数理解析研究所講究録983 巻 1997 年 156-166

156

On Boundary Value Problems for Micro-hyperbolic
Systems of Differential Equations

MOTOO UCHIDA
Osaka University, Graduate School of
Science, Department of Mathematics
内田素夫 (大阪大学大学院理学研究科)

In [KK], Kashiwara and Kawai formulate boundary value problems for
elliptic systems of differential equations from a microlocal point of view, where
they describe the obstruction of extension beyond the boundary in terms of asystem of micro-differential equations induced on the boundary. In this shortpaper, we prove the same formula as established in [KK] for (semi-)microhyperbolic systems of differential equations. This enables us to understandboundary value problems for elliptic systems and for semi-hyperbolic systems
in a unified manner.
The results proved in this paper1 are more or less known to specialists, but
are not found in the literature.

Notations. In this paper, we freely use the notations of [KS1] for sheaves
and functors. For a complex manifold $X,$ $T^{*}X$ denotes the cotangent bundleof X.
denotes the sheaf of holomorphic functions on $X,$
the sheaf ofrings of differential operators, andthe sheaf of rings of microdifferential$M$
operators. If
is a closed real submanifold of $x,$
denotes the conormal
bundle of $M,$:
the projection to the base space. We denote$H$by
the Hamiltonian map $T^{*}T^{*}Xarrow TT^{*}X$ . If $M$ is a real submanifold of$X,$ $H$ induces an isomorphism $\tau^{*}\tau_{M}^{*}Xarrow T_{T_{M}X}*T*X$ , which is also denoted
simply by $H$ .
$\mathcal{O}_{X}$$D_{X}$$\mathcal{E}_{X}$$\tau_{M}^{*}X$$\pi_{M}$$\tau_{j1}^{*}X/Iarrow M$

1Its original version is in Research Reports in Mathematics 96-04, Osaka University(March 1996). The contents of this paper are not related to the author’s seminar talkat RIMS; the author would like to thank the editor of this volume who has given theopportunity of reproducing here the preprint.

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1. Main Theorems
Let $M$ be a real analytic manifold of dimension $n\geq 1,$ $N$ a submanifold
of $M$ of codimension 1 defined by equation $f=0$ for a real-valued analytic
functionwith $df|_{N}\neq 0$ . Let
denote the closed subset $\{f\geq 0\}$ of$M$ ; thenis a real analytic submanifold of $M$ with boundary. We set$N^{+}=\{k\cdot df(x)|x\in N, k>0\}$ ; then $N^{+}\subset T_{N}^{*}M$ . Let $X$ be a complex
neighborhood of $M,$ $Y$ a closed complex submanifold of $X$ of codimension 1such that $M\cap Y=N$ . Denote by the closed embedding $Yarrow X$ .
Let
be a coherent-module.
denotes the characteristic variety. We assume the following conditions :
of
(A.1)
.
: $Yarrow X$ is non characteristic for,(A.2) At any point of$f$$Z_{+}$$Z_{+}$$\varphi$$D_{X}$$\Lambda 4$$\mathrm{C}\mathrm{h}(\mathcal{M})$$\mathcal{M}$$\mathcal{M}$$\varphi$$(T_{M}^{*}X\cap T_{N}^{*}X\backslash N)\cap \mathrm{C}\mathrm{h}(\mathcal{M})$

$p$

,

(1.1)

$-H(\pi^{*}df)\not\in C_{p}(\mathrm{C}\mathrm{h}(\mathcal{M}), Z_{+}\cross_{M}\tau_{M}^{*}X)/\tau_{p}\tau_{M}^{*}x$

where

$\pi$

: $T_{M}^{*}Xarrow M$ and$\pi^{*}$:$T_{\pi(p)}^{*}Marrow\tau_{p}^{*}\tau_{M}^{*}X$.

