{j}x_{j}+f_{2}(x,u,\{x..\partial_{x_{j}}u\}..,j=1,2,\ldots,n)u(0)=0\end{array}$where. Our equation (1.4) which is areduced form
from (1.1) is similar with (1.4) for linear part but is amore weaker form for nonlinear terms in the sense of vanishing order. Our theory can be said to be atrial of aclassification of singular equations from the general point view.
$f_{2}(x, u, \xi)=\sum_{|\alpha|+r+|\kappa|\geq 2}f_{\alpha r\kappa}x^{\alpha}u^{r}\xi^{\kappa}$

2

Refinement of Theorem.

After alinear transformation of variables which reduces the matrix
to its Jordan
canonical form, we can obtain more precise estimates of the Gevrey order in each variable.
In order to state the result, we prepare some notation and definitions$A$

97
Definition 1( -Borel transformation) Let$\{x\in \mathrm{R};x\geq 1\}$ . For formal power series
tion $B_{S}(f)(x)$ of $f(x)$ is defined by$s$$f(x)= \sum_{|\alpha|\geq 0}f_{\alpha}x^{\alpha}$

(2.1)

where
transforma-Borel, the$\mathrm{R}_{\geq 1}=$$s=(s_{1}, \ldots, s_{n})\in(\mathrm{R}\geq 1)^{n}$$B_{S}(f)(x)= \sum_{|\alpha|\geq 0}f_{\alpha}\frac{|\alpha|!}{(s\cdot\alpha)!}x^{\alpha}$$s$.

, if the s-Borel) We say that
Definition 2(Gevrey classtransformation $B_{S}(f)(x)$ converges in aneighborhood of the origin, and is called theGevrey order.
$f(x)= \sum_{|\alpha|\geq 0}f_{\alpha}x^{\alpha}\in \mathcal{G}_{x}^{S}$$\mathcal{G}_{x}^{S}$$s$

Remark 1(i) If two Gevrey orders $s=\{sj\}$ and $s’=\{s’\}j$ SatiSw $Sj\leq S’j$ for all$j=1,2$ ,$\ldots$(ii) If, , then$n$$\mathcal{G}_{x}^{S}\subset \mathcal{G}_{x}^{S’}$.

$s’=(s’, s’, \ldots, s’)\in(\mathrm{R}_{\geq 1})^{n}$, then$\mathrm{f}(\mathrm{x})\in \mathcal{G}_{x}^{S’}$

if and only if$\sum\frac{f_{\alpha}}{|\alpha|!^{s’-1}}x^{\alpha}$converges in aneighborhood of the origin, Moreover, for all linear transformations.
invertible matrix),
and $M$ is an$\xi=xM$ (, then we have $u(x)\in$, if(iii) For aformal power series$g(\xi):=f(\xi M^{-1})\in \mathcal{G}_{\xi}^{(S’)}$$n$$n\mathrm{x}$$\xi\in \mathrm{C}^{n}$$B_{S}(u)(x)\in \mathcal{G}_{x}^{\hat{S}}$$u(x)\in \mathrm{C}[[x]]$$\mathcal{G}_{x}^{S+\hat{\mathit{8}}-1_{n}}$with $1_{n}=(1,1, \ldots, 1)\in \mathrm{N}^{n}$ .

Let us give arefined form of Theorem 2. Let assume the vanishing order of $v(x)$ bein$K\geq 2$ . Then by alinear change of independent variables which brings the matrix(1.5) to the Jordan canonical form, the equation (1.4) is reduced to the following form:
$A$

,

(2.2)

$Pv(y, z)= \sum_{|\beta|+|\gamma|=K}c_{\beta,\gamma}y^{\beta}z^{\gamma}+f_{K+1}(y, z, v, \partial_{y}v, \partial_{z}v)$

with $v(y, z)=O((|y|+|z|)^{K})$ , where(2.3)$\delta$,$P= \sum_{i=1}^{p}\sum_{j=1}^{n_{i}-1}\delta y_{i,j+1}\partial_{yi,j}+c$$c\in \mathrm{C}\backslash \{0\}$,$y=\{\mathrm{u}\mathrm{k}\}y^{2}$$z=(z_{1}, \ldots, z_{q})\in \mathrm{C}^{q}$,$\ldots$,$y^{p}$),$\in \mathrm{C}^{n_{1}+\cdots+n_{p}}$$c=f_{u}(0,0, \xi^{0})$

