数理解析研究所講究録983 巻 1997 年 175-187

175

GOURSAT PROBLEM FOR A
MICRODIFFERENTIAL OPERATOR OF
FUCHSIAN TYPE AND ITS APPLICATIONSUSUMU YAMAZAKI(山崎晋)

Graduate School of Mathematical Sciences, The University of Tokyo\S 0. INTRODUCTION.
The Goursat problem in the holomorphic (or the real analytic) category

is treated by several authors and studied in depth. Moreover, C. Wagschal[W] extended the problem to the case of a system of integro-differential

operators and obtained the Cauchy-Kovalevskaja type (the unique solv-

a bility) theorem. However, it seems that the study of the Goursat problem is not so satisfactory from the microlocal point of view. Therefore

in this article, we treat a microdifferential operator of Fuchsian typewith respect to several variables and consider the Goursat problem in theframework of holomorphic (or micro-) functions.
The notion of Fuchsian type (with respect to one variable) was intro-

duced by M. S. Baouendi and C. Goulaouic [Ba-G] for a partial differential
operator. This includes non characteristic type as a special case, and the

Cauchy-Kovalevskaja type theorem was proved in [Ba-G]. Seeing this result, N. S. Madi [M] generalized Fuchsian type to several variables case

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by the name of “a Goursat operator of several Fuchsian variables” andobtained the Cauchy-Kovalevskaja type theorem for the Goursat prob-

lem in the framework of holomorphic functions. Note that Y. LaurentT. Monterio Fernandes [La-MF] and Z. Szmydt and B. Ziemian [Sz-Zi]

gave different definitions of Fuchsian type with respect to several variables respectively. On the other hand, succeeding to Baouendi-Goulaouic[Ba-G], many mathematicians have obtained almost sufficient results inFuchsian type with respect to one variable. For example, H. Tahara[Ta] treated a Fuchsian system in the sense of Volevi\v{c} and proved the

Cauchy-Kovalevskaja type theorem in the complex domain. Further, as
an application he obtained the existence and uniqueness theorem on an
initial value problem for a Fuchsian hyperbolic system in the frameworkof hyperfunctions. Moreover, he proved the existence theorem on a ho-mogeneous initial value problem for a Fuchsian microhyperbolic systemof microdifferential operators in the framework of microfunctions. On theother hand, T.

$\hat{\mathrm{O}}$aku proved the existence theorem on an inhomogeneous

initial value problem for a Fuchsian hyperbolic$\mathrm{n}\mathrm{l}\mathrm{i}\mathrm{C}\Gamma \mathrm{o}\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}$tor in [O1] and the uniqueness theorem under the$\mathrm{F}$opera--mildness condition(but without the hyperbolicity assumption) in [O3] in the framework of

microfunctions (cf. [O2]).
In this article, we define a matrix of microdifferential operators of Fuch-sian type with respect to several variables as a natural generalization of

one variable case due to Tahara [Ta] or non-microlocal case due to Madi

177[M]. Moreover, we prove the Cauchy-Kovalevskaja type theorem for theGoursat problem in the space of holomorphic functions under the action of microdifferential operators due to J. M. Bony and P. Schapira[Bo-Sc]. As an application we solve the Goursat problem in the frame-

work of; we prove the existence theorem for$\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{o}-(\mathrm{o}\mathrm{r}\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r}-)\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}$sufficiently “regular” initial data under suitable assumptions.

\S 1. STATEMENT OF MAIN THEOREM.
In this article, we use the following notation:

natural numbers (not containing ) and$0$$\mathrm{N}$denotes the set of$\mathrm{N}_{0}:=\mathrm{N}\cup\{0\}$. For a subset$D$of some topological space, $[D]$ denotes the closure. For natural numbers$M,$ $N\in \mathrm{N}$, and a linear space$L$

we denote by Mat $(M\cross N;L)$ the spaceof matrices of size $N\cross N$ whose components are in$\ln$addition, if$A$