denotes the norIn the right-hand side of (1.1),, and), which is a closed cone inmal cone at (cf.
for short.
the image of the normal cone inLetbe an open subset ofdefined by $(\tau_{N}^{*}x)^{+}=q^{-1}(N^{+})$ ,. Let:with being the canonical projection
the induced map of
the projection induced fromon $N$ .
the, with : $T^{*}Xarrow X$ . Denoting by
Let
induced-module ofon , we have :$c_{p}(\mathrm{c}\mathrm{h}(\mathcal{M}), Z_{+}\cross_{M}\tau_{M}^{*}x)$$[\mathrm{K}\mathrm{S}1,$$p$$\tau_{p}\tau^{*}x$$\mathrm{D}\mathrm{e}\mathrm{f}.4.1.1]$$C_{p}(\cdot, \cdot)/T_{pM}\tau^{*}X$$(\tau_{T_{M}X}*\tau*x)_{p}$$(\tau_{N}^{*}x)^{+}$$T_{N}^{*}X$$T_{N}^{*}Xarrow T_{N}^{*}M$$q$$T^{*}\mathrm{Y}$$\varphi,$$T^{*}X\cross_{X}\mathrm{Y}arrow$$\rho:T_{N}^{*x}arrow T_{N}^{*}\mathrm{Y}$$\overline{\mathcal{M}}=\mathcal{E}_{X}\otimes_{\pi^{-1}D_{\underline{X}}}\pi^{-1}\mathcal{M}$$\mathcal{M}$$\mathcal{E}_{Y}$${}^{t}\varphi’$${}^{t}\varphi’$$\varphi^{*}\overline{\mathcal{M}}$$\pi$$\mathrm{Y}$Lemma 1.1. If we assume (A.1) and (A.2), there exists a coherent$uleN^{+}$-homomorphism
and andefined on
that$T_{N}^{*}\mathrm{Y}\backslash N$$(1.2)$$N^{+}arrow\varphi^{*}\overline{\mathcal{M}}$$\mathcal{E}_{Y}$$\oplus$$N_{q}^{+}\cong$$\mathcal{E}_{Y^{-mo}}d-$such$(\mathcal{E}_{Yarrow X}\otimes_{\mathcal{E}_{X}}\overline{\mathcal{M}})_{p}$$p\in(\tau_{N}^{*}X)+\cap \mathrm{c}\mathrm{h}(\mathcal{M})\cap\rho^{-}1(q)$for any$q\in T_{N}^{*}Y\backslash N$.

be the sheaf of hyperfunctions on $M,$
the sheaf of microfunctions$N$(cf. [SKK]). Letonbe the relative orientation sheaf of $N$ in $M$ as
C-module.

Let$B_{M}$$C_{N}$$\mathrm{o}\mathrm{r}_{N|M}$158

Theorem 1.2. Assume (A.1) and (A.2). There is an isomorphism
(1.3)

where

,

$\mathrm{R}\Gamma z_{+^{\mathrm{R}}}\mathcal{H}om_{D}X(\mathcal{M}, B_{M})|N\otimes \mathrm{o}\mathrm{r}_{N|M}[1]\cong \mathrm{R}\dot{\pi}_{N*}\mathrm{R}\mathcal{H}om_{\mathcal{E}}Y(N^{+}, c_{N})$$\dot{\pi}_{N}$:$T_{N}^{*}Y\backslash Narrow N$.

Remark 1. Theorem 1.2 is first proved for elliptic-modules by Kashiwara
and Kawai [KK]. Note that (A. 1) and (A.2) are automatically satisfied ifiselliptic. Let
be a system of local coordinates of $M,$ $Z_{+}=\{x_{1}\geq$. A classical example of non-elliptic differential operators which satisfy$k\geq 2$ , wherecondition (A.2) is $D_{1}^{2}-X^{k}A1(x, D’)$ , withand $A(x, D’)$ is a differential operator of order 2 such that $[x_{1}, A]=0$ andits principal symbol
is negative valued on
being the zero section of(i.e. $\sigma(A)(x,$ $i\eta’)<0$ if).
$D_{X}$$\mathcal{M}$$(x_{1}, \ldots , x_{n})$$0\}$$D_{1}=\partial/\partial x_{1}$$k\in \mathrm{Z},$$\sigma(A)$$T_{M}^{*}X\cap T_{N}^{*}X\backslash \rho^{-1}(\mathrm{o}_{N}),$$0_{N}$$\eta’\neq 0$$T_{N}^{*}Y$

Remark 2. Condition (1.1) is an analogue of micro-hyperbolicity [KS2] andnaturally appears in microlocal study of boundary value problems (cf. [S2,. It is well known that, if we assume$\mathrm{S}\mathrm{Z}])$$+H(\pi^{*}df)\not\in C_{p}(\mathrm{C}\mathrm{h}(\mathcal{M}), Z_{+}\cross_{M}\tau_{M}^{*}X)/\tau_{p}\tau_{M}^{*}x$

at $p\in T_{M}^{*}X\cap T_{N}^{*x}$ , this entails propagation of regularity up to the boundary
point from the positive side of $N$ (see [Kt2, Sl, S2,).
$p$$\mathrm{S}\mathrm{Z}]$Let
be the sheaf of real analytic functions on $M$ . In place of (A.1) and(A.2), consider the following slightly stronger assumption. (is the sameas (A.1).)(B. 1): $Yarrow X$ is non characteristic for.
(B.2)
is micro-hyperbolic forat all:For,$A_{M}$$(\mathrm{B}.1)$$\mathcal{M}$$\varphi$$\mathcal{M}$$p\in T_{M}^{*}x\cap\tau*x\backslash NN[\mathrm{K}\mathrm{S}2, \mathrm{D}\mathrm{e}\mathrm{f}.2.1.2]$$\varphi$$\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{h}\pm$$\pm H(\pi^{*}df)\not\in C_{p}(\mathrm{c}\mathrm{h}(\mathcal{M}), \tau_{M}*X)/\tau_{p}\tau_{M}^{*}X$.

