数理解析研究所講究録 1211 巻 2001 年 96-104

96

A Sharp Existence and Uniqueness Theorem for
Linear Fuchsian Partial Differential Equations
Jose Ernie C. LOPE
Abstract
This paper considers the equation $Pu=f$ , where and are continuous with respect to and holomorphic with respect to , and is thelinear Fuchsian partial differential operator
$f$

$u$

$t$

$z$

$\mathcal{P}$$P$ $=(tD_{t})^{m}+ \sum_{j=0}^{m-1}\sum_{|\alpha|\leq m-j}aj,\alpha(t, z)(\mu(t)D_{z})^{\alpha}(tD_{t})^{j}$.

We will give asharp form of unique solvability in the following sense: we
can find adomain such that if is defined on , then we can find aunique solution also defined on Q.
$\Omega$$f$$\Omega$$u$

1Introduction and Result
Denote by the set of nonnegative integers, and let $(t, z)=(t, z_{1}, \ldots, z_{n})\in$. Let $R>0$ be sufficiently small, and for $\rho\in(0, R]$ , let be the polydisk; $|z:|<\rho$ for $i=1,2$ ,{}.
Given any bounded, open subset $D$ of , we define by $A(D)$ the Banach
space of all functions $g(z)$ holomorphic in and continuous up to
the normin this space is given by. Let $T>0$ . Then we denote by$C^{0}([0, T], A(D))$ the set of functions continuous on the interval $[0, T]$ and valued
in the space $A(D)$ .
We say that acontinuous, positive-valued function
on the interval $(0, T)$
is aweight function if
is increasing and the function$\mathrm{N}$$\mathbb{R}\cross \mathbb{C}^{n}$$B_{\rho}$$z\in \mathbb{C}^{n}$$\ldots,n$$\mathbb{C}^{n}$$D$$\overline{D},\cdot$$||g||_{D}= \max_{z\in\overline{D}}|g(z)|$$\mu(t)$$\mu(t)$(1.1)$\varphi(t)=\int_{0}^{t}\frac{\mu(s)}{s}ds$

is well-defined on (0, T), i.e., the integral on the right is finite. (See Tahara [7].)Consider now the linear partial differential operator$\mathcal{P}=(tD_{t})^{m}+\sum_{j=0}^{m-1}\sum_{|\alpha|\leq m-j}a_{j,\alpha}(t, z)(\mu(t)D_{z})^{\alpha}(tD_{t})^{j}$Here,
the coefficients$D_{t}=\partial/\partial t$

and $D_{z}=$z) belong in the space$a_{j,\alpha}(t,$$(\partial/\partial z_{1},$$\partial/\partial z_{n});\mu(t)$\ldots ,$C^{0}([0,$

.

(1.2)

is aweight function; andT],$A(B_{R}))$ , i.e., for any

97when viewed as afunction of z,, each of the functions
is holomorphic in B. and continuous up to Br. We associate apolynomial withthis operator, called the characteristic polynomial of , and we define it bysE [0,$a_{\mathrm{j}_{\mathrm{t}}\mathrm{o}}(’?z)_{7}$$\ovalbox{\tt\small REJECT} \mathrm{I}]$$7^{\ovalbox{\tt\small REJECT}}$$C(\lambda, Z)=\lambda^{m}+a_{m-1,0(\mathrm{o},z)\lambda^{m-1}+\cdots+a_{0,0}(0,z)}$

.

(1.3)

,
,
will be referred to as characteristic exponents. In
Its roots
what follows, we will assume that there exists apositive number such that$\lambda_{1}(z)$$\ldots$$\lambda_{m}(z)$$L$for all $z\in B_{R}$ and $1\leq j\leq m$ .

$\Re\lambda_{j}(z)\leq-L<0$(1.4)