where

,

$y^{i}=(y_{i,1}, \ldots, y_{i,n:})\in \mathrm{C}^{n_{j}}$,,,$f_{K+1}(y, z, v, \eta, \zeta)=\sum_{|\beta|+|\gamma|+Kr+(K-1)(|\mu|+|\nu|)\geq K+1}f_{\beta\gamma r\mu\nu}y^{\beta}z^{\gamma}v^{r}\eta^{\mu}\zeta^{\nu}$

where$|\beta|$,$|\gamma|$,$\gamma=\{\gamma_{k}\}\in \mathrm{N}^{q}$$|\mu|$,

and$|\nu|$

denote the length of multi-indices, respectively
and$\mu=\{\mu_{i,j}\}\in \mathrm{N}^{n_{1}+\cdots+n_{p}}$$\beta=\{\beta_{i,j}\}\in \mathrm{N}^{n_{1}+\cdots+n_{\mathrm{p}}}$$\nu=\{\nu_{k}\}\in \mathrm{N}^{q}$,98
Remark 2 We may assume that the constant is as small as we want. Indeed, we
introduce new independent variables
by. Then ischanged by . Therefore, by choosing
small enough, we may assume that thecoefficient is arbitrary small.
$\delta$$\eta=\{\eta_{i,j}\}$$\epsilon\delta$$\eta_{i,j}=\epsilon^{n_{1}+\cdots+n_{i-1}+j}y_{i,j}$$\delta$$\epsilon>0$$\delta$

For $p=(p_{1},p_{2}, \ldots,p_{d})(d\geq 1)$ and a constant , we define $p(a)$ by
$a$

(2.4)

$p(a)=(p_{1}+a,p_{2}+a, \ldots,p_{d}+a)$ .

Then Theorem 2is obtained immediately from the following:
Proposition 1 The equation (2.2) has a unique fomal solution which belongs to theGevrey class of order with
$s$

(2.5)

$s$

where $s^{i}=$ 1, 2,
$($

(2.6)

$\ldots$,$=(s^{1}(\sigma), s^{2}(\sigma),$$n_{i})\in \mathrm{N}^{n}:$$\ldots$,$s^{p}(\sigma)$,$1_{q}(\sigma))$,, $1_{q}=(1,1, \ldots, 1)\in \mathrm{N}q$ and$\sigma=\max_{\gamma(\beta,,r,\mu,\nu)}\{\frac{A(\mu,\nu)}{|\beta|+|\gamma|+Kr+(K|-1)(|\mu|+|\nu|)-K}$;$f_{\beta\gamma r\mu\nu}\neq 0\}$,$A(\mu, \nu)=\{\max\{j..01\mu_{i,j}\neq 0\}ififif|\mu|=|\nu|=0|\mu|=0,|\nu|\geq|\mu|\geq 1,.1$

Proof of Theooem 2. As mentioned above, the equation (2.2) is the one which is obtained
from (1.4) by alinear change of independent variables. The Gevrey order of the formal
solution $v(x)$ of (1.4) is estimated by the maximal value of components of . Since$A( \mu, \nu)\leq N=\max\{n_{1}, \ldots, n\}p$ and the determination of, we see that the Gevrey
order of is estimated by $2N$ .
$s$$s^{i}$’$v$$\blacksquare$3

Sketch of the Proof of Proposition 1.

In this section, we shall prove Proposition 1by assuming the lemmas below, since we are
not permitted enough space to write down the complete proofs of lemmas. The complete
proofs will be found in aforthcomming paper [5].
The uniqueness of formal solutions is easily proved by using the following lemma:
Lemma 1 (i) Let
be the set of homogeneous polynomials of degoee in andvariables. Then for all
the operator
is invertible.
(ii) Let $:=(s^{1}, \ldots, s^{p}, 1_{q})=s-\sigma_{n}$ with,and $u_{L}(y, z)$ ,. We consider the following equation:$\mathrm{C}[y, z]_{L}$$z$$L\geq 2_{f}$

$L$

$y$

$P:\mathrm{C}[y, z]_{L}arrow \mathrm{C}[y, z]_{L}$$\hat{s}$$\sigma_{n}=1_{n}(\sigma)-1_{n}=(\sigma, \sigma, \ldots, \sigma)\in(\mathrm{R}_{\geq 1})^{n}$$f_{L}(y, z)\in \mathrm{C}[y, z]_{L}$$Pu_{L}(y, z)=f_{L}(y, z)$.