Theorem 1.3. Assume (B.1) and (B.2). There is an isomorphism(1.4)$\mathrm{R}\Gamma_{z}\mathrm{R}\mathcal{H}om_{D}(+xM)\mathcal{M},$$A|_{N}\otimes \mathrm{o}\mathrm{r}N|M[1]\cong \mathrm{R}\dot{\pi}_{N*}\mathrm{R}\mathcal{H}om_{\mathcal{E}_{Y}}(N+, c_{N})$as well as isomorphism (1.3), where
given in Lemma 1.1.

$N^{+}$

is the coherent$\mathcal{E}_{Y}$-module on$T_{N}^{*}Y\backslash N$

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2. Proof of Theorem 1.2 and 1.3

the
As in [KK], the proof of Theorem 1.2 is divided into two steps.
first step, we relate the left-hand side of (1.3) to a differential complex with
. In the second step, proving Lemma 1.1,
coefficients in $C_{N|X}$ induced fromwe complete the proof of Theorem 1.2.
due to Kataoka [Ktl] and-moduleLet us recall the notion of the
Schapira [S2]. Following [S2], let$\ln$$\mathcal{M}$$c_{z_{+}|X}$$\mathcal{E}_{X}$$C_{Z_{+}|}x=\mu \mathrm{h}_{\mathrm{o}\mathrm{m}}(\mathrm{C}_{Z}+’ \mathcal{O}_{X})\otimes \mathrm{o}\mathrm{r}M|x[n]$.

Then all the cohomology groups $H^{k}(C_{z_{+}|X}),$ $k\neq 0$ , are zero and $H^{0}(C_{z_{+}|X})$with its zero-th cohomology $H^{0}(C_{z_{+}|X})$ .
is an-module. We identify
For the-module $C_{N|X}$ , refer to [KK], [KS2] and also [Sl, S2]. (In this] :paper, we follow the definition of [KK,We prepare two lemmas.
$c_{z_{+}|X}$$\mathcal{E}_{X}$$\mathcal{E}_{X}$$\mathrm{K}\mathrm{S}2$

Lemma 2.1.
(1)
.
.
(2)
(3) There is an-homomorphism.
an isomorphism on$C_{N|X}=H^{n}\mu_{N}(\mathcal{O}_{X})\otimes \mathrm{o}\mathrm{r}_{N|X}.)$$\mathrm{R}\pi_{*}C_{Z|X}|_{M}+\mathrm{R}\cong\Gamma Z_{+}BM$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(C_{Z_{+}}|x)\cap T_{N}^{*x}\subset(T_{N}^{*}X)^{+}$$\mathcal{E}_{X}$$C_{N|X}\otimes \mathrm{o}\mathrm{r}_{N|M}arrow c_{z_{+}|X\mathrm{z}}$and this is$(\tau_{N}^{*}x)^{+}$For the proof, see [Kt3, Sect.4] and [S2, S3].
Lemma 2.2. If we assume (1.1) at a point

$p$

of

$T_{M}^{*}X\cap T_{N}^{*}X$, we have$\mathrm{R}\mathcal{H}om_{\mathcal{E}x}(\overline{\mathcal{M}}, Cz_{+}|x)|T_{N}^{*}x=0$

in a neighborhood of .
$p$

Proof. (Cf. the proof of Corollary 3.3 of [SZ].) Let

function defined onsuch that $g|_{M}=f$ . We set
X. From (1.1), we have
$X$

$g$

be a real-valued smooth, with : $T^{*}Xarrow$$h=g\circ\pi$$-H(dh)\not\in C_{p}(\mathrm{C}\mathrm{h}(\mathcal{M}), Z_{+}\cross_{M}\tau_{M}^{*}X)$

Hence we can find an open subset$U$

of $T^{*}X$ so that

$\pi$

.

$U\cap \mathrm{C}\mathrm{h}(\mathcal{M})=\otimes$$-H(dh)\not\in C_{p}(\tau^{*}X\backslash U, Z_{+}\mathrm{X}_{M}\tau_{M}^{*}X)$

,

,

160

$\mathrm{a}\mathrm{n}\mathrm{d}-H(dh)\not\in C_{p}(T^{*}X\backslash U, U)$of the sheaf
on $X$ (cf.
on a neighborhood of , we$\mathrm{C}_{Z_{+}}$