has a norm $||||$ , for$L$. Further set$P=(P^{(\mu,\nu}))_{\mu}N,\nu=1\in \mathrm{M}\mathrm{a}\mathrm{t}(M\cross$$N;A)$ we set $||P||:= \max\{||P^{(\mu,\nu})||;1\leq\mu\leq M, 0\leq\nu\leq N\}$natural numbers$z=$$d,$ $n\in \mathrm{N}$$(z_{1}, \ldots , z_{n})\in \mathbb{C}^{n}$, we use coordinates. For$\tau=(\tau_{1}, \ldots, \tau_{d})\in \mathbb{C}^{d}$. Moreover for multi-indices$\gamma=(\gamma_{1}, \ldots , \gamma_{n})$

and
and

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$\alpha=$$(\alpha_{1}, \ldots , \alpha_{d})$, we set

as usual. For vectors $R=(R_{1}, \ldots, R_{d})$ and$R’=(R_{1}’, \ldots , R_{d}’)\in \mathbb{R}^{d}$, wedefine an order relation as follows:$R’\leq R\Leftrightarrow \mathrm{d}\mathrm{e}\mathrm{f}$$R_{j}’\leq R_{j}$$R’<R\Leftrightarrow \mathrm{d}\mathrm{e}\mathrm{f}$$R’\leq R$$R’\prec R\Leftrightarrow \mathrm{d}\mathrm{e}\mathrm{f}$$R_{j}’<R_{j}$.

.
.

For a vector $r=$$(r_{1}, \ldots , r_{d})\in \mathbb{R}^{d}$$(m^{(1)}, \ldots , m^{(N)})$andfor all$j$.

, we set $[r]_{+}:=([r_{1}]_{+}, \ldots , [r_{d}]_{+})$ ,$(m_{1}^{(\nu)}, \ldots , m_{d}^{(N)})$and $k^{(\nu)}=$with $m^{(\nu)}\geq k^{(\nu)}(1\leq\nu\leq N)$ and set $m=$$k=(k^{(1)}, \ldots , k^{(N)})\in(\mathrm{N}_{0^{d}})^{N}$(generalized) functions$O(\tau^{m-k})$$j$and $R’\neq R$ ,where $[r_{j}]_{+}= \max\{r_{j}, 0\}$ . We fix $m^{(\nu)}=$$(k^{(1)}, \ldots , k^{(N)})\in \mathrm{N}_{0^{d}}$for all ,$f(z, \tau)={}^{t}(f_{1}(Z, \mathcal{T}),$$\ldots$. For any $N$ -tuple of, $f_{N}(z, \tau))$ , we mean $f=$by$\partial_{\tau}if_{\nu}j|_{\mathcal{T}=0}\mathrm{j}=0$$(1 \leq j\leq d, 1\leq\nu\leq N, 0\leq i\leq m_{j}^{(}-\nu)k(\nu)-j\mathrm{I})$.

Set $1_{d}:=(1, \ldots , 1)\in \mathrm{N}^{d}$ . For a vector $R=(R_{1}, \ldots , R_{d})\in \mathbb{R}^{d}$ with$\mathrm{O}\prec R$, we set $B(R):=\{\tau\in \mathbb{C}^{d};|\tau_{j}|<R_{j}(1\leq j\leq d)\}$ . Let$V\subset \mathbb{C}^{n}$

be

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a relatively compact open neighborhood of the origin and$h_{0}$

a positive

number. We set$U=\{(z;()\in T^{*n}\mathbb{C} ; z\in V, \zeta_{1}=1, |\zeta_{j}|<h_{0}(2\leq j\leq n)\}$.

We denote the sheaf of rings of microdifferential operators of finite
order (resp. of order at most ) by$\mathcal{E}$$\nu$

1.1 Definition. Lettrix in Mat$P(Z, \tau;\partial_{z}, \partial_{\tau})$(resp.

$\mathcal{E}(\nu)$) as usual.

$=(P^{(\mu,\nu)}(Z, \tau;\partial z’\partial\tau))_{\mu,\nu=1}^{N}$.

