数理解析研究所講究録 1211 巻 2001 年 86-95

86

Microlocalizationof Topological Boundary Value Morphism
and Regular-Specializable Systems
Susumu YAMAZAKI (山崎 晋)*
Graduate School of Mathematical Sciences, the University of Tokyo,
8-1 Komaba 3-chome, MegurO-ku, Tokyo 153-8914, Japan

Introduction
In microlocal analysis, it is one of the main subjects to give an appropriate formulationof the boundary value problems for hyperfunction or microfunction solutions to asystemof linear partial differential equations with analytic coefficients (that is, acoherent (left)-Module, here in this article, we shall write Module with acapital letter, instead ofsheaf of modules). If the system is regular-specializable, the nearby-cycle of the system
can be defined in the theory of -Modules. After the results by Kashiwara and Oshima [K-O], Oshima [Os] and Schapira [Sc 2], [Sc3], for any hyperfunction solutions toregular-specializable system Monteiro Fernandes [MF1] defined aboundary value morphism which takes values in hyperfunction solutions to the nearby-cycle of the system
instead of the induced system. This morphism is injective (cf. [MF2]) and ageneralization of the non-characteristic boundary value morphism (for the non-characteristic
case, see Komatsu and Kawai [KO-K], Schapira [Sc 1] and further Kataoka [Kat] . Moreover recently Laurent and Monteiro Fernandes [L-MF2] reformulated this boundary valuemorphism and discussed the solvability under akind of hyperbolicity condition (the nearhyperbolicity). However, since this morphism is defined only for hyperfunction solutions,amicrolocal boundary value problem is not considered. Therefore in this article, we
state amicrolocalization of their result in the framework of Oaku [Oa2] and Oaku$\mathcal{D}$$\mathcal{D}$$)$$5\mathrm{h}\mathrm{a}\mathrm{l}\mathrm{l}$Yamazaki [O-Y].
The details of this article will be given in our forthcoming paper [Y].

Research Fellow of The Japan Society for The Promotion of Science87
1NotationandWe denote the set of integers, of real numbers and of complex numbers by ,.
andrespectively as usual. Moreover we set
All the manifolds are assumed to be paracompact. Let $\tau:Earrow Z$ avector bundle
and the restriction of to . Let $M$ be anover amanifold . Then, set$(n+1)$ -dimensional real analytic manifold and $N$ aone-codimensional closed real analytic
submanifold of $M$ . Let $X$ and be complexifications of $M$ and $N$ respectively such thatis aclosed submanifold of $X$ and that $\mathrm{Y}\cap M=N$ . Moreover, we assume the existenceof apartial complexification of $M$ in $X$ ;that is, there exists a $(2n+1)$ -dimensional realsuch that the triplet $(N, M, L)$analytic submanifold of $X$ containing both $M$ andby alocal coordinate system $(z, \tau)=$is locally isomorphic to$(x+\sqrt{-1}y, t+\sqrt{-1}s)$ of $X$ around each point of $N$ . We say such acoordinate system
admissible. We shall mainly follow the notation in Kashiwara-Schapira [K-S]; we denoterespectively and regard
andthe normal deformations of $N$ and in $M$ and by. We have the following commutative diagram:as aclosed submanifold of$\mathbb{Z}$$\mathrm{N}:=\{n\in \mathbb{Z};n\geq 1\}$$\mathbb{C}$$\mathrm{N}_{0}:=\mathrm{N}\mathrm{U}\{0\}$$\dot{E}$$\dot{E}:=E\backslash Z$$Z$$\mathbb{R}$$\dot{\tau}$$\tau$$\mathrm{Y}$$\mathrm{Y}$$L$$\mathrm{Y}$$(\mathbb{R}^{n}\cross\{0\}, \mathbb{R}^{n+1}, \mathbb{C}^{n}\cross \mathbb{R})$$L$$\mathrm{Y}$$\overline{M}_{N}$$\tilde{L}_{\mathrm{Y}}$$\tilde{L}_{\mathrm{Y}}$$\overline{M}_{N}$and by admissible coordinates we have locally the following relation:$N=\mathbb{R}_{x}^{n}\mathrm{J}|\cross\{0\}arrowarrow M=\mathbb{R}_{x}^{n}i\mathrm{J}|\cross$$\mathrm{Y}=\mathbb{C}_{z}^{n}\cross\{0\}\frac{\tau i_{Y}}{\prime}L=\mathbb{C}_{z}^{n}\cross$With these coordinates, we often identify $TyX$ andand ::The projection$\tau_{\mathrm{Y}}$$T_{\mathrm{Y}}Larrow \mathrm{Y}$$=\mathbb{C}_{z}^{n}\cross \mathbb{C}_{\tau}$with $X$ and respectively.
induce natural mappings:$L$$T_{\mathrm{Y}}L$$T_{\mathrm{Y}}Larrow\overline{L}_{\mathrm{Y}}$$s_{L}$.