$p$

. Let

$[\mathrm{K}\mathrm{S}1,$$\tau_{z_{+}}^{*}x$$\mathrm{S}\mathrm{e}\mathrm{c}\mathrm{t}.5.1]$

denote the micro-support). Since. This yields$\mathrm{S}\mathrm{S}(\mathrm{C}z_{+})$$\tau_{z_{+}}^{*}x\subset Z_{+}\cross_{M}T_{M}^{*x}\cup U$$\mathrm{h}\mathrm{a}\mathrm{v}\mathrm{e}-H(dh)\not\in C_{p}(T^{*}X\backslash U, T_{z_{+}}^{*}X)$$-H(dh)\not\in C_{p}(\mathrm{C}\mathrm{h}(\mathcal{M}), \tau_{z}*X)+\cdot$

Since$\mathrm{s}\mathrm{s}(\mathrm{R}\mathcal{H}_{om_{Dx}}(\Lambda 4, C_{Z}+|x))\subset C(\mathrm{C}\mathrm{h}(\mathcal{M}), \tau_{z_{+}}^{*}x)$,

it follows from the definition of micro-supports that$\mathrm{R}\Gamma_{\{h\geq 0\}}\mathrm{R}\mathcal{H}om_{D_{X}}(\mathcal{M}, CZ+|X)|\{h=0\}=0$

in a neighborhood of . Since, we have$p$$C_{Z_{+}|}x$

is supported on$\tau_{z_{+}}^{*}x$

and$\tau_{z_{+}}^{*}x\subset\{h\geq$

$0\}$

.

$\mathrm{R}\mathcal{H}om_{D_{X}}(\mathcal{M}, CZ+|x)|_{\{}h=0\}\cong \mathrm{R}\Gamma_{\{h\geq 0\}}\mathrm{R}\mathcal{H}om_{D}\mathrm{x}(\mathcal{M}, cz+|x)|\{h=0\}\cong 0$

Q.E.D.

Since$\mathrm{c}_{z_{+}}$

is cohomologically constructible, if we set$F=\mathrm{R}\mathcal{H}om_{D_{X}}(\mathcal{M}, \mathcal{O}_{X})$it follows from [$\mathrm{K}\mathrm{S}1$,, Prop.4.4.2] that$\mathrm{R}\pi_{**}\mathrm{R}\Gamma_{\tau_{\mathrm{x}}}x\mathrm{R}\mathcal{H}om_{D}x(\Lambda 4, c_{z_{+}|X})|N\cong \mathrm{R}\pi*\mathrm{R}\Gamma*\tau \mathrm{x}^{\mu}x\mathrm{h}\mathrm{o}\mathrm{m}(\mathrm{c}_{z}+’ F)|_{N}[n]$$\cong \mathrm{R}\mathcal{H}om\mathrm{c}(\mathrm{c}_{z}+’ \mathrm{C}_{X})\otimes F|_{N}[n]$$\cong F\otimes \mathrm{C}z_{+\backslash }N|_{N}$$\cong 0$.

Hence, from Lemma 2.1, we have$\mathrm{R}\mathrm{r}_{z_{+^{\mathrm{R}\mathcal{H}m_{D_{X}}}}(}o\mathcal{M},$$B_{M})|_{N}\cong \mathrm{R}\pi*\mathrm{R}\mathcal{H}om_{D}\mathrm{x}(\mathcal{M}, C_{Z_{+}|}\mathrm{x})|_{N}$$arrow \mathrm{R}\pi_{*\tau X\backslash +}\sim \mathrm{R}\mathrm{r}*X\mathrm{R}\mathcal{H}omDX(\mathcal{M}, C_{z}|\mathrm{x})|_{N}$$\cong \mathrm{R}\pi_{*}’(\mathrm{R}\mathcal{H}om_{\mathcal{E}}(XC_{Z|x}\overline{\mathcal{M}},)+|_{T_{N}X}*\backslash N)$

,

161

where
that

$\pi’$:$T_{N}^{*}X\backslash Narrow N$. It then follows from Lemma 2.1(2), (3) and 2.2$\mathrm{R}\mathcal{H}om_{\mathcal{E}_{X}}(\overline{\mathcal{M}}, C_{Z|}\mathrm{x})+|_{T}*_{X}\backslash N\cong \mathrm{R}N(\tau_{N}*\mathrm{r}(X)+\mathrm{R}\mathcal{H}om_{\mathcal{E}}(\overline{\mathcal{M}}, cz_{+}|X)|\tau^{*}X\backslash \mathrm{x}N)N$$\cong \mathrm{R}\Gamma_{(T_{N}^{*_{X)}}}\mathrm{R}+\mathcal{H}om_{\mathcal{E}\mathrm{x}}(\overline{\mathcal{M}}, CN|X)\otimes \mathrm{o}\mathrm{r}_{N}|M$.