$(N;\tau([U\mathrm{X}B(R)];\epsilon_{\mathbb{C}}n+d))$; that is, each$P^{(\mu,\nu)}$

be a ma-

is a microdiffer-ential operator of finite order defined in some neighborhood of $[U\cross B(R)]$ .
Then, $P$ is said to be of Fuchsian type with weight $(k, m)$ (with respect$\tau$-variables) if it has the following form:,$P^{(\mu,\nu)}(Z, \tau;\partial z’\partial\tau)=\sum_{0\leq\alpha\leq m(\nu)}P^{(}\mu,\mathcal{U})(\alpha Z, \tau;\partial_{z})\partial_{\mathcal{T}}\alpha$where eachrameters$\tau$$P_{\alpha}^{(\mu,\nu)}$

is a microdifferential operator with holomorphic pa-

and satisfies the following:(1) The order ord(2) There exist$P_{\alpha}^{(\mu,\nu)}$$P_{\alpha}^{1,(\mu,\nu}$such that ord)of$P_{\alpha}^{(\mu,\nu)}$$(z, \tau;\partial z)$$P_{\alpha}^{1,(\mu,\nu)}\leq 0$

is at mostand

$P_{\alpha}^{2,(\mu,\nu}$$|m^{(\nu)}|-|\alpha|$;) $(z, \tau;\partial z)(0\leq\alpha\leq m^{(\nu)})$and$P_{\alpha}^{(\mu,\nu)}(z, \tau;\partial_{z})=\tau^{[-m+]}\alpha(\nu)k(\nu)+P^{1,()}\mu,\nu(\alpha;z, \mathcal{T}\partial z)$$+\tau^{[}\alpha-m^{(\nu})+k(\nu)+1_{d}]_{+}P_{\alpha}^{2,()}\mu,\nu(z, \mathcal{T};\partial z)$

.

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1.2 Remark. (1) The Fuchsian property above is invariant under any

coordinate change of -variables, or more generally an arbitrary quantized$z$contact transformation for$(z;\zeta)$-variables.

(2) The Fuchsian type defined in Definition 1.1 is a natural generaliza-tion of differential operators of Fuchsian type introduced by Madi [M];that is, if $P$ is a differential operator of Fuchsian type in the sense of

Definition 1.1, then $P$ is of Fuchsian type in the sense of Madi. Further if$d=N=1$ , a microdifferential operator of Fuchsian type is nothing but
of Fuchsian type defined by Tahara (see [Ta]).

Let $T^{(\nu)}=$$(T_{1}^{(\nu)}, \ldots , T_{d}^{(\nu)})(1\leq\nu\leq N)$$\overline{T}:=(T^{(}1),$$\ldots,$

be indetermminates and set$T(N))$.

If $P$ is of Fuchsian type with weight $(k, m)$ , we define the indicial poly$\mathrm{n}$omial of $P$ by$\mathcal{I}_{P}(z;\zeta;\overline{T}):=\det(\sum_{k(\nu)\leq\alpha\leq m(\nu)}\sigma_{0}(P_{\alpha}1,(\mu,\nu))(z, 0;\zeta)\mathcal{I}_{\alpha}(\tau(\nu)))m(\nu)-$where$\mathcal{I}_{\alpha_{j}}(T_{j}(\nu))$

Let

$:=$

$\mathcal{I}_{\alpha}(T^{(\nu)})=\prod_{j=1}^{d}\mathcal{I}_{\alpha}j(T_{j}^{(U)})$$A(z, \tau;\partial_{z})$

,

with

be a microdifferential operator of finite order with holo-

morphic parameters$\tau$

defined in a neighborhood of $[U\cross B(R)]$ . Let181$c\in \mathbb{C}$and set $\Sigma:=\{z\in \mathbb{C}^{n}; z_{1}=c\}$ . Let

and assume that$z\in\Omega,$ $w\in\Sigma$$w\in\Omega\cap\Sigma$

If$p\in \mathrm{N}$$\Omega$is$h_{0}-\Sigma$$\Omega\subset V$

be an open convex set-flat in the sense of Bony-Schapira; that is, if

and $h_{0}|zj-wj|\leq|z_{1}-w_{1}|(2\leq j\leq n)$ , then it follows that

. Let $f(z, \tau)$ be a holomorphic function defined on$\Omega\cross B(R)$.

