数理解析研究所講究録983 巻 1997 年 47-59

47

TOEPLITZ OPERATORS IN THE ANALYTIC GOURSAT PROBLEM
MASATAKE MIYAKE (三宅正武)

Nagoya University (名古屋大学、 多元数理科学研究科)$0$. Introduction- Goursat problem and the spectral condition.
Let $P(x, D)$ be a partial differential operator of order $m$ ,$P(x, D)= \sum|\alpha \mathrm{I}\leq ma(\alpha X)D^{\alpha}$,

are assumed to be in , the set of holomorphic functions ina neighborhood of the origin $x=0$ of }. Here we use the usual notation such aswhere the coefficients$O$$a_{\alpha}(x)$$\mathrm{C}’$$x=(x_{1},x_{2}, \cdots,x_{n})\in \mathrm{C}^{n}$,$D=(D_{1}, D_{2}, \cdots, D_{n})(D_{j}=\partial/\partial x_{j})$.

be a given multi-index of nonnegative integers with$|\gamma|=m$ . Then the Goursat problem ( $P,$
is formulated as follows.
Let$\gamma\in \mathrm{N}^{n}(\mathrm{N}=\{0,1,2, \cdots\})$$\mathcal{O},\gamma\rangle$$P(X, D)u(x)=f(x)\in \mathrm{O}’$$(P, \mathcal{O}, \gamma)$,$u\langle x$) $-v(x)=O(x^{\gamma})$ in

$O$

,

.
in $O$ means that $w(x)x^{-}\gamma\in O$ , or equivalently,
is unquely solvable if for any $(f(x), v(x))\in$
We say that the Goursat problem

where$w(x\rangle$$w(x)\in x^{\gamma}\cdot \mathcal{O}$$=O(x^{\gamma})$$(P, \mathcal{O},\gamma)$of the problem above.
there exists a unique solution
The most general result on the unique solvability of the Goursat problem$O\cross O$$u(x)\in \mathcal{O}$$(P, \mathcal{O},\gamma)$

isproved under the spectral condition which is stated by$(\mathrm{S}\mathrm{p})$$|a_{\gamma}(0)|> \sum_{|\alpha|=m.\alpha\neq\gamma}|a_{\alpha}(0)|\mathrm{w}\gamma’-\alpha$where$\mathrm{R}_{+}=(0,$$\infty\rangle$

, and

,

for

$\exists_{\mathrm{W}}=(w_{1},w_{2}, \cdots ’ w_{n})\in \mathrm{R}_{+}^{n}$$\mathrm{w}^{\gamma-\alpha}=w_{\mathrm{I}}^{\alpha_{1}-\gamma}1\ldots uf?\gamma_{n}-\alpha,\mathrm{t}l$. (Cf.

$\mathrm{G}^{\mathrm{o}}\mathrm{a}\mathrm{r}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}[\mathrm{G}]$,, Wagschal [W]).

The spectral condition enable us to employ the contraction mapping theorem or its
variation by introducing a Banach space associated with the Goursat problem, and this48

condition seems to be best possible so far as we employ the contraction mapping theorem.
In fact, while there are many variations of the Goursat problem, for example, there havebeen made soIne attempts for the generalization of function spaces and also for the

method of proofs, in all those papers the spectral condition is posed as a fundamental
assumption (see Persson [P], Miyake [M] and references cited there).
$l\mathrm{n}$a very recent paper by the author with M. Yoshino ([M-Y]), we have succeeded

to relax this condition, and proved the spectral property of the Goursat problem by$\mathrm{e}\mathrm{n}\mathrm{l}\mathrm{p}1_{0}\mathrm{y}\mathrm{i}\mathrm{n}\mathrm{g}$the Toeplitz operator method in two dimensional case, which is attempted togive a generalization of results by Leray$([\mathrm{L}])$

where a very simple example of operators

was studied.
In this report, we shall give more general results for the Goursat problem in thefunction spaces, and remove the restriction on the dimension of$x$variables from theview point of the Toeplitz operators, which is a generalization of results in [M-Y].

We note that this is a joint work with M. Yoshino of Chuo University, and the detailwill be published elsewhere.
1. Goursat problem in Gevrey space.

First, we introduce the Gevrey space where we consider the Goursat problem.
Let
space

$\mathrm{s}=(s_{1}, s_{2}, \cdot\cdot, , s_{n})\in \mathrm{R}_{+}^{n}$$\mathcal{G}^{\mathrm{s}}(\mathrm{w})$and$\mathrm{w}=(w_{1},w_{2}, \cdot, .

, w_{n})\in \mathrm{R}_{+}^{n}$. Then an s-Gevrey

is defined by the following isomorphism which is called the formal Borel

transformation.
(1)$u(X)=$$\sum_{n,\alpha\epsilon \mathrm{N}}u_{\alpha}\frac{x^{\alpha}}{\alpha!}\in \mathcal{G}^{\mathrm{s}}(\mathrm{w})\Leftrightarrow \mathrm{d}\mathrm{e}\mathrm{f}$.

$v(z):= \sum_{t\overline{X}\in \mathrm{N}^{n}}u_{\alpha}\frac{z^{\alpha}}{(\mathrm{s}\cdot\alpha)!}\in O(|z|<\mathrm{w}^{\mathrm{s}})$where $\{|z|<\mathrm{w}^{s}\}:=\prod_{j=1}^{n}\{|z_{j}|<w_{j}^{s_{j}}\}$ , andfunctions on$\Omega\subset \mathrm{C}^{n}$

By the definition,. Here,$\mathcal{G}^{s}(\mathrm{w})$$\mathrm{s}\cdot\alpha:=\sum_{j=1}^{n}S_{j}\alpha_{j}$$O(\Omega)$and,denotes the set of holomorphic$(\mathrm{s}\cdot\alpha)!:=\Gamma(\mathrm{s}\cdot\alpha+1)$.

has a structure of Fr\’echet space induced fromequipped with uniformly convergent topology on every compact set of$\mathcal{O}(|z|<\mathrm{w}^{\mathrm{s}})$$\{|z|<\mathrm{w}^{s}\}$.