Thus we have
(2.0)$\mathrm{R}\Gamma z_{+}\mathrm{R}\mathcal{H}om_{D}X(\mathcal{M}, B_{M})|_{N}\otimes \mathrm{o}\mathrm{r}_{N|M}$$\cong \mathrm{R}\pi_{*\mathrm{x}}^{\prime_{\mathrm{R}\Gamma_{()}}}\tau_{N}^{*}+X\mathrm{R}\mathcal{H}om_{\mathcal{E}}(\overline{\mathcal{M}}, CN|X)$

Since$T_{Y}^{*}X\cap \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(\overline{\mathcal{M}})\subset T_{X}^{*}X$.

, we havethe right-hand side of (2.0)$\cong \mathrm{R}\dot{\pi}_{N*}[\mathrm{R}\rho_{*}\mathrm{R}\mathrm{r}_{()}*+^{\mathrm{R}}\mathcal{E}X\tau NX\mathcal{H}om(\overline{\mathcal{M}}, C_{N}|\mathrm{x})|\tau_{N}^{*}Y\backslash N]$$=\mathrm{R}\dot{\pi}_{N*}[\mathrm{R}\rho_{*}^{+}(\mathrm{R}\mathcal{H}om_{\mathcal{E}_{X}}(\overline{\mathcal{M}}, CN|X)|_{()}\tau^{*_{X}}+)N|_{T_{N}^{*}Y\backslash }N]$

where we denote bysummary, we have2(2.1)$\rho^{+}$:$(\tau_{N}^{*}x)^{+}arrow T_{N}^{*}Y$,

the restriction of . Hence, in$\rho$$\mathrm{R}\Gamma z_{+^{\mathrm{R}}}\mathcal{H}om_{D}X(\mathcal{M}, B_{M})|_{N}\otimes \mathrm{o}\mathrm{r}_{N|M}$$\cong \mathrm{R}\dot{\pi}_{N*}\mathrm{R}\rho+*(\mathrm{R}\mathcal{H}om\mathcal{E}\mathrm{x}(\overline{\mathcal{M}}, cN|X)|_{()}T_{N}+)*_{X}\cdot$

In the rest of this section, we prove(2.2)$\mathrm{R}\rho_{*}^{+}(\mathrm{R}\mathcal{H}om_{\mathcal{E}x}(\overline{\mathcal{M}}, C_{N|X})|(T^{*}x)N+)[1]\cong \mathrm{R}\mathcal{H}om_{\mathcal{E}_{Y}}(N^{+}, C_{N})$. Combining (2.1) and (2.2), we get isomorphism (1.3).
We prepare two lemmas for the second part of the proof. Lemma 1.1 follows
from the following Lemma 2.3 with $I=T_{N}^{*}Y\backslash N$ .

on

$T_{N}^{*}\mathrm{Y}\backslash N$

Takeuchi also proves (2.1) in the case where (B.1) and (B.2) are fulfilled; see K.
Takeuchi : Edge of the wedge type theorems for hyperfunction solutions, preprint (Jan.
is non characteristic for $F$ on $N^{+}(\subset T^{*}M)$ , and we1996). If we assume (B.2),
immediately obtain (2.0) by applying Theorem 6.7.1 of [KS1] (see also Corollary 6.7.3).
2$M\mathrm{c}_{-\succ}X$

162

Lemma 2.3. Let I be a conic open subset of. Let
be a coherent-module on a conic neighborhood of, with :Assumethe following.
(a.1): $Yarrow X$ is non characteristic foron a neighborhood of I in thesense of [$SKK,$ $II$, Def.3.5.4].
(a.2) For a conic neighborhoodof $\rho^{-1}(I)\cap T_{M}^{*}X$ ,$T_{N}^{*}Y\backslash N$$\rho^{-1}(I)$$\mathcal{E}_{X}$$\mathcal{M}$$T_{N}^{*}Xarrow T_{N}^{*}Y.$$\rho$$\mathcal{M}$$\varphi$$U$

Then (1)$\rho$

is finite on$U\cap(T^{*}X)N\mathrm{n}\mathrm{s}_{\mathrm{u}}\mathrm{p}+(\mathrm{P}\mathcal{M})=\emptyset$.

$\rho^{-1}(I)\cap(\tau_{N}^{*}x)^{+}\cap \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(\mathcal{M})$. (2) If we set$N^{+}=\rho_{*}((\mathcal{E}_{Y}arrow X^{\otimes_{\mathcal{E}x}}\mathcal{M})\otimes \mathrm{C}_{()}*_{X}+)\tau_{N}$$N^{+}$

is a coherent$\mathcal{E}_{Y}|_{I}$’-module.

(We omit the proof. Cf. [SKK, II, Thm.3.5.3].)