99does hold $with|y|= \sum_{i=1}^{p}\sum^{n_{i}}j=1y_{i,j}$
and $|z|= \sum_{k=1}^{q}z_{k}$ , then there exists a positive constant C $>0$ independent of L suchthat
$B_{\hat{S}}(u_{L})(y, z)=B_{\hat{S}}(P^{-1}f_{L})(y, z)<<CF_{L}\cross(|y|+|z|)^{L}$.
(3.1)

If a majorant relation$B_{\hat{S}}(f_{L})(y, z)<<F_{L}\cross(|y|+|z|)^{L}$

In fact, the uniqueness of formal solutions is implied from this lemma as follows.
$K\geq 2)$ . By substituting this into
We put $v(y, z)= \sum_{L>K}v_{L}(y, z)$ ,are determined uniquely. Thus
, we can see that
(2.2), and by Lemma
the uniqueness is proved.
$(v_{L}(y, z)\in \mathrm{C}[y, z]_{L},$$\{v_{L}(y, z)\}_{L\geq K}$$\overline{1}(\mathrm{i})$$\blacksquare$.
polynomial
In order to prove Lemma 1(ii), we introduce anorm for ahomogeneous$\bullet Idea$of the proof of Lemma 1. Lemma 1 (i) is obvious, since$u_{L}(y, z)\in \mathrm{C}[y, z]_{L}$$c=f_{u}(0,0, \xi^{0})\neq 0$

by$||u_{L}||_{S}:= \inf\{C>0;B_{S}(u_{L})(y, z)\ll C(|y|+|z|)^{L}\}$ ,$s\in(\mathrm{R}_{\geq 1})^{n}$.

We may assume that the constant in is as small as we want by alinear change ofindependent variables. Then by this assumption, we can prove that the operator norm
by apositive constant $C>0$ which is independentis estimated byofimplies Lemma 1.
of . In fact, it is easily proved that$\delta$$P^{-1}$$P$$||P^{-1}||_{\hat{S}}\leq C$$||y_{i,j+1}\partial_{yij}u_{L}||_{\hat{S}}\leq||u_{L}||_{\hat{S}}$$L$$\mathrm{T}\mathrm{l}\dot{\mathrm{u}}\mathrm{s}$

(ii),

II

We put $U(y, z)=Pv(y, z)$ as
new unknown function. Then $U(y, z)$ satisfies the following equation:
Next, we shall give aestimate of the Gevrey order.

(3.2)$U(y, z)= \sum_{|\beta|+|\gamma|=K}c_{\beta\gamma}y^{\beta}z^{\gamma}+f_{K+1}(y, z, P^{-1}U, \partial_{y}P^{-1}U, \partial_{z}P^{-1}U)$$\mathrm{a}$,with $U(y, z)=O((|y|+|z|)^{K})$ . By applying the -Borel transformation to the equation(3.2), we have$\hat{s}$(3.3)$B_{\hat{S}}(U)(y, z)$$=$$\sum_{|\beta|+|\gamma|=K}c_{\beta\gamma}\frac{(|\beta|+|\gamma|)!}{\{\hat{s}\cdot(\beta,\gamma)\}!}y^{\beta}z^{\gamma}$$+B_{\hat{S}}\{f_{K+1}(y, z, P^{-1}U, \partial_{y}P^{-1}U, \partial_{z}P^{-1}U)\}$.

In order to construct amajorant equation of this equation, we prepare the following
lemma.
Lemma 2(i) For trno arbitrary formal power series, we have$u(y, z)= \sum_{|\beta|+|\gamma|\geq 0^{u}\beta\gamma}y^{\beta_{Z}\gamma}$$v(y, z)= \sum_{|\beta|+|\gamma|\geq 0}v_{\beta\gamma}y^{\beta}z^{\gamma}$$B_{\hat{S}}(uv)(y, z)\ll C_{0}B_{\hat{S}}(|u|)(y, z)B_{\hat{S}}(|v|)(y, z)$,$C_{0}= \max\{s_{ij}\}\geq 1$

,

and

100
where
.
(ii) If $B_{\hat{S}}(u)(y, z) \ll W(T)=\sum_{L\geq 0}W_{L}T^{L}(T=|y|+|z|)$ , then there exists a positive
constant M $>0$ independent of i,j and k such that$|u|(y, z):= \sum_{|\beta|+|\gamma|\geq 0}|u_{\beta\gamma}|y^{\beta}z^{\gamma}$$B_{\hat{S}}( \partial_{y.,j}.P^{-1}u)(y, z)\ll M\frac{d}{dT}(T\frac{d}{dT})^{j-1}W(T)$$B_{\hat{S}}( \partial_{z_{k}}P^{-1}u)(y, z)\ll M\frac{d}{dT}W(T)$,,for $(i,j)\in J$,
for $k=1,2$ ,$\ldots$, ,$q$where $J=\{(i,j) ; i=1,2, \ldots,p, j=1,2, \ldots, n_{i}\}$ .