Baouendi and Goulaouic [1] studied the above operator in the case when$\mu(t)=t^{a}(a>0)$ . They called such operator aFuchsian partial differential
operator, which for them is the “natural” generalization of aFuchsian ordinary
differential operator. In their paper, they gave some generalizations of theclassical Cauchy-Kowalewski and Holmgren theorems for this type of operators.
Their method has been applied and extended to various cases as can be seen,for example, in Tahara [6], Mandai [5] and Yamane [8].
In aprevious paper [4], the author proved existence and uniqueness theorems. Essentially, he proved thesimilar to those given in [1], but for generalfollowing unique solvability result.
$\mu(t)$Theorem 1. Let $P$ be as in (1.2). Then given any $\rho\in(0, R)$ , there exists an$\epsilon\in(0, T]$ such thatfor any $f(t, z)\in C^{0}([0, T], A(B_{R}))$ , the equation $Pu=f$
has a unique solution $u(t,z)\in C^{0}([0,\epsilon], A(B_{\rho}))$ satisfying for $1\leq p\leq m$ therelation $(tD_{t})^{p}u\in C^{0}([0,\epsilon], A(B_{\rho}))$ .
,We remark that although $f(t, z)$ , viewed as afunction of , is defined on, with $\rho<R$ .
the existence of the solution $u(t, z)$ is only guaranteed up toMoreover, any two solutions of $Pu=f$ can only be shown to coincide in aneighborhood of the origin which is smaller than the neighborhood on which
the two are defined.
In this paper, we shall present aformulation leading to an existence anduniqueness result sharper than the one given above. The result is sharper in thesense that the solution $u(t, z)$ of the equation Pu $=f$ will now have the same
domain of definition as the inhomogeneous part $f(t, z)$ .
To proceed, we will need the following definitions.
$B_{R}$$z$$B_{\rho}$

Definition 1. Let $\tau\in(0,T)$ , $\gamma>0$ and
(i)
(ii)

$\omega_{\tau}[\gamma]=$$\mathrm{Q}\mathrm{T}[\mathrm{y}]=${{$z\in \mathbb{C}^{n}$$(\tau,$; $|z_{i}|<R-\gamma\varphi(\tau)$$\varphi(t)$for $i=1,2$ ,$z)\in \mathbb{R}\cross \mathbb{C}^{n};0\leq\tau\leq T$

Definition 2. Let p$\in \mathrm{N}$be the one in (1.1). We define

and$\ldots$$z\in\omega_{\tau}[\gamma]$, }, and$n$}.

and $\gamma>0$ .

(i) We say that $f(t, z)$ belongs in$f(t)\in C^{0}([0,\tau], A(\omega_{\tau}[\gamma]))$.

$\mathcal{K}_{0}(\Omega_{T}[\gamma])$

if for each $\tau\in[0, T]$ , we have

98$(\mathrm{i}\ovalbox{\tt\small REJECT})$We say thathave$\mathrm{w}(\mathrm{t},$r],z) belongs in ($\ovalbox{\tt\small REJECT} 4(\mathrm{c}\mathrm{p}.[\mathrm{t}]))$$\ovalbox{\tt\small REJECT}([0,$if for all 0$\ovalbox{\tt\small REJECT}$j$\ovalbox{\tt\small REJECT}$

p, we

.

$(\ovalbox{\tt\small REJECT}_{t})^{j}w(t)’-\ovalbox{\tt\small REJECT}([0_{\ovalbox{\tt\small REJECT}}\mathrm{r}]_{:}\mathrm{J}(\mathrm{u}.[\mathrm{y}]))$(iii) We say that $u(t, z)$ belongs in$u(t)\in C_{p}^{0}([0,\tau],A(\omega_{\tau}[\gamma]))$$\mathcal{K}_{p}(\Omega_{T}[\gamma])$.

if for each $\tau\in[0,T]$ , we have

Under the above assumptions, we now state the following main result.
Theorem 2. Let be the operator given in (1.2). The there exist constants, thedepending on $P$ such that for any
andequation(1.5)
in$Pu=f$$P$$n$$f(t, z)\in \mathcal{K}_{0}(\Omega_{T_{0}}[\gamma_{0}])$$\gamma_{0}>0$$T_{\mathrm{O}}>0$$\Omega_{T_{0}}[\gamma_{0}]$.
has a unique solution $u(t,$z) in
Moreover the solution satisfies the a priori estimate$\mathcal{K}_{m}(\Omega_{T_{0}}[\gamma 0])$$\sum_{p=0}^{m}\max_{\Delta}|(tD_{t})^{p}u|\leq C\max_{\Delta}|f|$