Examples. Let $s=(s, s, \cdots, s)(0<s\leq 1)$ .
[1]$\mathcal{G}^{1}(\mathrm{w})=^{o(}||X||/\mathrm{W}<1)=,\cap o(||X\mathrm{W}<\mathrm{W}||/\mathrm{W}^{;}\leq 1)$,

where $\{||X||/\mathrm{w}<1\}=\{|x_{1}|/w_{1}+|x_{2}|/w_{2}+\cdots+|x_{n}|/w_{n}<1\}$ , and $(w_{1}’, \cdots, w_{7?}’)<$$(w_{1}, \cdots, w_{n})$means that $w_{j}’<w_{j}((j=1,2, \cdots,n)$ . Here, $O(||x||/\mathrm{W}\leq 1):=O(||x||/\mathrm{W}<$49$1)\cap C0(||X||/\mathrm{W}\leq 1)$$1\}$

is a Banach space by the uniform convergence topology on$\{||x||/\mathrm{w}\leq$.
[2] For $0<s<1$ ,$\mathcal{G}^{\mathit{8}}(\mathrm{w})=,\cap A1/\mathrm{t}1-s)(\mathrm{w}’)\mathrm{W}<\mathrm{W}$where$A^{1/\langle-S}1$)$(\mathrm{w})$’is a Banach space of entire functions of exponential order $1/(1-s)$

defined by$A^{1/(1-s\rangle}(\mathrm{w})=\{u(x)$;$\exists c=C(\mathrm{W})\geq 0\}$$|u(_{X)|} \leq C\exp[(1-S)_{S}s/(1-s)(\frac{||x||}{\mathrm{W}^{S}})^{1/()}1-S],$

where $||x||/\mathrm{w}^{S}=|x_{1}|/w^{S}1+\cdots|x_{n}|/w^{s}n$ . Here, the norm,in the above inequality.
defined by the infimum of such$||u||$,

for $u(x)\in A^{1/(1}-s)(\mathrm{W})$ , is$C’ \mathrm{s}$Let $P(x, D)$ be a partial differential operator with polynomial coefficients and we write

it by

,

(2)

$a_{\alpha\beta}\in$$P(x, D)= \mathrm{f}\mathrm{i}\alpha_{\backslash }\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{S}\mathrm{u}\mathrm{m}\beta\sum_{\mathrm{N}\in}a\alpha\beta XD\beta\alpha$For a given multi-index$(P, \mathcal{G}^{\mathrm{s}}(\mathrm{W}),\gamma)$

where$\gamma\in \mathrm{N}^{n}$, the Goursat problem$P(_{X,D})u(x)=f(X)\in \mathcal{G}^{\mathrm{s}}(\mathrm{w})$$f(x),v(x)\in \mathcal{G}^{\mathrm{s}}(\mathrm{w})$Here $w(x)=O(x^{\gamma})$ in,

C.

$(P, \mathcal{G}^{\mathrm{s}}(\mathrm{w}),$$\gamma)$

is formulated by$u(x)-v(x)=^{o(x^{\gamma}})$in$\mathcal{G}^{\mathrm{s}}(\mathrm{w})$are given functions, and $u(x)$ is the unknown function in.
means that,$\mathcal{G}^{\mathrm{s}}(\mathrm{w})$.

$w(x)_{X^{-\gamma}}\in \mathcal{G}^{\mathrm{s}}(\mathrm{w})$$\mathcal{G}^{\mathrm{s}}(\mathrm{w})$

2. Gevrey filtlation and Assumptions.
Let $P(x, D)$ be the operator given by (2). For, is defined byoperator $P(x, D)$ , denoted by$\mathrm{s}\in \mathrm{R}_{+}^{\tau x}$, the -Gevrey order of the$\mathrm{s}$$\mathrm{o}\mathrm{r}\mathrm{d}_{\mathrm{s}}(P)$(3)$\mathrm{o}\mathrm{r}\mathrm{d}_{\mathrm{s}}(P)=\max\{\mathrm{s}\cdot\alpha+(1-\mathrm{s})\cdot\beta;a_{\alpha\beta}\neq 0\}$.

Remark. The assumption that the operator $P(x, D)$ is of polynomial coefficients is made. In fact, in the Goursat problem
for every
only to assure that
we may assume that the coefficients of the operator are homomorphic in a.
thatneighborhood of the origin in such variables$\mathrm{o}\mathrm{r}\mathrm{d}_{s}(P)<+\infty$$\mathrm{s}\in \mathrm{R}_{+}^{n}$$(P, \mathcal{G}^{\mathrm{s}}(\mathrm{w}),\gamma)$$x_{j}$$s_{j}\geq 1$

50

In the Goursat problem$(P, \mathcal{G}^{\mathrm{s}}(\mathrm{w}),\gamma)$, the multi-index$\gamma$

is assumed to be taken so

that

(4)

$\mathrm{s}\cdot\gamma=\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{s}(P)$.

Also, as a fundamental assumption we suppose that(5)$[mathring]_{\mathrm{s}}_{P}(D):= \sum_{\mathrm{s}\cdot\alpha=\mathrm{s}\gamma}.a_{\alpha 0}D(x\neq 0$

For the Goursat problem(6)$(P, \mathcal{G}^{s}(\mathrm{w}),\gamma)$.