, there exists a unique holomorphic function $g(z, \tau)$ on

Then, we define $(\partial_{z_{1}}-p)\Sigma f(Z, \tau):=g(Z, \tau)$ ; that is,$( \partial_{z_{1^{-p}}})\Sigma f(_{Z}, \tau):=\int_{c}^{z_{1}}\frac{(z_{1}-w_{1})^{p-}1}{(p-1)!}f(w_{1}, z’)dw_{1}$where $z’:=$$(z_{2}, \ldots , z_{n})$,. We write formally$A(z, \tau;\partial_{z})=\sum_{n\gamma_{1}}\in\mathbb{Z},$$\gamma2,\ldots,\gamma\in \mathrm{N}0A_{\gamma}(_{Z}, \tau)\partial_{z}^{\gamma}$.

Then, applying the argument as in Bony-Schapira [Bo-Sc] regarding$\tau$

asholomorphic parameters, we find that
$A_{\Sigma}f(_{Z}, \tau)$$:= \sum_{0\gamma 1,\ldots,\gamma_{n}\in}A_{\gamma}(Z, \tau)\partial zf\gamma(Z, \mathcal{T})\mathrm{N}$$+ \sum_{\mathrm{N}\gamma_{1}<0,\gamma 2,\ldots,\gamma n\in 0}A_{\gamma}(_{Z}, \mathcal{T})(\partial_{z}\gamma 1)1\Sigma\partial z’\gamma\prime f(z, \mathcal{T})$is holomorphic on $\Omega\cross B(R)$ . Let
fix a point$z_{0}\in\Omega\cap\Sigma$$s$be a parameter with $0<s<1$ . We

and set$\Omega_{s}:=\{s(z-Z_{0})+z0\in \mathbb{C}^{n}; z\in\Omega\}$

Consider the following condition:

182[A-1]. There exist a positive constant $C>0$ and a neighborhood $W$ ofsuch th at for any $(z;\zeta)\in[W]$ and

$[U]$

with

$\beta^{(\nu)}\geq m^{(\nu)}-k^{(\nu}$)$\beta=(\beta^{(1)}, \ldots , \beta^{(N)})\in(\mathrm{N}_{0^{d}})^{N}$(I $\leq\nu\leq N$ )$| \mathcal{I}_{P}(z;\zeta;\beta)|\geq c\prod_{1\nu=}^{N}(\beta^{()}\nu+1d)m^{(\nu)}$

Note that if $N=1$ , then [A-1] is a natural generalization of Madi’s

condition which is similar to the “Fuchsian ellipticity condition” due to
Szmydt-Ziemian [Sz-Zi].

1.3 Theorem. Let $P$ be a ma trix of microdifferential opera

in a neighborhood of $[U\cross B(R)].$ A$\mathrm{s}s\mathrm{u}\mathrm{m}e$

with

Let

$\Omega$be anyand$h$$h-\Sigma$$co\mathrm{n}$stants $r_{0}>0$ andsuch th at the following hold:$0\prec R^{\circ}\leq R$

Take arbitrary$r$with $0<h<h_{0}$ and $0<r<r_{0}$ resp $ec$ tively.

-flat open convex $su$ bset of $V$ with dia

dia den es the diameter. Then, there exists a constant$o\mathrm{t}$any$\overline{R}$with$0\prec\overline{R}\leq R^{\circ}$$f(z, \tau)={}^{t}(f_{1}(Z, \mathcal{T}),$on$\Omega\cross B(\overline{R})$$\ldots$, $f_{N}(z, \tau))$ and$\delta$$g(z, \tau)={}^{t}(g_{1}(Z, \mathcal{T}),$, there exists a unique holom orphic solutionof the Goursat problem$\Omega\leq r$, wheresuch th at for

it follows that for any holomorphic functions$u(z, \tau)={}^{t}(u_{1}(z, \tau),$$(G.P.)$

definedth at $P$ is of Fuchsi an type withweight $(k, m)$ . and satisfies [A-1]. Then, there exist$R^{\circ}$$\mathrm{t}ors$$\ldots,$$u_{N}(z, \mathcal{T}))$$\ldots$, $g_{N}(z, \tau))$