,

(1.6)

and C $>0$ is some constant dependent on thewhere Ais the closure of.
above equation and on the domain$\Omega_{T_{0}}[\gamma_{0}]$$\Omega_{T_{0}}[\gamma 0]$

as their domain of definition.
Note that $f(t, z)$ and $u(t, z)$ both haveThis fact allows us to restate our theorem in the following manner: for any, respec, $\gamma>0$ , let
andbe the spaces
andsuch
consisting of functions
tively. Let
be the subspace ofwith. Define alinear operator Ifrom
tothat Pu belongs in$=Vu$ . Letbydomain,denote the maximum norm in the closure,. Thenandare Banach spaces; given
andof, respectively. Note
we define their norms by
,and
. The
to
further that the operator is aclosed linear operator fromabove theorem can now be stated as$\Omega_{T_{0}}[\gamma_{0}]$$T$$X_{T,\gamma}$$\mathcal{K}_{0}(\Omega_{T}[\gamma])$$\mathcal{K}_{m}(\Omega_{T}[\gamma])$$\mathrm{Y}_{T,\gamma}$$u\in X_{T,\gamma}$$X_{T,\gamma}$$W_{T,\gamma}$$X_{T,\gamma}$$\mathrm{Y}_{T,\gamma}$$W_{T,\gamma}$$\Omega_{T}[\gamma]$$\Psi u$$X_{T,\gamma}$$\mathrm{Y}_{T,\gamma}$$|||\cdot|||\tau_{\gamma}$$u\in X_{T,\gamma}$$\mathrm{Y}_{T,\gamma}$$\sum_{p=0}^{m}|||(tD_{t})^{p}u\mathrm{N}|\tau_{\gamma}$$f\in \mathrm{Y}_{T,\gamma}$$|||f|||\tau_{\gamma},$$\Psi$$X_{T,\gamma}$$\mathrm{Y}_{T,\gamma}$

Theorem 2’. There eist , $\gamma 0>0$ depending on $P$ such that the operator.
onto
is one-One, closed linear operator from
$T_{0}$$X_{T_{0},\gamma 0}$$a$$\Psi$$\mathrm{Y}_{T_{0},\gamma 0}$

exists and is also closed. The Closed Graph
Since is an injection,is continuous. The estimate given in (1.6) is
Theorem further implies that.
just aconsequence of the continuity of$\Psi^{-1}$$\Psi$$\Psi^{-1}$$\Psi^{-1}$2Preliminary Discussion
We can rewrite the operator

P $=Q$$\mathcal{P}$as$+ \sum_{j=0}^{m-1}\sum_{|\alpha|\leq m-j}c_{j,\alpha}(t, z)(\mu(t)D_{z})^{\alpha}(tD_{t})^{j}$,

99
where the operator
$Q$

$Q$

is defined by(2.1)$=(tD_{t})^{m}+a_{m-1,0}(0,z)(tD_{t})^{m-1}+\cdots+a_{0,0}(0, z)$

and
,
if
if $|\alpha|=0$ .
$|\alpha|\neq 0$$a_{j,\alpha}(t, z)$$c_{j,\alpha}(t, z)=\{$$aj,\alpha(t, z)-aj,\alpha(0, z)$Note that the coefficients of. Noteare holomorphic functions of in
further that the characteristic exponents of are the same as that of $P$ , andhence satisfy (1.4).
$Q$$B_{R}$$z$$Q$

Lemma 1. Fix $\tau>0$ and let $g(t)\in C^{0}([0, \tau], A(\omega_{\tau}[\gamma]))$ . Then the equation$Qu$ $=g$ has a unique solution
given by$u(t)\in C_{m}^{0}([0, \tau], A(\omega_{\tau}[\gamma]))$$u(t)$$=$$\frac{1}{m!}\sum_{\sigma\in S_{m}}\int_{0}^{ts_{2}}\int_{0}^{s_{m}}\ldots\int \mathrm{o}(\frac{s_{m}}{t})^{-\lambda_{\sigma(m)}}(\frac{s_{m-1}}{s_{m}})^{-\lambda_{\sigma(m-1)}}\ldots$$\cross(\frac{s_{1}}{s_{2}})^{-\lambda_{\sigma(1)}}g(s_{1})\frac{ds_{1}}{s_{1}}\frac{ds_{2}}{s_{2}}\cdots\frac{ds_{m}}{s_{m}}$Here,$S_{m}$

is the group of per mutations of 1, 2,
$\{$

.

(2.2)

, $m\}$ .