, we define a function$f_{\mathrm{s}_{\}\gamma}(z)(z\in \mathrm{C}^{n})$by$f_{\mathrm{s}.\gamma}(Z):=[mathring]_{P} \mathrm{S}(_{Z}-1)_{Z}\gamma=\sum_{\mathrm{s}\cdot\alpha=\mathrm{s}\gamma}.a_{\alpha 0}z\gamma-\alpha$which is called the Toeplitz symbol associated with the Goursat problem. Here, $z=$
.
and
for
.
$(z_{1}, \cdots, z_{\tau\tau})\in \mathrm{C}^{n}$$\alpha\in \mathrm{Z}^{7l}$$z^{\alpha}=z_{1}\cdots z_{n}\alpha_{1(\rangle_{- 7t}}$

3. Theorems.

Under the preparations as above, we can state our theorems as follows.

Theorem 1.

$S\mathrm{u}$ppose that

(7)

$0\not\in \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{x}$

where

–hull

$\{|z|=\mathrm{w}^{\mathrm{s}}\}--\prod_{j=}^{n}1\{|z_{j}|=w_{j}^{s_{j}}\}$$\{f_{s,\gamma}(z);|z|=\mathrm{w}^{\mathrm{s}}\}$. Then Go ursa pro em$t$$\mathrm{b}l$,$(P, \mathcal{G}^{\mathrm{s}}(\rho \mathrm{w}),\gamma)$

is uniquelysolvable for sufficiently small $\rho>0$ .
Furthermore, suppose that the -principal part of $P(x, D)$ is of constant coefficients,$\mathrm{s}$i.e.,$\mathrm{s}\cdot\alpha+(1-\mathrm{s})\cdot\beta<\mathrm{o}\mathrm{r}\mathrm{d}_{\mathrm{s}}(P)$$(P, \mathcal{G}^{\mathrm{s}}(p_{\mathrm{W}}),$ $\gamma)$

if $\beta\neq 0$ for$a_{\mathrm{c}\mathrm{x}\beta}\neq 0$

is uniquely solvable for any $p>0$ .

In the theorem, we do not assume a priori that$(P, \mathcal{G}^{\mathrm{s}}(\mathrm{w}),$$\gamma)$. Then the Goursat problem$a_{\gamma 0}\neq 0$. Concerning this we can prove the following,

Corollary. In the$Go\mathrm{u}rs\partial tP^{T}$

oblem$(P, \mathcal{G}^{s}(\mathrm{w}),$$\gamma)$, the condition (7) implies that

Proof. The condition (7) implies that the Goursat problem
solvable. On the other hand, if$O(x^{\gamma})$in the Goursat problem$([mathring]_{\mathrm{s}}_{P}(D), \mathcal{G}^{\mathrm{S}}(\mathrm{w}),\gamma)$$a_{\gamma 0}\neq 0$.

is uniquely$0$$a_{\gamma 0}=0$then $u(x)=x^{\gamma}$ satisfies $P_{\mathrm{s}}(D)u(x)=0,$ $u(x)=$, which menas that $u(x)$ is a non trivial solution with

contradiction to the unique solvability.

$0$

Goursat data which is a

51

Theorem 2. Let $n=2$ . Suppose$f_{\mathrm{s}.\gamma}(z)\neq 0$on$\{|z|=\mathrm{w}^{\mathrm{s}}\}$, and suppose,(8)$\oint_{|z_{1}}|=w_{1}^{s}1d\log f_{\mathrm{S}},\gamma(\mathcal{Z}, z21)=0$

for any fixed $z_{2}(|z_{2}|=w_{2}^{s_{2}})$ . Then the $ind$ ex of the Goursat probl
to for su fficiently small . Furt herm ore, if the condition (7) is satisfied onis, then the Goursat problem
choice offor some uita$\mathrm{e}m(P,\mathcal{G}^{\mathrm{s}}(\rho \mathrm{w}),\gamma)$$0$$e\mathrm{q}u\mathrm{a}l$$\rho$$s$$\{|_{\sim}^{\gamma}|=\mathrm{r}^{\mathrm{s}}\}$$\mathrm{r}=(r_{1},r_{2})\in \mathrm{R}_{+}^{2}$$\mathrm{b}le$all . In the case where the -Gevrey
is uniquely solvable for fficien
principal part is of const ant coefficients, the assertions as in Theorem 1 hold for any $\rho>0$ .
$s\mathrm{u}$$(P, \mathcal{G}^{\mathrm{s}}(\rho \mathrm{w}),\gamma)$$\mathrm{t}lysm$$\mathrm{s}$$\rho$

It is obvious that the condition (8) is replaced by(8’)$\oint_{|\mathcal{Z}_{2}\mathrm{I}=w^{S}}2=2d\log f_{s.\gamma}(Z_{1}, z_{2})\mathrm{o}$for any fixed$\sim’ 1(|z_{1}=w_{1}^{s_{1}})$,.

Examples.
[1] Let $P(D)=D_{1}-D_{2}^{2}$ be the heat operator, and consider the following two Cauchy

problems,(1)$(P, \mathcal{G}^{(2_{S},s})(w_{1},w2),$$(1,0))$ ,(2)$(P, \mathcal{G}^{()}2s,S(w_{1}, w2), (\mathrm{o}, 2))$,where $s>0$ . The Toeplitz symbols corresponding to each Cauchy problem are given by

(1)
where

$|_{\sim 1}^{\gamma}|=w_{1}^{2s},$$f_{s.(1,0} \rangle(Z)=1-\frac{z_{1}}{z_{2}^{2}}$$|z_{2}|=u_{2}^{s}$,(2)$f_{B,(0.2} \rangle(Z)=\frac{z_{2}^{2}}{z_{1}}-1$,. Hence the problem (1) is uniquely solvable if $w_{1}<w_{2}$ , andthe problem (2) is uniquely solvable if $w_{1}>w_{2}$ .
is a complex[2] (Example by Leray [L]) Let $P(D)=\lambda D_{1}D_{2^{-}}D^{2}1^{-D_{2}^{2}}$ ’ whereparameter. Let $\mathrm{w}=(w, 1)(w>0)$ and $\gamma=(1,1)$ , and consider the Goursat problem$\lambda\in \mathrm{C}$$(P(D), O(||X||/\mathrm{w}<\rho))\gamma)$. Then this Goursat problem is uniquely solvable if$\frac{({\rm Re}\lambda)^{2}}{(1+w)^{2}}+\frac{({\rm Im}\lambda)^{2}}{(1-w)^{2}}>1$,$w\neq 1$$\lambda\not\in[-2,2]$

,

$w=1$

$\{$

[3] Let$P(D)=- \frac{9}{4}D_{1}^{3}+3D_{1}D_{2}^{2}+\frac{4}{3}D_{2}^{3}$.