183

and each $u_{\nu}(z, \tau)(1\leq\nu\leq N)$ is holom orphic on$\bigcup_{0<s<1}(\Omega_{s}\cross\{\tau\in B(\overline{R});\prod_{=j1}^{d}|\tau_{j}|<\delta(\mathrm{I}-s)^{1}m|\})$

where

,

).

$|m|:= \sum_{\nu=1}^{N}\sum_{=j1}dm_{j}^{(\nu}$

We can prove Theorem 1.3 by applying techniques of [O1] and [W].

1.4 Remark. Assume that $P$ is a differential operator. Then Theorem
1.3 is (essentially) obtained by Madi [M] (cf. [La-MF]).

\S 2. APPLICATIONS.
Let $M$ beand$\pi_{M}$$X:=\mathbb{C}_{z}^{n}\cross \mathbb{C}_{\tau}^{d}=\mathrm{Y}\cross \mathbb{C}^{d}$the canonical projection $T_{M}^{*}Xarrow M$ . Set $N:=\mathbb{R}^{n}=-M\cap\{t=$$0\}arrow\succ M,$

andwith its complexification$\mathbb{R}_{x}^{n}\cross \mathbb{R}_{t}^{d}$$L:=X\cap\{1\mathrm{m}Z=0\}=\mathbb{R}^{n}\cross \mathbb{C}^{d},\overline{\Lambda}:=\tau_{L^{*}}x=-T_{N}*\mathrm{Y}\mathrm{X}\mathbb{C}^{d}$. We denote the sheaf of nicrofunctions on$\Lambda:=T_{M}^{*}X\cap\overline{\Lambda}$(resp.

$T_{N}^{*}\mathrm{Y}$) by$\mathrm{C}_{M}$(resp.

$\mathrm{C}_{N}$) as usual. Further, letof microfunctions with holomorphic parameters on$\mathrm{C}\mathrm{t}9_{L}:=\mu L((9\mathrm{x})\otimes orN/\mathrm{Y}[n]$where$\mu_{L}$$\mathrm{C}\mathrm{t}9_{L}$$T_{M}^{*}X$

be the sheaf, that is,$\overline{\Lambda}\cdot$,

denotes Sato’s microlocalization functor along$L$and $or_{N/Y}$

denotes the relative orientation sheaf (see [K-Sc] and [S-K-K]). The sheaf$\mathfrak{B}_{M}$of hyperfunctions on $M$ and the sheaf$\mathfrak{B}\mathrm{t}9_{L}$of hyperfunctions with

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holomorphic parameters on

ce$L|_{L}$

respectively. Let$\rho$$L$are defined by$\mathfrak{B}_{M}:=\mathrm{C}_{M}|_{M}$

and

$:=$

$\mathfrak{B}\mathrm{t}9_{L}$be a natural mapping$N\cross T_{M}^{*}X\ni M(x, \mathrm{o};\sqrt{-1}(\langle\xi, dX\rangle+\langle\eta, dt\rangle))\mapsto(x;\sqrt{-1}\langle\xi, d_{X}\rangle)\in T_{N}^{*}$Y.

Then, we have the following canonical morphisms:

CO$L|_{\Lambda}\mapsto \mathrm{C}_{M}|_{\Lambda}$,$\rho_{!}(\mathrm{C}M|N\cross T_{M}^{*}X)Marrow \mathrm{C}_{N}$

Set $p_{0}:=(0;\sqrt{-1}d_{X_{1}})\in T_{N}^{*}\mathrm{Y}$ and assume that.