$\ldots$

Aresult in symmetric entire functions asserts that $u(t, z)$ is holomorphic
with respect to . The fact that it belongs in
is seen in the
integral expression, but may actually be obtained a priori. (See [1].)To facilitate computation, we define forthe function$C_{m}^{0}([0,\gamma],A(\omega_{\tau}[\gamma]))$$z$$\lambda=(\lambda_{1}, \ldots, \lambda_{m})$

,

(2.3)

$G_{\theta}{}^{t}( \lambda)\mathrm{d}\mathrm{e}\mathrm{f}=\frac{1}{m!}\sum_{\sigma\in S_{m}}(\frac{s_{m}}{t})^{-\lambda_{\sigma(m)}}(\frac{s_{m-1}}{s_{m}})^{-\lambda_{\sigma(m-1)}}\ldots(\frac{\theta}{s_{2}})^{-\lambda_{\sigma(1)}}$for some dummy variables$\int_{[t_{j}\theta]}^{(m)}g$$s_{2}$,$\ldots$,$s_{m}$. Define, too, the integral operator(2.4)$\mathrm{d}\mathrm{e}\mathrm{f}=\int_{0}^{ts_{2}}\int_{0}^{s_{m}}\cdots\int \mathrm{o}g(\theta)\frac{d\theta}{\theta}\frac{ds_{2}}{s_{2}}\ldots\frac{ds_{m}}{s_{m}}$Using the above, we can now write the solution $u(t)$ of the equation $Qu$ $=g$ as$u(t)= \int_{[t_{j}s]}^{(m)}G_{s}^{t}(\lambda)g$.

In our proof of the main theorem, it will be necessary to consider the actionof the differential operator
on integral expressions similar to the one in(2.2). One can easily verify the following$(tD_{t})^{p}$

Lemma 2. Let $u(t)$ be the solution of $Qu$ $=g$ . Then for a natural number
less than $m$ , we have$(tD_{t})^{p}u$

$=$

$p$

$\sum_{i=m-p}^{m}\int_{[t_{j}s_{1}]}^{(i)}g\cross\{\frac{1}{m!}\sum_{\sigma\in S_{m}}h_{i}(\sigma, \lambda)(\frac{s_{i}}{t})^{-\lambda_{\sigma(i)}}$$\cross(\frac{s_{i-1}}{s_{i}})^{-\lambda_{\sigma(:-1)}}\cdots(\frac{s_{1}}{s_{2}})^{-\lambda_{\sigma(1)}}\}$

,

(2.5)

100

t here the functions

istic exponents

A) are suitable polynomial functions of the character, A.(s)$7^{\ovalbox{\tt\small REJECT}_{\mathrm{g}}}\ovalbox{\tt\small REJECT}(_{\mathrm{c}\mathrm{r}},$$\mathrm{A}_{\mathrm{Z}}(z)_{\ovalbox{\tt\small REJECT}}\ldots$

For brevity, let us set, for anatural number k,(2.6)$H_{\theta}{}^{t}(k, \lambda)=\frac{1}{m!}\sum_{\sigma\in S_{m}}h_{k}(\sigma, \lambda)(\frac{s_{k}}{t})^{-\lambda_{\sigma(k)}}(\frac{s_{k-1}}{s_{k}})^{-\lambda_{\sigma(k-1)}}.\ldots(\frac{\theta}{s_{2}})^{-\lambda_{\sigma(1)}}$By symmetry, the functions
are holomorphic with respect to and thus
belong in $A(B_{R})$ .
The next lemma is useful in evaluating some integral expressions in the proof.
$H_{s}^{t}(k, \lambda)$

Lemma 3. Let
(a)
(b)

$k$

$z$

be natural number. Then the following equalities hold:$\int_{0}^{s_{k}}\int_{0}^{s_{k-1}}.\ldots\int 0(\frac{s_{0}}{s_{k}})^{L}\frac{ds_{0}}{s_{0}}\ldots\frac{ds_{k-1}}{s_{k-1}}\epsilon_{1}=\frac{1}{L^{k}}$$\int_{0}^{t\epsilon_{k}\epsilon_{1}}\mathit{1}_{0}\ldots \mathit{1}0\frac{\mu(s_{k})}{s_{k}}\frac{\mu(s_{k-1})}{s_{k-1}}\ldots\frac{\mu(s_{1})}{s_{1}}$x$( \frac{s_{0}}{t})^{L}\frac{s_{0}^{-1}}{[\varphi(t)-\varphi(s_{0})]^{k}}ds_{0}\ldots ds_{k}=\frac{1}{Lk!}$

The first equality is obvious. The second can be proved by reversing the orderof integration and recalling that $t\varphi’(t)=\mu(t)$ .
To estimate the derivatives with respect to , we have the following lemma.
(For aproof, see H\"ormander [3], Lemma 5.1.3.)$z$Lemma 4. Let the function $v(z)$ be holomorphic in
positive constants $K$ and such that

$B_{R}$, and suppose there are$c$$||v||_{\rho} \leq\frac{K}{(R-\rho)^{c}}$

.

(2.7)

for every $\rho\in(0,R)$ .