52

Let $\mathrm{w}=(w, 1)$ and $\gamma=(2,1)$ , and consider the Goursat problem $(P(D),$ $O(||x||/\mathrm{w}<$$\rho),$$\gamma)$, where$D_{1}^{2}D_{2}$Then for any$w$

is absent from the operator $P(D)$ .

with $3/4<w<5/2$ , this Goursat problem has an index

with 1$0$dimensional kernel and cokernel. In fact, in this case the Toeplitz symbol is given by$f_{1.\gamma}(_{Z}1,1)=- \frac{9}{4}z^{-1}+13z_{1}+\frac{4}{3}Z_{1}^{2}=(\frac{4}{3}-\frac{1}{z_{1}})(z_{1}+\frac{3}{2})2$,

and the condition (8) is satisfied only for $3/4<w<3/2$ , but the condition (7) can not
be satisfied on$\{|z|=\mathrm{r}^{\mathrm{s}}\}$

for any$\mathrm{r}=(r_{1}, r_{2})$. It is easily seen that$u(x)=x_{1}^{2_{X_{2}}}$is thebase of the kernel, and $(v(x),f(x))=(0,1)$ is the base of the cokernel of the Goursat
problem.

4. Toeplitz operators.
Theorems are proved by employing the following elementary results on the Toeplitz
operators.

Let $T^{n}= \prod_{j=1}^{n}\{|z_{j}|=1\}$ be theof square integrable functions onwith finite norm$n$$T^{n}$-dimensional torus. We denote by, that is, the set of functions$||u||:= \sum_{\alpha\in \mathrm{Z}^{n}}|u_{\alpha}|^{2}<\infty$. Let$T_{f}$:$H^{2}(\tau^{n})arrow H^{2}(\tau^{n})$(9)$T^{n}$. Let$T^{n}$, that:, the Toeplitz operator$\alpha\not\in \mathrm{N}^{n}$be the natural projection. Then for $f(z)$ continuous on

the set$u(z)= \sum_{\alpha\in \mathrm{Z}^{n}}u_{\alpha}Z^{\alpha}$be the Hardy space on$H^{2}(\tau^{n})$is, the set of functions $u(z)\in L^{2}(T^{?l})$ such that $u_{\alpha}=0$ for$H^{2}(\tau^{n})$$L^{2}(T^{7\mathrm{z}})$$\pi$$L^{2}(T^{?l})arrow$is defined by$T_{f}(u)=\pi(f(z)u(z))$ ,$u(z)\in H^{2}(\tau^{n})$Here $f(z)$ is called the Toeplitz symbol of the operator$\tau_{f}$.

.

Now we have,

Proposition 1. (i) Let
(10)
For

$\sigma(T_{f})$$\sigma(T_{f})\subset$$\lambda\not\in\Gamma(f)$, the $op$ era(11)

be the spectrum ofConvex–hull$t_{0\Gamma \mathrm{n}}o\mathrm{r}\mathrm{m}$

of

(12)

:

$H^{2}(\tau^{n})arrow H^{2}(\tau^{n})$$\{f(z);z\in T^{n}\}=_{\mathrm{t}}\mathrm{r}(\mathrm{p}\mathrm{u}f)$$\lambda I-T_{f}$. Then,.

is estimated by$||\lambda I-\tau_{f}||\geq \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(\lambda, \Gamma(f))$(ii) Let $n=1$ . Then if $f(z)\neq 0(|z|=1),$$\chi(T_{f})$$\tau_{f}$$\tau_{f}$.

is a Fredholm operator with an index

given by$\chi(T_{f})=\frac{-1}{2\pi i}\oint_{|z|1}=\mathrm{o}d\mathrm{l}\mathrm{g}f(_{Z)}$.

53

Next we introduce the notion of finite section Toeplitz operators.

Suppose that $n=1$ . For $N\geq 0$ , we define$H_{N}^{2}(T):=H2(T)\cap \mathcal{P}_{N}$where$\mathcal{P}_{N}$$H_{N}^{2}(T)$denotes the set of polynomials of$z$,of degree at most $N$ . Let$\pi_{N}$: $L^{2}(T)arrow$be the natural projection. Then an $N$ -th finite section Toeplitz operator $T_{f}(N)$ :$H_{N}^{2}(T)arrow H_{N}^{2}(T)$is defined by$T_{f}(N)(u)=\pi_{N}(f(Z)u(Z))$ ,(13)$u(z)\in H_{N}^{2}(T)$.

Proposition 2. Let $f(z)\in \mathrm{C}[z,z^{-1}]$ . Then we have:(i) In order that there exists $N_{0}\geq 0$ such that$T_{f}(N)$is inverti$\mathrm{b}le$: $H_{N}^{2}(T)arrow H_{N}^{2}(T)$for all $N\geq N_{0}$ with the uniform norm estim ate from below$||T_{f}(N)||\geq d>0$

for some$co\mathrm{n}$stant$\mathrm{a}\mathrm{n}d_{/}\chi(T_{f}\cdot)=0$$d$$\mathrm{O}\not\in\Gamma(f)$The Toepli$\chi(T_{J}\cdot)=0$

in epend$\mathrm{e}nt$of$N\geq N_{\zeta\}_{f}}$it is necessary and sufficient th at $f(z)\neq 0$.