Lemma 2.4. Let
tative diagram on$\mathcal{M},$$N^{+}$

be as in Lemma 2.3. Then there exists a commu-$I$$\mathrm{R}\mathcal{H}om_{\mathcal{E}_{Y}}(N^{+}, \mathcal{E}_{Y})$$rightarrow\sim \mathrm{R}\rho_{*}^{+}(\mathrm{R}\mathcal{H}om\mathcal{E}x(\mathcal{M}, \mathcal{E}_{Xarrow}Y)|_{(}T_{N}^{*_{X)}}+)[1]$$\uparrow$$\uparrow$$\mathrm{R}\mathcal{H}om_{\mathcal{E}_{Y}}(\varphi \mathcal{M}, \mathcal{E}_{Y}*)arrow\sim$$\mathrm{R}\rho_{*}(\mathrm{R}\mathcal{H}om\epsilon \mathrm{x}(\mathcal{M}, \mathcal{E}_{X}arrow Y)|_{\tau_{N}}*_{X})[1]$and every horizontal arrow is an isomorphism, where

Proof. This follows from the definition of

Q.E.D.

Since
is coherent overby Lemma 2.4, we have$N^{+}$$\mathcal{E}_{Y}|_{T_{N}^{*_{Y}}}$and$\rho^{+}$$N^{+}$$\rho^{+}=\rho|(\tau_{N}^{*}x)^{+}$and [SKK,

is finite on$1\mathrm{I}$.

, Thm.3.5.6].

$\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(\overline{\mathcal{M}})\cap(\tau_{N}^{*}x)^{+}$,$\mathrm{R}\mathcal{H}om\mathcal{E}_{Y}(N^{+}, cN)\cong \mathrm{R}\mathcal{H}om\epsilon Y(N+, \mathcal{E}_{Y})\otimes_{\mathcal{E}_{Y}}^{L}CN$$\cong\rho_{*}^{+}[\mathrm{R}\mathcal{H}om_{\mathcal{E}\mathrm{x}}(\overline{\mathcal{M}}, \mathcal{E}_{xarrow Y})|_{(\tau^{*_{X)}}}+]\otimes^{L}\mathcal{E}_{Y}CN[1N]$$\cong\rho_{*}^{+}[\mathrm{R}\mathcal{H}om\epsilon x(\overline{\mathcal{M}}, \mathcal{E}_{xarrow Y})|_{(}\tau_{N}^{*_{X)}}+\otimes^{L1}\rho-1\mathcal{E}Y\rho c-]N[1]$Using the$\mathcal{E}_{X}$-homomorphism$\mathcal{E}_{Xarrow Y}\otimes_{\rho}-1\epsilon_{Y}\rho^{-}C_{N}1arrow C_{N|X}[\mathrm{K}\mathrm{K}, \mathrm{I}\mathrm{I}]$$\mathrm{R}\mathcal{H}om_{\mathcal{E}_{Y}}(N^{+}, c_{N})$(2.3)$arrow\rho_{*}^{+}[\mathrm{R}\mathcal{H}om_{\mathcal{E}\mathrm{x}}(\overline{\mathcal{M}}, \mathcal{E}xarrow Y\otimes^{L}\mathcal{E}_{Y}\rho\rho^{-1})|(T-1C_{N}*_{X)}+]N[1]$$arrow\rho_{*}^{+}[\mathrm{R}\mathcal{H}om\mathcal{E}_{X}(\overline{\Lambda 4}, CN|X)|_{(}T_{N}*+]X)[1]$.

.

, we have

163

Let

$q\in T_{N}^{*}Y\backslash N$. For$k\in \mathrm{Z}$, looking at the stalk on , we have from (2.3)$q$$\mathcal{E}xt_{\mathcal{E}^{+}x}^{k1}(\overline{\mathcal{M}}p’(c_{N|X})_{p})$$\oplus$$\mathcal{E}xt_{\epsilon}^{k}Y(N_{q}^{+}, C_{Nq})arrow$.

$P\in(T_{N}^{*_{X)\cap}}+\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{P}(\overline{\lambda 4})\cap\rho^{-1}(q)$, Prop.3;-moduleIt follows from the division theorem for thethat this is an isomorphism for any
and the definition of. This completes the; therefore (2.3) is an isomorphism inproof of Theorem 1.2.
$C_{N|X}[\mathrm{K}\mathrm{K},$$\mathcal{E}_{X}$$1\mathrm{I}$$N^{+}$$\mathrm{K}\mathrm{S}2,6.3.1]$$k\in \mathrm{Z}$$\mathrm{D}^{\mathrm{b}}(T_{N}^{*}Y\backslash N)$

Proof of Theorem 1.3. If$\varphi$is micro-hyperbolic for$\overline{\mathcal{M}}$at $p\in T_{M}^{*}X\cross_{M}N$ , wehave [KS2].

$\mathrm{R}\Gamma_{\pi_{M}^{-1}(Z+})\mathrm{R}\mathcal{H}om_{\mathcal{E}}(XC_{M}\overline{\mathcal{M}},)_{p}--0$

Since this holds at all
an isomorphism$p\in(T_{M}^{*}.