(2.8)

for every$\rho\in(0, R)$

Then we have$||D_{z}^{\alpha}v||_{\rho} \leq\frac{Ke^{|\alpha|}(c+1)_{|\alpha|}}{(R-\rho)^{c+|\alpha|}}$In the above, we define $(c)_{p}=(c)(c+1)\cdots(c+p-1)$ .

3Proof of Main Theorem
Let be any element of. Here, the constants $T_{0}>0$ andsatisfy some conditions which will later be specified. For convenience, we will
drop the subscript in both and instead use
and ;we will again use thesubscript upon stating the conditions that these constants need to satisfy$f$$\gamma_{0}>0$$\mathcal{K}_{0}(\Omega_{T_{0}}[\gamma_{0}])$$T$$\gamma$

101

We will use the method of successive approximations to solve the equation
Pu f. Define the approximate solutions as follows:$\ovalbox{\tt\small REJECT}$(3.1)$u_{0}(t)= \int_{[tS]}^{(m)}G_{s}^{t}(\lambda)f$

and for $k\geq 1$ ,$u_{k}(t)= \int_{[t_{j}s]}^{(m)}G_{s}{}^{t}(\lambda)[f-S(s)u_{k-1}]$

.

(3.2)

Here, $t\in[0,T]$ , and for brevity, we have set $S(t)= \sum_{j=0}^{m-1}\sum_{|\alpha|\leq m-j}c_{j,\alpha}(t, z)$ .
. Note that for all , the approximate solutions $u_{k}(t, z)$ aredefined on. Furthermore, they are continuous with respect to andholomorphic with respect to on this region.
For each , we also define the sequence of functions $v_{k}(t)=u_{k}(t)-u_{k-1}(t)$ ,with $u_{-1}\equiv 0$ . Then theare also defined on the same region as the, and are also continuous with respect to and holomorphic with respect
to . Using the expression for, we haveand for $k\geq 1$ ,$(\mu(t)D_{z})^{\alpha}(tD_{t})^{j}$$k$$\Omega_{T_{0}}[\gamma_{0}]$

$t$

$z$

$k$

$v_{k}(t, z)’ \mathrm{s}$$uk(t, z)’ \mathrm{s}$$t$$u_{k}(t)$$z$$v_{0}(t)= \int_{[t_{j}s]}^{(m)}G_{s}{}^{t}(\lambda)f$$v_{k}(t)=- \int_{[t_{j}s]}^{(m)}G_{s}{}^{t}(\lambda)S(s)v_{k-1}$

.

(3.3)

To prove that the approximate solutions converge to the real solution, we
will henceforth fix one $t\in[0, T]$ , and estimate the functions. Let $C$ be the
$K$
$[0,
T]$
bound on, andof allbe the bound in.
ofAs, we have for $1\leq k\leq m$ and for some $D>0$ :and$v_{k}(t)$$\cross\overline{B}_{R}$$c_{j,\alpha}(t, z)$$\overline{\overline{\Omega_{T}[\gamma]}}$$\mathrm{S}(\mathrm{t})z)$$H_{s}^{t}(k, \lambda)$$G_{s}{}^{t}(\lambda)$$z \in_{R}\mathrm{s}\mathrm{u}_{\frac{\mathrm{p}}{B}}|G_{s}{}^{t}(\lambda)|\leq(\frac{s}{t})^{L}$and$z \in_{R}\mathrm{s}\mathrm{u}_{\frac{\mathrm{p}}{B}}|H_{\mathit{8}}^{t}(k, \lambda)|\leq D(\frac{s}{t})^{L}$(3.1)

We can easily see thatis bounded by $KL^{-m}$ for any $0\leq t\leq T$ .
Here, we have written for conveniencein place of. For general ,we note thatis given by the iterated integral$||v_{0}(t)||_{\omega_{t}}$$||\cdot||_{\omega_{t}}$$k$$||\cdot||_{\omega_{t}[\gamma]}$$v_{k}(t)$$v_{k}(t)$$=$$(-\mathrm{l})^{}$$\int_{[ts_{k}]}^{(m)}G_{s_{k}}^{t}(\lambda)S(s_{k})\int_{[s_{k},s_{k-1}]}^{(m)}.G_{s_{k-1}^{k}}^{\mathit{8}}(\lambda)S(s_{k-1})\cdots$...

$\int_{[s_{2j}s_{1}]}^{(m)}G_{s_{1}^{2}}^{s}(\lambda)S(s_{1})\int_{[\epsilon_{1j}s\mathrm{o}]}^{(m\rangle}G_{\mathit{8}0}^{s_{1}}(\lambda)f(s_{0})$

.