Moreover, if

(ii)

$d$

. And th$\mathrm{t}z$, then we an take $N_{0}=0$ .
$c$operator$\mathrm{e}op$era$\tau_{f}$is invertible if and only if $f(z)\neq 0(|z|=1)$ andis estimated $by||T_{f}||\geq d>0$ , where depen $ds$
$d$$tor\mathrm{n}oT\mathrm{m}o\mathrm{f}T_{f}$on $\min\{|z-z_{j}|;|z|=1, f(z_{j})=0\}$ .
5. Sketch of the proof.

We shall give an outline of the proofs deviding several steps.
5.1. Reduction to an integro-differetial equation.
Let$u(x)= \sum_{\alpha\in \mathrm{N}^{nC}}ux^{c}xX/\alpha!\in \mathrm{C}[[x]]$over C. Then an integration(14)$D^{-\gamma}u(x)$, the set of formal power series of -variables

for

$x$

$\gamma\in \mathrm{N}^{n}$is defined by$D^{-\gamma}u(X)= \sum\alpha\epsilon \mathrm{N}^{n}u_{\Omega}\frac{x^{\alpha+\gamma}}{(\alpha+\gamma)!}$.

54

Then by the definition we have thathave$D^{\gamma}D^{-\gamma}=id$. on$\mathrm{C}[[x]]$, andfor all$D^{\alpha}D^{-\gamma}=D^{\mathfrak{a}-\gamma}$$D^{-\gamma}D^{\gamma}=id$. on$\alpha,$$\gamma\in \mathrm{N}^{n}$for all$x^{\gamma}\cdot \mathrm{C}[[x]]$. Especially we$\gamma\in \mathrm{N}^{n}$.

It is easily proved that
$D^{-\gamma}$:$\mathcal{G}^{\mathrm{s}}(\rho \mathrm{w})arrow x^{\gamma}\cdot \mathcal{G}^{\mathrm{s}}(\rho \mathrm{w})$is isomorphic.

We notice that in the Goursat problem$(P, \mathcal{G}^{\mathrm{s}}(p\mathrm{w}),$

for the Goursat data (i.e., $u(x)=O(x^{\gamma})$ in, we may assume that $v(x)=0$, by taking $w(x)=u(x)-v(x)$$\gamma)$$\mathcal{G}^{\mathrm{s}}(p\mathrm{w}\rangle)$

as a new unknown function. Hence the unique solvability of the Goursat problem
isivalent to the bijectivity of the mapping$(P, \mathcal{G}^{\mathrm{s}}(p\mathrm{w}),$$\gamma)$$\mathrm{e}\mathrm{q}u$$P(x, D)$:$x^{\gamma}\cdot \mathcal{G}^{\mathrm{s}}(\rho \mathrm{w})arrow \mathcal{G}^{\mathrm{s}}(\rho \mathrm{w})$.

These observations show that the study of the Goursat problem$(P, \mathcal{G}^{\mathrm{s}}(p\mathrm{W}),\gamma)$

is reduced

to the study of the following mapping of integro-differential operator(15)$L(x, D)\equiv P(x, D)D^{-\gamma}$:$\mathcal{G}^{\mathrm{s}}(p\mathrm{w})arrow \mathcal{G}^{\mathrm{s}}(\rho \mathrm{w})$.

We notice that
$L(x, D)= \mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{e}\sum_{\alpha,\beta}^{\mathrm{t}}a_{JX/\mathrm{i}^{XD\sum_{\beta}D}}-\gamma=\mathrm{s}\mathrm{u}\mathrm{m}\rho\alpha \mathrm{p}\mathrm{u}\mathrm{t}\delta,a_{\}}‘\beta^{X^{\beta\delta}}$,$(\delta\in \mathrm{Z}^{n})$satisfies,$a_{\mathit{6}\beta}\neq 0\Rightarrow \mathrm{s}\cdot\delta+(1-\mathrm{s})\cdot\beta\leq 0$i.e.,$\mathrm{o}\mathrm{r}\mathrm{d}_{\mathrm{s}}(L)\leq 0$.

Also by the assumption (5) we have.

$[mathring]_{\mathrm{s}}_{L}(D):= \sum_{s\cdot\delta=0}a\delta 0^{D=}[mathring]_{P}\mathrm{s}(\delta D\mathrm{I}D-\gamma\neq 0$

To end this section, we remark that the Fredholm property of the Goursat problem$(P, \mathcal{G}^{\mathrm{s}}(\mathrm{w}),$$\gamma)$means that of the mapping (15) and an index of the Goursat problem isjust given by that of the the mapping (15).

5.2. Hilbert space
We define a Hilb

$G^{\mathrm{s}}(\rho \mathrm{w})$$e\mathrm{r}\mathrm{t}$

space

.
$G^{\mathrm{s}}(\rho \mathrm{w}\rangle$

by$u(x)= \sum_{\in\alpha \mathrm{N}^{\eta}}u_{\alpha^{\frac{x^{\alpha}}{\alpha!}}}\in G^{s}(p_{\mathrm{W})}$$\Leftrightarrow v(z)(\mathrm{l}\mathrm{e}\mathrm{f}.:=$$\sum_{n,\alpha\in \mathrm{N}}\frac{u_{\alpha}}{(\mathrm{s}\cdot\alpha)!}((p\mathrm{W})^{\mathrm{s}_{Z)}}\alpha 2\in H(T\}\mathrm{t})$

,

55

where

$(\rho \mathrm{w})^{\mathrm{s}}Z=((\rho w_{1})^{s_{1}}z_{1}, \cdots, (pw_{n})^{s}l\gamma z)n$

. Then it is easily seen that.