X\backslash M)\cross_{M}N$by assumption (B.2), we have$\mathrm{R}\Gamma_{Z_{+}}\mathrm{R}\mathcal{H}om_{D}x(\Lambda t, A_{M})|_{N}arrow \mathrm{R}\Gamma z_{+}\mathrm{R}\mathcal{H}\mathit{0}\sim m_{D}\mathrm{x}(\mathcal{M}, B_{M})|_{N}$Combining this and (1.3), we get (1.4).

.
$\mathrm{Q}.\mathrm{E}$.D.

3. Application
Let $M_{+}=Z_{+}\backslash N$ . Isomorphism (1.3) gives a description of the structure
in terms of a system of micro-differentialof the sheaf
equations on the boundary.
$\mathcal{E}xt_{Dx}^{k}(\mathcal{M}, \Gamma_{M}B_{M})+|_{N}$Theorem 3.1. Let

Assume moreover(3.1)$\mathcal{M}$be a coherent-module. Assume (A.1) and (A.2).
for all $k>0$ . Then

$D_{X}$$\mathcal{E}xt_{Dx}^{k}(\mathcal{M}, A_{M})=0$$\mathcal{E}xt_{D}^{0}X(\mathcal{M}, \Gamma_{M_{+}}\beta_{M})|_{N}\cong \mathrm{K}\mathrm{e}\mathrm{r}(\mathcal{H}om_{D_{Y}}(\varphi^{*}\mathcal{M}, B_{N})$$arrow\dot{\pi}_{N*}\mathcal{H}onx\epsilon_{Y}(N+, cN))$

where$\varphi^{*}\mathcal{M}=D_{Yarrow X}\otimes_{\varphi^{-1}Dx}\varphi^{-1}\mathcal{M}$and$N^{+}$

is the coherent$\mathcal{E}_{Y}$-module on$T_{N}^{*}Y\backslash N$

given in Lemma 1.1, and(3.2)$\mathcal{E}xt_{Dx}^{k}(\mathcal{M}, \Gamma_{M}B_{M})+|_{N}\cong H^{k}\mathrm{R}\dot{\pi}N*\mathrm{R}\mathcal{H}om\epsilon Y(\varphi^{*}\overline{\mathcal{M}}/N^{+}, CN)$.
for
Proof. Let us first recall that, if,$k\neq 0$

module$\mathcal{M}$$\varphi$: $Yarrow X$ is non characteristic for a, we have a canonical isomorphism$\mathrm{R}\Gamma_{N}\mathrm{R}\mathcal{H}om_{D_{X}}(\mathcal{M}, B_{M})|_{N}\otimes \mathrm{o}\mathrm{r}_{N|M}[1]\cong \mathrm{R}\mathcal{H}om_{D_{Y}}(\varphi^{*}\mathcal{M}, e_{N})$$D_{X^{-}}$

164[SKK, II, Cor.3.5.8]. By the proof of Theorem 1.2, the following diagram iscommutative.’$\mathrm{R}\Gamma_{N}\mathrm{R}\mathcal{H}om_{D_{X}}(\mathcal{M}, \beta_{M})|_{N^{\otimes \mathrm{o}\mathrm{r}_{N}}}|M[1]$$arrow \mathrm{R}\dot{\pi}_{N*}\mathrm{R}\mathcal{H}om_{\mathcal{E}_{Y}}(\varphi^{*}\overline{\mathcal{M}}, CN)$$\downarrow$$\downarrow$$\mathrm{R}\Gamma z_{+}\mathrm{R}\mathcal{H}om_{D}(X\mathcal{M}, B_{M})|_{N}\otimes \mathrm{o}\mathrm{r}_{N}|M[1]arrow(1.3)\sim \mathrm{R}\dot{\pi}_{N*}\mathrm{R}\mathcal{H}om_{\mathcal{E}_{Y}}(N^{+}, C_{N})$.

Hence, from the Mayer-Vietoris cohomological sequence, we have a long exact
sequence...