(3.5)

The expression above can be expanded using Lemma 2, and thus obtain afinite
sum whose number of terms is less than $(mJ)^{k}$ , where is the cardinality ofthe set {( , ) $;0\leq j\leq m-1$ and $|\alpha|\leq m-j$ }. Each term of the finite sum
$J$

$j$

$\alpha$102
has the form
$I$

$=$

$(-\mathrm{l})^{}$$\int_{1t_{j\mathit{8}]}}^{(m_{k})}G_{s_{k}}^{t}(\lambda)c_{j_{k},\alpha_{k}}(\mu D_{z})^{\alpha_{k}}\int^{(i_{k})}[s_{kj}s_{k-1}H_{\partial_{k-1}}^{s_{k}}(i_{k}, \lambda)c_{j_{k-1},\alpha_{k-1}}(\mu D_{z})^{\alpha_{k-1}}]$. ..

$\int_{[\epsilon_{2j\partial_{1}]}}^{(i_{2})}H_{s_{1}}^{s_{2}}(i_{2}, \lambda)c_{j_{1},\alpha_{1}}(\mu D_{z})^{\alpha_{1}}\int_{[\epsilon_{1}\epsilon 0]}^{(i_{1})}H_{\epsilon_{0}}^{s_{1}}(i_{1}, \lambda)f(s_{0})$where for each , the relations $m-j_{p}\leq i_{p}\leq m$ andis amulti-index and should not be confused with the
The above is further equal to$\alpha_{p}$

$=$

(3.6)

hold. (Here,component of )$|\alpha_{p}|\leq m-j_{p}$

$p$

$I$

,

$(-\mathrm{l})^{}$$p\mathrm{t}\mathrm{h}$$\alpha.$$\int_{[t_{j\partial}]}^{(m_{k})}\int_{[_{\mathit{8}_{k}j\partial_{k-1}}]}^{(i_{k})}\ldots\int_{[\epsilon_{1j}\mathrm{s}\mathrm{o}]}^{(\dot{l}_{1})}G_{k}^{t},c_{j_{k\prime}\alpha_{k}}(s_{k})(\mu(s_{k})D_{z})^{\alpha_{k}}$$\cross H_{\epsilon_{k-1}}^{\epsilon_{k}}c_{j_{k-1\prime}\alpha_{k-1}}(s_{k-1})(\mu(s_{k-1})D_{z})^{\alpha_{k-1}}\cdots$.

$\cross H_{\epsilon_{1}}^{\mathit{8}2}c_{j_{1},\alpha_{1}}(s_{1})(\mu(s_{1})D_{z})^{\alpha_{1}}H_{\epsilon_{0}}^{s_{1}}f(s_{0})$

(3.7)

Let
denote the integrand of the above integral. Let.
Then all the functions above, when viewed as afunction of , belong in.
(This explains the necessity of the assumption that the coefficients be defined
up to, for all in the interval $[0, T].)$
We can therefore aPPly Lemma 4repeatedly, starting from the rightmost
expression, to obtain the following estimate: for any, we have$F_{k}(s)$$R_{\epsilon_{0}}=R-\gamma\varphi(s_{0})$$z$$B_{R}$$A(\omega_{\epsilon_{0}}[\gamma])$$t$$\rho\in(0, R_{\epsilon_{0}})$$\leq$$||F_{k}(s)||_{B_{\rho}}$$K(CD)^{k} \mu(s_{1})^{|\alpha_{1}|}\cdots\mu(s_{k})^{|\alpha_{k}|}(\frac{s_{0}}{t})^{L}\mathrm{x}$$( \frac{e}{R_{\epsilon 0}-\rho})^{|\alpha_{1}+\cdots+\alpha_{k}|}|\alpha_{1}+\cdots+\alpha_{k}|!$

.