$\mathcal{G}^{\mathrm{s}}(p\mathrm{w})=\bigcap_{\hslash<_{\mathit{1}}j}G^{\mathrm{s}}(\kappa \mathrm{w})(= \mathrm{p}\mathrm{r}\mathrm{o}_{\wedge}\mathrm{j}\lim_{\beta h\nearrow}G\mathrm{s}(\kappa \mathrm{w}))$Hence by taking the procedure of projective limit the Goursat problem
is reduced to the following mapping instead of (15),(16)$L(x, D)$:$0<\kappa<p$$G^{\mathrm{s}}(\kappa \mathrm{W}\ranglearrow G^{\mathrm{s}}(\kappa \mathrm{W}),$$(P, \mathcal{G}^{s}(p\mathrm{W}),\gamma)$.

Now we decompose the operator $L(x, D)$ in the form$L(x,D)=[mathring]_{\mathrm{s}}_{L}(D)+ \sum_{\neq \mathrm{s}\cdot\ell,+(1-\mathrm{s}\rangle\cdot\beta=0;\beta 0}a_{\delta\beta}XD\beta\delta+\sum_{<\mathrm{s}\cdot b+(1-\mathrm{s}\rangle\cdot\beta 0}a\delta\beta xD\beta\delta$$\mathrm{p}\mathrm{u}\mathrm{t}=[mathring]_{\mathrm{s}}_{L}(D)+Q(x, D)+R(_{X,D)}$.

Then we haveis a unded opera
Proposition 3. (1)es ated $ind$ ependently on $p>0$ by$L_{\mathrm{s}}(D)\circ$$b\mathrm{o}$$to\mathrm{r}$on$G^{\mathrm{s}}(\rho \mathrm{w})$and th opera$e$$to\mathrm{r}$norm is$ti\mathrm{m}$$||[mathring]_{\mathrm{s}}_{L}(D)|| \leq\sum_{s\cdot\delta=0}|a\delta|_{\mathrm{W}}5\delta$

where
(2)

$\mathrm{w}^{\mathrm{s}b}=w_{1}^{s_{1}b_{1}}\cdots wn^{\mathrm{t}}g\delta \mathrm{b}1\iota$$Q(D)$,.

is a onded opera $tor$ on$\mathrm{b}$$G^{\mathrm{s}}(p\mathrm{w})$, and its operator norm is evaluated by$||Q(X, D)||=o(1)$ as $p\searrow 0$ .

(3)

$R(x, D)$ is a compact operator on$||R(x, D)||=\mathit{0}\langle 1)c?sp\searrow 0$$G^{\mathrm{s}}(p\mathrm{w})$, and its operator norm is evaluated by.

As an immediate corollary to this proposition, we get the spectral condition as follows.
Corollary. Suppo se for$\mathrm{w}$the foll owing$\mathrm{c}on$

dition is sa tisfi$(\mathrm{S}\mathrm{p})$$|a_{\gamma 0}|> \sum_{\gamma_{\backslash }\mathrm{s}\cdot\alpha=\mathrm{s}\cdot\cdot\alpha\neq\gamma}|a_{\alpha 0}|\mathrm{w}^{\mathrm{s}(}\gamma-\alpha\rangle$

Then the Goursat probl em $(P(x, D),$$p$.

$\mathcal{G}^{\mathrm{s}}(p\mathrm{w}),$$\gamma)$$\mathrm{e}d$.

is uniq uely olvable for$\mathrm{s}$$s\mathrm{t}\mathrm{I}\mathrm{f}\mathrm{f}\mathrm{i}ciently$

small

56

Furthermore, if th$\mathrm{e}\mathrm{s}$-Gevrey principal part of $P(x, D)$ is of$co\mathrm{n}$stant $co$ efficients, i.e.,$Q(x, D)=0$ in th decomposition of $L(x, D)$ , then the Goursa problem$t$$\mathrm{e}$$(P, \mathcal{G}^{\mathrm{s}}(\rho_{\mathrm{W}),\gamma})$is un iq uely solvable for any $p>0$ .

Proof. As the considerations above the problem is reduced to the invertibility of theintegro-differential operator $L(x, D)$ on$G^{\mathrm{s}}(p\mathrm{w})$for all small $p>0$ . Since,$[mathring]_{\mathrm{c}}_{L}9(D)=a_{\gamma}0+ \sum_{\neq \mathrm{s}\cdot\alpha=\mathrm{s}\cdot\gamma\cdot\alpha\gamma},a_{CX0}D\alpha-\gamma$

the operator norm on$G^{\mathrm{s}}(\rho_{\mathrm{W}})$is estimated by$||L_{\mathrm{s}}( \circ D)||\geq|a_{\gamma}0|-\sum_{(\mathrm{s}\cdot\alpha=\mathrm{s}\cdot\gamma;x\neq\gamma}|a\alpha 0|\mathrm{w}^{(\gamma)}\mathrm{s}-r\chi>0$

which implies the invertibility offor$L_{\mathrm{s}}(D\circ)^{-1}$$G^{\mathrm{s}}(\rho \mathrm{w})$as$L_{\mathrm{s}}(D)\circ$on$G^{\mathrm{s}}(\rho \mathrm{w})$,with uniform norm estimates on $p>0$. Now the result follows from the operator norms for $Q(xD)$ and $R(x, D)$ on$\rhoarrow 0$

in the above proposition.

To prove the latter half we notice that $R(x,D)$ is a compact operator on$G^{\mathrm{s}}(\rho \mathrm{w})$

all $p>0$ . Hence $L(x, D)$ is a compact perturbation from the invertible operator
and therefore the index of $L(x, D)$ on

the injectivity of $L(x, D)$ on
the inclusionon$G^{\mathrm{s}}(\rho’\mathrm{w})$$G^{\mathrm{s}}(\rho \mathrm{w})$for sufficiently small$0$$L_{\mathrm{s}}(D)$,. Hence it is sufficient to show

for all $\rho>0$ . By the definition we easily see that

$G^{\mathrm{s}}(\rho \mathrm{w})$$G^{\mathrm{s}}(\rho \mathrm{w})arrow G^{\mathrm{s}}(\rho’\mathrm{w})$is equal to

for

$0$

is injective for any $\rho’<p$ . The invertibility of $L(x, D)$$p’\mathrm{i}\mathrm{m}_{\mathrm{P}^{\mathrm{l}\mathrm{i}\mathrm{s}}}\mathrm{e}$the injectivity of $L(x, D)$ on$G^{\mathrm{s}}(\rho \mathrm{w})$, whichproves the assertion.
5.3. Reduction to the theory of Toeplitz operators.