Idea of the proof of Lemma 2. In order to prove Lemma 2, it is sufficient to estimate
the product of the Gamma functions by using the Stirling formula.
$\bullet$$\blacksquare$

Next we consider the following ordinary differential equation which is called the
majorant equation of (3.3):(3.4)$W(T)=( \sum_{|\beta|+|\gamma|=K}|c_{\beta\gamma}|\frac{(|\beta|+|\gamma|)!}{\{\hat{s}\cdot(\beta,\gamma)\}!})T^{K}$$+|f_{K+1}|$

(

$T$

,

$\ldots$, $T$, $C_{1}W$, $\{C_{2}\frac{d}{dT}(T\frac{d}{dT})^{j-1}W\}_{(:,j)}$ ,$\{C_{2}\frac{d}{dT}W\}_{k}$),where $C_{1}=CC_{0}$ , $C_{2}=MC_{0}$ .
Let us explain how the equation (3.4) is derived ffom (3.3). By Lemmas 1and 2,
we can show that amajorant relation $B_{\hat{S}}(U)(y, z)<<G(T)$ implies $B_{\hat{S}}(P^{-1}U)(y, z)\ll$$CG(T)$ and$B_{\hat{\mathit{8}}}\{f_{K+1}(y, z, P^{-1}U,\partial_{y}P^{-1}U,\partial_{z}P^{-1}U)\}$$\ll$$|f_{K+1}|$

(

$T$

,

$\ldots$, , $C_{1}G$ ,$T$

Indeed, it is sufficient to notice that
$B_{\hat{S}}(y^{\beta}z^{\gamma}(P^{-1}U)^{r}.

$\{C_{2}\frac{d}{dT}(T\frac{d}{dT})^{j-1}G\}_{(:,j)}$$B_{\hat{S}}(U^{2})\ll\{C_{0}B_{\hat{S}}(|U|)\}^{2}$,

by$\{C_{2}\frac{d}{dT}G\}_{k}$$C_{0}\geq 1$).

, etc., and that

\cdot,\prod_{j,k}(\partial_{y\dot{.},\mathrm{j}}P^{-1}U)^{h.\mathrm{j}}(\partial_{z_{k}}P^{-1}U)^{\nu_{k}})$$\ll y^{\beta}z^{\gamma}\{C_{0}B_{\hat{S}}(|P^{-1}U|)\}^{r}.\cdot,\prod_{j,k}\{C_{0}B_{\hat{S}}(|\partial_{y.,\mathrm{j}}P^{-1}U|)\}^{\mu_{\mathrm{j}}}\cdot\{C_{0}B_{\hat{S}}(|\partial_{z_{k}}P^{-1}U|)\}^{\nu_{k}}$$\ll T^{|\beta|+|\gamma|}(CC_{0}G)^{r}\prod_{i,j,k}\{MC_{\sigma}\frac{d}{dT}(T\frac{d}{dT})^{j-1}G\}^{\mu:,\mathrm{j}}\{MC_{0}\frac{d}{dT}G\}^{\nu_{k}}$101
Therefore by the above construction of the equation (3.4), the formal solution $W(T)$ of, that is,(3.4) is amajorant series of$B_{\hat{S}}(U)(y, z)$(3.5)$W(|y|+|z|)>>B_{\hat{S}}(U)(y, z)$

For the equation (3.4), we havefollowing result:.
, because we can prove the$W(|y|+|z|)\in \mathcal{G}_{y,z}^{1_{n}(\sigma)}$Lemma 3Let $s_{k}\geq 0$ $(k=1,2, \ldots, n)$ be non negative real numbers and $D_{T}=d/dT$by. We define the formal differentiation$(TD_{T})^{s_{k}}$$(T\in \mathrm{C})$(3.6)$(TD_{T})^{s_{k}}(T^{L}):=L^{s_{k}}T^{L}$.