(3.8)

If $|\alpha_{1}+\cdots+\alpha_{k}|=0$ , then for sufficiently small $T=To$ , the bound for any$Cj,0(t, z)=aj,0(t, z)-aj,0(0, z)$ is actually small, since $a_{j,0}(t, z)$ is continuouswith respect to . In other words, by choosing asmall, we could find a$T=T\mathrm{o}$$t$small constant$\delta$such that for any $t\in[0, T_{0}]$ and $0\leq s\leq t$ , the following holds:$||F_{k}(s)||_{\omega_{\ell}} \leq K\delta^{k}(\frac{s_{0}}{t})^{L}$(3.9)

Going back to the integral, we have$||I||_{\omega_{t}}$$= \leq\int_{K}[t_{j}\ldots\int_{[\epsilon_{1j}\epsilon_{0}]}^{(i_{1})}K\delta^{k}(\frac{s_{0}}{t})^{L}\frac{(m)\epsilon_{k}]\int^{(\dot{l}_{k})}[s_{kj}\epsilon_{k-1}]\delta^{k}}{L^{m+\dot{l}_{1}+\cdots+\dot{l}_{k}}}\leq K(\frac{\delta}{L_{0}})^{k},$(3.10)

for some constant$L_{0}$

dependent on L. This is possible since $i_{p}\leq m$ for all p.

103
If

$|\alpha_{1}+\cdots+\alpha_{k}|\neq 0$$||F_{k}(s)||_{\omega_{t}}$$\leq$, set the$\rho$in (3.8) to be equal to$R-\gamma\varphi(t)$. This gives$K(CD)^{k} \mu(s_{1})^{|\alpha_{1}|}\cdots\mu(s_{k})^{|\alpha_{k}|}(\frac{s_{0}}{t})^{L}$(3.11)$\cross|\alpha_{1}+\cdots+\alpha_{k}|!(\frac{e}{\gamma[\varphi(t)-\varphi(s_{0})]})^{|\alpha_{1}+\cdots+\alpha_{k}|}$.
, , we have
By renaming if necessary, assume that for $p=1$ ,$a_{j,0}(t,z)$$q\geq1$. We will again use the continuity ofto estimate those
Note that.
, i.e., the $k-q$ cases when
expressions which are not acted upon by
Just like before, we can show that for small ,$\ldots$$|\alpha_{p}|\neq 0$$q$$|\alpha_{p}|=0$$D_{z}$$\delta$$||F_{k}.(s)||_{\omega_{t}}$$\leq$$K(CD)^{q} \delta^{k-q}\mu(s_{1})^{|\alpha_{1}|}\cdots\mu(s_{q})^{|\alpha_{q}|}(\frac{s_{0}}{t})^{L}$(3.11)$\cross|\alpha_{1}+\cdots+\alpha_{q}|!(\frac{e}{\gamma[\varphi(t)-\varphi(s_{0})]})|\alpha_{1}+\cdots+\alpha_{q}|$Thus, the integral I can now be estimated as follows:$||I||_{\omega_{t}}$$\leq$$K(CD)^{q} \delta^{k-q}(\frac{e}{\gamma})^{|\alpha_{1}+\cdots+\alpha_{q}|}|\alpha_{1}+\cdots+\alpha_{q}|!$$\cross\int_{[t_{j}s_{k}]}^{(m)}\int_{[s_{kj}s_{k-1}]}^{(i_{k}\rangle}\ldots\int_{[\sigma_{1j}s_{\mathrm{O}}]}^{(i_{1})}(\frac{s_{0}}{t})^{L}\frac{\mu(s_{1})^{|\alpha_{1}|}\cdots\mu(s_{q})^{|\alpha_{q}|}}{[\varphi(t)-\varphi(s_{0})]^{|\alpha_{1}+\cdots+\alpha_{g}|}}$. (3. 8)

Let $d=m+i_{1}+\ldots+i_{k}$ and $b=|\alpha_{1}+\ldots+\alpha_{q}|$ . Note that $b\geq q$ . Since forandeach , we have $|\alpha_{p}|\leq m-j_{p}\leq i_{p}$ , and using the fact that both$(0,
T_{0})$,
we haveare increasing on$\mu(t)$$\varphi(t)$$p$$||I||_{\omega_{t}} \leq K(CD)^{q}\delta^{k-q}(\frac{e}{\gamma})^{b}b!$$\cross\int_{0}^{t}\int_{0}^{\xi_{b}}\ldots\int_{0}^{\xi_{1}}\frac{\mu(\xi_{b})}{\xi_{b}}\cdots\frac{\mu(\xi_{1})}{\xi_{1}}(\frac{\xi_{0}}{t})^{L}\frac{1}{[\varphi(t)-\varphi(\xi_{0})]^{b}}\frac{d\xi_{0}}{\xi_{0}}d\xi_{1}\cdots d\xi_{b}$(3.14)$\cross\int_{0}^{\xi_{0}}\int_{0}^{\eta 1}\ldots\int_{0}^{\eta d-b-2}(\frac{s_{0}}{\xi_{0}})^{L}\frac{ds_{0}}{s_{0}}\cdots\frac{d\eta_{1}}{\eta_{1}}$By (a) of Lemma 3, the second integral is equal to $L^{-d+b+1}$ . Thus, the above
simplifies into$||I||_{\omega_{t}} \leq K(CD)^{q}\delta^{k-q}(\frac{e}{\gamma})^{b}L^{-d+b+1}b!$. (3.15)$\cross\int_{0}^{t}\int_{0}^{\xi_{b}}\ldots\int_{0}^{\xi_{1}}\frac{\mu(\xi_{b})}{\xi_{b}}\cdots\frac{\mu(\xi_{1})}{\xi_{1}}(\frac{\xi_{0}}{t})^{L}\frac{\xi_{0}^{-1}}{[\varphi(t)-\varphi(\xi_{0})]^{b}}d\xi_{0}\cdots d\xi_{b}$The last integral is equal to $(Lb!)^{-1}$ , by (b) of Lemma 3. Meanwhile, since.
, depending on , such that$d\leq m(k+1)$ , we can find aconstant$L_{1}$$L$$L^{-d}\leq L_{1}^{k}$