$P(x, t;\partial_{x}, \partial_{t})$is amatrix of microdifferential operators of Fuchsian type with weight $(k, m)$defined in some neighborhood of$\rho^{-1}(p\mathrm{o})$, then the following morphism

is induced:$P:\rho_{!}(\mathrm{e}_{M}|_{N\cross,M}\tau_{M}*X)_{p0}arrow\rho_{!}(\mathrm{C}M|N\cross MT_{M}^{*)}Xp0’$

where$\rho!(\mathrm{C}_{M}|_{N\cross T}*x)Mp_{0}$denotes the stalk at$p_{0}$.

Consider the following condition:$\tau;\sim$[A-2]. $\det(\sigma|m(\nu)|(P(\mu,\nu))(Z, \tau; (, \eta))=\tau^{k}\overline{P}(Z,$$( \overline{k}:=\sum_{\nu=1}^{N}k^{(}\nu)\in \mathrm{N}_{0^{d}})$$\overline{P}(z, t;\zeta, \eta)$

for a function$\overline{P}$which satisfies the following condition:

There exist positive constants

such that$(, \eta)$$h_{0},$$M$and$\nu_{i}$with $\nu_{i}\geq 1(1\leq i\leq d)$

never vanishes on the set$\{(Z, t;\zeta, \eta)\in \mathbb{C}^{n}\cross \mathbb{R}^{d}\cross \mathbb{C}^{n}\mathrm{x}\mathbb{C}^{d};|z|,$$|\zeta_{j}|<h_{0}|\zeta_{1}|(2\leq j\leq n),$for$|t|<h_{0}$,$|{\rm Im}(\eta_{i}/\zeta_{1})|=\nu_{i}\lambda(1\leq i\leq d)$$\exists\lambda>M(_{j=}\sum_{1}^{n}|1\mathrm{m}zj|+\sum_{2j=}^{n}|1\mathrm{m}(\zeta_{j}/\zeta 1)|)\}$

.

185

2.1 Remark. Condition [A-2] is satisfied if[A-3].

$\overline{P}(x, t;\xi, \eta)=\prod_{j=1}^{d}P_{j}(x, t;\xi, \eta_{j})$and eachand hyperbolic with respect to the direction$\mathrm{K}])$$t_{j}$$P_{j}$

is of degree$\sum_{i=1}^{N}m^{(}ji$)(cf. Kashiwara-Kawai [K-.

2.2 Theorem.

$\mathrm{A}ss\mathrm{u}me$that $P$ satisfies [A-1] and [A-2]. Then, for any

microfun ctions with A olomorphic parameters$f(x, t),$$g(X, t)\in\rho*(\mathrm{c}\mathcal{O}_{L}|N\mathrm{x}MT_{M}^{*}x)^{\oplus}p0N$,

there exists a microfunction$u(x, t)\in\rho_{!}(\mathrm{c}_{M}|_{N}M\cross T_{M}^{*x})^{\oplus}p0N$

such that

$u$

is a

$sol\mathrm{u}$tion of the Goursat problem$(G.P.)$

Outline of Proof of Theorem 2.2 is as follows: First, choosing suitabledefining functions, we can solve $(G.P.)$ in a complex open set by using
Theorem 1.3. Next, we can apply the holomorphic continuation method
due to Kashiwara-Kawai [K-K] by assumption [A-1].

2.3 Remark. The author does not know how to prove the uniquenessof $u(x, t)$ in Theorem 2.2 (cf. [O2] and [O3]).

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2.4 Corollary. Let $P$ be a matrix of an analytic differential opera$\mathrm{t}ors$of Fuchsian type defined on a neighb orhood of $(x, t)=(0,0)$ . Assum$e$th at [A-1] and [A-3]. Then, for any holomorphic hyperfun ctions withholomorphic parameters$f(x, t),$there exis$\mathrm{t}s$$g(x, t)\in(\mathfrak{B}\{9L|M)_{0}^{\oplus N}$,a hyperfunction$u(x, t)\in(\mathfrak{B}_{M})_{0}\oplus N$

such that

$u$

has

$t$

as real analytic parameters an is a solu tion of the$\mathrm{d}$Goursat problem$(G.P.)$REFERENCES[Ba-G]

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