$arrow \mathcal{E}_{Xt_{D}^{k}}(X\mathcal{M}, \Gamma_{M_{+}}B_{M})|_{N}\otimes \mathrm{o}\mathrm{r}_{N}|Marrow \mathcal{E}xt_{D_{Y}}(k\varphi \mathcal{M}*, B_{N})$$arrow H^{k}\mathrm{R}\dot{\pi}_{N*}\mathrm{R}\mathcal{H}om\mathcal{E}Y(N^{+}, cN)arrow\cdots$,

where the second arrow is factorized as follows :$\mathcal{E}xt_{D_{Y}}^{k}(\varphi^{*}M, \beta_{N})arrow H^{k}\mathrm{R}\dot{\pi}_{N*}\mathrm{R}\mathcal{H}\mathit{0}\alpha m_{\mathcal{E}_{Y}}(\varphi^{*}\overline{\Lambda 4}, cN)$

$arrow H^{k}\mathrm{R}\dot{\pi}_{N}*\mathrm{R}\beta \mathcal{E}\mathcal{H}omY(N+, C_{N})$

Since

.

for $k>0$ by assumption, is surjective for all
and is an isomorphism for $k>0$ . On the other hand, since
is adirect summand ofas an -module, is surjective and$\mathcal{E}xt_{D_{Y}}^{k}(\varphi^{*}\mathcal{M}, A_{N})=0$$\alpha$$N^{+}$$k\in \mathrm{Z}$$\varphi^{*}\overline{\mathcal{M}}$$\mathcal{E}_{Y}$$\beta$$\mathrm{K}\mathrm{e}\mathrm{r}(\beta)=H^{k}\mathrm{R}\dot{\pi}_{N}*\mathrm{R}\mathcal{H}_{\mathit{0}}m_{\mathcal{E}_{Y}}(\varphi^{*+}\overline{\mathcal{M}}/N, cN)$

Hence, using an isomorphism(3.1) and (3.2).

$\mathrm{o}\mathrm{r}_{N|M}\cong \mathrm{C}_{N}$.

(see Remark 1 below), we obtain.D.
$\mathrm{Q}.\mathrm{E}$Remark 1. The following diagram is commutative and every vertical arrow is
an isomorphism:$arrow$$\mathrm{C}_{M}+$$\mathrm{c}_{z_{+}}$$\downarrow$$\downarrow$.

$\mathrm{R}\mathcal{H}om_{\mathrm{C}}(\mathrm{c}_{z}+’ \mathrm{C}_{M})arrow \mathrm{R}\mathcal{H}om\mathrm{c}(\mathrm{c}_{M}+’ \mathrm{C}_{M})$

Hence we have an isomorphism$\mathrm{c}_{z_{+}}$$\downarrow$$\eta$$arrow$:$\mathrm{C}_{N}arrow \mathrm{o}\mathrm{r}_{N|M}$$\mathrm{C}_{N}$$\downarrow\eta$$arrow$such that$\mathrm{C}_{M}[+1]$$\downarrow$$\mathrm{R}\mathcal{H}om\mathrm{c}(\mathrm{c}_{M}+’ \mathrm{C}_{M})arrow \mathrm{o}\mathrm{r}_{N|M}arrow \mathrm{R}\mathcal{H}om\mathrm{c}(\mathrm{C}z_{+}, \mathrm{C}_{M})[1]$165

becomes commutative. (This corresponds to choosing a non-degenerate secas positive orientation.) Note that the following diagram istion $df$ of:then commutative for$T_{N}^{*}M$$F\in \mathrm{O}\mathrm{b}(\mathrm{D}^{\mathrm{b}}(M))$$arrow$$arrow 1\otimes\eta$$\mathrm{R}\Gamma_{M}F|_{N}+0\otimes \mathrm{r}_{N}|M$$\mathrm{R}\Gamma_{M}F|_{N}+$$\downarrow\cong$$\mathrm{R}\mathcal{H}om_{\mathrm{C}(}\mathrm{c}_{M}+’ F)|_{N}arrow \mathrm{R}\mathcal{H}om\mathrm{c}(\mathrm{o}\mathrm{r}_{N}|M[-1], F)\mathrm{t}Narrow\cong$$arrow \mathrm{R}\Gamma_{N}F|_{N}[1]\otimes \mathrm{o}\mathrm{r}_{N}|M$$\downarrow\cong$$arrow\cong \mathrm{R}\Gamma_{N}F|_{N}[1]\otimes \mathrm{o}\mathrm{r}N|\mathrm{v}M$.
, which is canonically isomorphic to
with(The topological boundary value morphism for $F$ is deflned [S2, S3] as antito’
clockwise composition of morphisms, fromin this diagram.)$\mathrm{o}\mathrm{r}_{N|M}$$\mathrm{o}\mathrm{r}_{N|M}^{}=\mathcal{H}om_{\mathrm{C}}(\mathrm{o}\mathrm{r}_{N|}M, \mathrm{C}_{N})$$\mathrm{R}\Gamma_{M_{+}}F|_{N}$$\mathrm{R}\Gamma_{N}F|_{N}[1]\otimes \mathrm{o}\mathrm{r}_{N|}\mathrm{v}M$

Remark 2. For single differential equations, Oaku [ , Sect.3] extends (3.1) to. If $N^{+}=0$ in thatthe case where condition (A.2) is satisfied locally on
case, this has been first treated by Kaneko [Kn].
$\mathrm{O}$$T_{N}^{*}Y$

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