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Substituting these results into the above equation, we get$||I||_{\omega_{t}}$$\leq$$K(CD)^{q} \delta^{k-q}(\frac{eL}{\gamma})^{b}L_{1}^{k}$(3.16)$=K( \frac{CD}{\delta})^{q}(\delta L_{1})^{k}(\frac{eL}{\gamma})^{b}$

By taking asufficiently small To, we can find a small enough such thatabove andin (3.10) are both less than $(mJ)^{-1}$ . Now, since $q\leq b$ , we canmake the remaining expression less than one by choosing alarge.
To summarize, we have shown that if
is sufficiently small and 70 is sufficiently large, some constants $K>0$ andexist such that for all , wehave.
(3.17)
for any $t\in[0, T_{0}]$ .
$\delta$$\delta L_{1}$$\delta L_{0}^{-1}$$\gamma=\gamma_{0}$$T_{0}$$k$$\delta_{0}<1$$||v_{k}(t)||_{\omega_{t}[\gamma 0]}\leq K\delta_{0}^{k}$

It follows that the series
is majorized by aconvergent geometricseries, and hence is itself convergent in.
for all
converges uniformly to $u(t)$ on
This means that.
By following the steps above, we can also show that for $1\leq p\leq m-1$ ,the sequence $(tD_{t})^{p}uk(t)$ converges uniformly to $(tD_{t})^{p}u(t)$ on. Thus,, the sequence
it follows that on acompact subset of
converges to. This implies the convergence of the approximate
solutions to the true solution $u(t)$ .
Uniqueness may be proved in asimilar manner.
$\sum_{k=0}^{\infty}.v_{k}(t, z)$$C^{0}([0, \tau], A(\omega_{\tau}[\gamma 0]))$$u_{k}(t)$$\tau\in[0, T_{0}]$$\Omega_{T_{0}}[\gamma_{0}]$$\Omega_{T_{0}}[\gamma_{0}]$$\Omega_{T_{0}}[\gamma_{0}]$$D_{z}^{\alpha}(tD_{t})^{p}u_{k}(t)$$D_{z}^{\alpha}(tD_{t})^{p}u(t)$References[1] Baouendi, M. S. and C. Goulaouic, Cauchy Problems with characteristic
initial hypersurface, Comm. Pure. Appl. Math. 26 (1973), 455-475.
[2] Fischer, E., Intermediate Real Analysis, Springer-Verlag, New YorkHeidelberg-Berlin, 1983.
[3] H\"ormander, L., Linear Partial Differential Operators, Springer-Verlag,
Berlin-Heidelberg-New York, 1969.
[4] Lope, J. E. C., Existence and Uniqueness Theorems for a Class of Linear
Fuchsian Partial Differential Equations, J. Math. Sci. Univ. Tokyo 6(1999),527-538.

[5] Mandai, T., Characteristic Cauchy problems for some non-Fuchsian Partial
differential operators, J. Math. Soc. Japan 45 (1993), 511-545.
[6] Tahara, H., On asystem of singular partial differential equations,J. Math. Soc. Japan 34 (1982), 279-288.
[7] Tahara, H., On the uniqueness theorem for nonlinear singular partial diferential equations, J. Math. Sci. Univ. Tokyo 5(1998), 477-506.
[8] Yamane, H. Singularities in Fuchsian Cauchy Problems with Holomorphic
Data, Publ. RIMS, Kyoto Univ. 34 (1998), 179-190$Volevi\check{c}$