Following the argument above, we study the following mapping,
(17)

Let

$L_{\mathrm{s}}(D)\circ$$\delta\in \mathrm{Z}^{n}$satisfy:$G^{\mathrm{s}}(\rho \mathrm{w})arrow G^{\mathrm{s}}(\rho \mathrm{w})$.

. Then the following commutative diagram is$\mathrm{s}\cdot\delta=\sum_{j=1}^{n}S_{jj}\delta=0$

examined easily.
Borel tran .
$S\mathrm{f}$$\sum_{\alpha\geq 0}u_{\alpha}\frac{x^{\alpha}}{\alpha!}$$\sum_{\alpha\geq 0}\frac{u_{\alpha}}{(\mathrm{s}\cdot\alpha)!}((p\mathrm{W})^{\mathrm{s}}Z)^{\alpha}$$D^{\delta}\downarrow$$\downarrow T_{((p\backslash \backslash r}))^{8}\approx-b$.

$\sum_{\alpha-\delta\geq 0}u_{\alpha^{\frac{x^{\alpha-\delta}}{(\alpha-6)!}}}\overline{\mathrm{B}_{\mathrm{o}\mathrm{r}\mathrm{e}}1\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{D}_{\mathrm{t}}\mathrm{s}\mathrm{f}.}\sum_{a-\delta\geq 0}\frac{u_{n}}{(\mathrm{s}\cdot\alpha)!}((p\mathrm{w})^{s}Z)^{\alpha-}\delta$

57

Here

$T_{((p}\mathrm{w}\rangle^{\mathrm{s}}z$)$-\delta$

assumption thatis the Toeplitz operator on$\mathrm{s}\cdot\delta=0$, we have$H^{2}(\tau^{n})$$((p\mathrm{w})^{\mathrm{s}}Z)^{-b}=(\mathrm{w}^{\mathrm{s}}z)^{-b}$$T_{((\rho \mathrm{w})Z)^{-b}}\mathrm{s}=T_{(\mathrm{w}^{8}z)}-b$

Here, we recall thatwith symbol,$f_{\mathrm{s},\gamma}(z)\equiv P(Z^{-})1L\mathrm{s}(z^{\gamma}=z^{-1})\mathrm{o}_{\mathrm{S}}\mathrm{o}$$((\rho \mathrm{w})^{s}z)^{-}\delta$. By the, and hence$z\in T^{n}$.

.

Thus we have shown that the mapping (17) is equivalent to the following Toeplitz
which is independent of the paramerter
operator with a symbol$f_{\mathrm{s},\gamma}(\mathrm{w}^{\mathrm{s}}Z)(z\in T^{t1})$$p>0$ .

(18)

$T_{f_{\mathrm{s},\gamma}(_{\mathrm{W}z)}}\mathrm{s}$:$H^{2}(\tau^{n})arrow H^{2}(\tau^{n})$.

5.4. Proof of Theorems.

Proof of Theorem 1. The assumption (6) is equivalent that-hull$\mathrm{O}\not\in \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{x}$.

$\{f_{\mathrm{s},\gamma}(\mathrm{w}^{\mathrm{S}}z);|z_{j}|=1\}\mathrm{P}^{\mathrm{u}\mathrm{t}}=\mathrm{r}(\mathrm{w})$Hence by Proposition 1 the Toeplitz operator
norm is estimated by$T_{J\epsilon,\gamma}.\mathrm{t}\mathrm{w}^{\mathrm{s}}z$

is invertible and its operator)$||T_{J_{\mathrm{S}},1}.(\mathrm{W}\mathrm{s}z)||\geq \mathrm{d}\mathrm{i}\mathrm{S}\mathrm{t}(0, \mathrm{r}(\mathrm{w}))$,

is invertible and theonwhich is independent of $p>0$ . This shows that
operator norm of inverse operator is estimated from above uniformly on $\rho>0$ . Hence by
Proposition 3, we can conclude that the integro-differential operator $L(x, D)$ is invertible$L_{\mathrm{s}}(D)\circ$$G^{\mathrm{s}}(p\mathrm{w})$for suffciently small $\rho>0$ , which proves he theorem.
Consider the case where the -principal part of $P(x, D)$ is of constant coefficients. In
this case, $Q(x, D)\equiv 0$ in the decomposition of $L(x, D)$ , and hence $L(x, D)$ is a compact

on

$t$

$G^{\mathrm{s}}(\rho \mathrm{w})$$\mathrm{s}$for any $p>0$ . Now theonperturbation from the invertible operator
reasoning below is the same with that of Corollary to Proposition 3, and the proof is$L_{\mathrm{s}}(D)\circ$$G^{\mathrm{s}}(p\mathrm{w})$completed.

Proof of Theorem 2.
The assumption (8) says that$\oint_{\mathrm{t}^{z_{1}}|1}==d\log f\mathrm{s},\gamma(w_{11,2}zw)s_{1}s20$.

58

We shall prove an index of the operator of $L(x, D)$ on$G^{\mathrm{s}}(p\mathrm{w})$is equal to$0$

for sufficiently

small $p>0$ under this condition.
For this purpose we first consider the following mapping,:$L_{\mathrm{s}}(D)\circ$$G^{s}(\rho \mathrm{w})arrow G^{\mathrm{s}}(p\mathrm{w})$In the following, we consider only the case where
since if otherwise we know that

$L_{\mathrm{s}}(D)\circ\equiv \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{S}t$$s_{1}/s_{2}$.

is rational for$\mathrm{s}=(S_{1}, s_{2})\in \mathrm{R}_{+}^{2}$,. which means that the above mapping isinvertible, and the conclusion is obvious.