We consider the following nonlinear equation:(3.7)$U(T)=aT^{K}+f_{K+1}(T, U, \{D_{T}(TD_{T})^{s_{k}}U\}_{k=1,2,\ldots,n})$$U(T)=O(T^{K})$ ,,where $K\geq 2$ and$f_{K+1}(T, U, \xi)=\sum_{V(i,j,\alpha)\geq K+1}f_{ij\alpha}T^{i}U^{j}\xi^{\alpha}$.

Here $V(i,j, \alpha)=i+Kj+(K-1)(\alpha_{1}+\cdots+\alpha_{n})$ which denotes the vanishing order of. Then the equation (3.7) has a unique formal solution
withhich belongs to$f_{ij\alpha}T^{i}U^{j} \prod_{k=1}^{n}\{D_{T}(TD_{T})^{s_{k}}U\}^{\alpha_{k}}$$\mathcal{G}_{T}^{1+\sigma}$$w$$\sigma=\max\{\frac{A(i,j,\alpha)}{V(i,j,\alpha)-K}$

where$A(i,j, \alpha)=\{$;$f_{ij\alpha}\neq 0\}$,$\max\{s_{k}\}+1$$(\alpha_{k}\neq 0)$0$(|\alpha|=0)$.

We remark that $A(i, j, \alpha)$ denotes the maximal order of differentiation in each term$f_{ij\alpha}T^{i}U^{j} \prod_{k=1}^{n}\{D_{T}(TD_{T})^{s_{k}}U\}^{\alpha_{k}}$$\bullet Idea$.

of proof of Lemma 3. Lemma 3is proved by the same manner as the proofof main theorem in [4, Theorem 1]. The most important point to prove Lemma 3is to
give aprecise estimate for the product of factorials of integers. In order to obtain such
apresice estimate, the following elementary inequality plays acrucial role:, $n_{k}\geq M(M\in \mathrm{N})$ , we haveFor ,$n_{1}$$\ldots$$n_{1}$! $\cdots n_{k}!\leq M!^{k-1}(n_{1}+\cdots+n_{k}-(k-1)M)!$ .

After some careful estimations based on this inequality, we can prove Lemma 3. The
detail of the proof can be found in [4], [5].

$\blacksquare$

102
Finally we return to the proof of Proposition 1. In our majorant equation (3.4), themaximal order of differentiation in each term is given by $A(\mu, \nu)$ which appeared in thestatement of Proposition 1, and the difference of vanishing order of each term and thatof $W(T)$ is given by$|\beta|+|\gamma|+Kr+(K-1)(|\mu|+|\nu|)-K$.

Therefore, by Lemma 3, we have.
By Lemma 1 (ii), the following majorant relation holds:$W(T)\in \mathcal{G}_{T}^{1+\sigma}$$B_{\hat{S}}(v)(y, z)=B_{\hat{S}}(P^{-1}U)(y, z)\ll CW(|y|+|z|)\in \mathcal{G}_{y,z}^{1_{n}(\sigma)}$

By Remark 1(iii), we haveThus Proposition 1 is proved.

$u(y, z)\in \mathcal{G}_{y,z}^{\hat{S}+1_{n}(\sigma)-1_{n}}=\mathcal{G}_{y,z}^{S}$.

.
$\blacksquare$

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G\’erard R. and Tahara H., Singular nonlinear partial differential equations, Vi eweg,

1996.

[2]

Hibino M., Divergence property of formal solutions for singular first order linear
partial differential equations, Publ.
. Inst Math. Sci. 35 (1999), No. 6, 893–${\rm Res}$

919.
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Miyake M. and Shirai A., Convergence of formal solutions of first order singularnonlinear partial differential equations in complex domain, Annales Polonic Mathmatici, 74 (2000), 215–228.

[4]

Shirai A., Maillet type theorem for nonlinear partial differential equations and Newton polygons, J. Math. Soc. Japan. Vol53,3 (2001), 565–587.

[5]

Shirai A., Maillet type theorem for first order singular nonlinear partial differential
equations, preprint.

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Yamazawa H., Newton polyhedrons and aformal Gevrey space of double indices forlinear partial differential operators. Funkcial. Ekvac. 41 (1998), No. 3, 337–345.

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Yamazawa H., Formal Gevrey class of formal power series solution for singular firstlinear partial differential operators. Tokyo J. Math. 23 (2000), No. 2, 537–$\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}$\prime\prime