Let $\mathit{8}1/s_{2}=q/p$ be the irreducible fraction. We recall that the Toeplitz$f_{\mathrm{s}.\gamma}(\mathrm{W}^{\cdot}z\backslash ^{\backslash })\in \mathrm{C}[z, z^{-1}]$$g(z_{1}, z_{2})$is -quasi
geneous of degree , that is,. Then $g(z)$ is wri tten in the form,$0$$\mathrm{h}\mathrm{o}\mathrm{m}o$$\mathrm{s}$$\mathrm{s}\mathrm{y}\mathrm{n}$)$\mathrm{b}\mathrm{o}\mathrm{l}\mathit{9}(Z)=$put$g(c^{s_{1}92}Z_{1}, Cz_{2})=$

for $c\neq 0(c\in \mathrm{C})$$g(z)= \sum_{=j\ell}b_{j2^{-}}mz_{1}Z\mathrm{P}jqj$,

Z.

$\exists\ell\leq\exists_{m\in}$As we have shown the above mapping is equivalent to the following Toeplitz operator,$T_{g}$: $H^{2}(\tau^{2})arrow H^{2}(\tau^{2})$ .

Let us show that this mapping is deconposed into a direct sum of finite section Toeplitz
operators as follows.

Let$\mathcal{P}^{\mathrm{s}}(k)=\{u(z)=\sum_{\mathrm{s}\cdot\alpha=}k.u_{\alpha^{Z\}}}\alpha$be the set of -quasi homogeneous polymonial$\mathrm{s}$-norm on $|z|=1$ . Then the -quasi homogeneity ofzero of $g(z)$ implies that the Teoplitz poerator . becomes an operator on. Henceof degreewith$k\in \mathrm{Q}_{+}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}_{\mathrm{P}\mathrm{p}\mathrm{e}\mathrm{d}}$$L^{9}arrow$$\mathrm{s}$$T_{q}$

by decomposing the space$H^{2}(\tau^{2})$$\mathcal{P}^{\mathrm{s}}(k)$into the direct sum of -quasi$\mathrm{s}$, the Toeplitz operator
on -quasi homogeneous spaces.
Let, and put$\{\mathcal{P}^{\mathrm{s}}(k)\}(k\in \mathrm{Q}_{+})$$T_{g}$

on$H^{2}(\tau^{2})$$\mathrm{d}\mathrm{i}\ln_{\gamma}\mathcal{P}^{\mathrm{s}}(k)=N$$h(z_{\mathrm{I}})= \sum_{=j-l}^{m}b_{j}z_{1}^{j}(=g(z_{1}^{1/}, 1\mathrm{P}))$

Then we can easily see that:$\mathcal{P}^{\mathrm{s}}(k)arrow \mathcal{P}^{\mathrm{s}}(k)$is equivalent to$T_{h}(N)$spacesis decomposed into the operators$\mathrm{s}$$T_{g}$$\mathrm{h}_{\mathrm{o}\mathrm{I}\mathrm{n}\mathrm{o}\mathrm{g}\mathrm{o}\mathrm{u}}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{S}$: $H_{N}^{2}(T)arrow H_{N}^{2}(T)$ .

.

59

The assumption (8) shows that

. Hence by Proposition 2, there$\oint_{|z\mathrm{I}=}1\mathrm{g}hd\mathrm{l}\mathrm{o}(z)=0$, and the norm insuch that for all $N\geq N_{0},$ $T_{h}(N)$ is invertible onequalities $||T_{h(N)}||\geq d>0$ hold by apositive constant independent of $N\geq N_{0}$ . This
exists$H_{N}^{2}(T)$$N_{0}$$d$imphies that$\chi(T_{g})=\chi([mathring]_{\mathrm{s}}_{L}(D), G^{s}(\rho \mathrm{W}))=0(^{\forall}p>0)$, and the first half of Theorem 2

follows from Proposition 3.
Recall the assumption of the latter half is that there eixsts
is invertible on. Hence,that
Convex-hull$\mathrm{r}=(r_{1}, r_{2})\in \mathrm{R}_{+}$$L_{\mathrm{s}}(D)\circ$$\{f_{\mathrm{s},\gamma}(\mathrm{r}^{\mathrm{s}_{Z}})|z_{j}|=1\}$$0\not\in$$G^{6}(\mathrm{r})$such. Thishas a unique solution
shows that for any $f(x)\in \mathrm{C}[[x]]$ , the equation$u(x)\in \mathrm{C}[[x]]$ . In fact, by decomposing
into the direct sum of -quasi homogeneouspolynomials, we see that on each space of -quasi homogeneous polynomials$L_{\mathrm{S}}(\mathrm{o}D)u(x)=f(x)$$\mathrm{C}[[x]]$$\mathrm{s}$$\mathcal{P}^{\mathrm{s}}(k)$$\mathrm{s}$is injective, and therefore isinvertible for any $\rho>0$ with uniform norn est\’imates on for the inverse operator$(k\in \mathrm{Q}_{+})L_{\mathrm{s}}(D)\circ$is. invert\’ible. Hence$L_{\mathrm{s}}(D)\circ$on$G^{\mathrm{s}}(p\mathrm{w})$$\rho$. Now by the same reasoning as above we can conclude the invertibil\’ity of$L(x, D)$ on
for sufficiently small $p>0$ .
$[mathring]_{\mathrm{s}}_{L}(D)^{-1}$$G^{\mathrm{s}}(\rho \mathrm{w})$The case where the -principal part \’is of constant coefficients is the same as inheorem 1. Thus the proof is completed.
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