,$T_{N}^{*} \mathrm{Y}T_{N}M\cross T_{N}^{*}\mathrm{Y}\overline{\tau_{Y\pi}}N\vec{\ell_{\mathcal{T}_{\acute{Y}}}}\overline{\overline{{}^{t}s_{\acute{L}}}}-T_{T_{N}hI}^{*}T_{\mathrm{Y}}LT_{N}MT\frac{*}{\mathrm{A}I}\tilde{L}_{\mathrm{Y}}\frac{\cross}{M}NN\vec{s_{L\pi}}NT\frac{*}{M}\tilde{L}_{\mathrm{Y}}$and by these mappings, we identify$T_{T_{N}h\mathrm{f}}^{*}T_{\mathrm{Y}}L$

with$T_{N}M\cross T_{N}^{*}\mathrm{Y}N$

and$T_{N}MT \frac{*}{M}\frac{\cross}{M}N\tilde{L}_{\mathrm{Y}}N$

88
has two components with respect to its fiber. We denote one of them by
and represent (at least locally) by fixing an admissible coordinate system$T_{\mathrm{Y}}L\backslash T_{\mathrm{Y}}\mathrm{Y}$$T_{\mathrm{Y}}L^{+}$$T_{\mathrm{Y}}L^{+}=\{(z, t)\in T_{\mathrm{Y}}L;t>0\}$.

Moreover set $T_{N}M^{+}:=T_{\mathrm{Y}}L^{+}\cap T_{N}M$ . Set an open embedding$f_{N}:=f|_{T_{N}M^{+}}:$ $T_{N}M^{+}\mapsto T_{N}M$ . We regard
as an open set of
Moreover induces mappings:$f:T_{\mathrm{Y}}L^{+}\mapsto T_{\mathrm{Y}}L$$T_{N}M^{+}\cross T_{N}^{*}\mathrm{Y}N$

and.

$T_{T_{N}M}^{*}T_{\mathrm{Y}}L$$f$$T_{T_{N}M^{+}}^{*}T_{\mathrm{Y}}L^{+}--T_{N}M^{+}\cross T_{T_{N}M}^{*}T_{\mathrm{Y}}Larrow T_{T_{N}M}^{*}T_{\mathrm{Y}}LT_{N}Mf_{n}$$\mathrm{O}$$|l$$|^{\iota}$.

$T_{N}M^{+}\cross T_{N}^{*}\mathrm{Y}N$

Hence we identify$T_{T_{N}M}^{*}+T_{\mathrm{Y}}L^{+}$:
Let
We denote as usual byrespectively.
$\pi_{N,M}$

with

’and

$f_{\pi}$with$f_{N}\cross \mathrm{i}\mathrm{d}$.

, be the natural projections.
:andand the Sato specialization and microlocalization functors$T_{\frac{*}{M}N}\tilde{L}_{\mathrm{Y}}arrow\overline{M}_{N}$$\nu$$T_{N}M^{+}\cross T_{N}^{*}\mathrm{Y}N$$N\cross T_{N}^{*}\mathrm{Y}$$T_{T_{N}M}^{*}T_{\mathrm{Y}}Larrow T_{N}M$$\pi_{N|M}$$\mu$

2General Boundary Values
By using an admissible coordinate system we define acontinuous section$(z, 1)$ . Similarly we definebyby. In general, let$\tau:EarrowZ$
acomplex manifold,acomplex vector bundle. Then, denote bysubcategory of
consisting of -conic objects.

$\sigma:\mathrm{Y}arrow\dot{T}_{\mathrm{Y}}X$${}^{t}\sigma:\mathrm{Y}arrow\dot{T}_{\mathrm{Y}}^{*}X$$z-\rangle$$z\vdash\Rightarrow(z, 1)$$Z$$\mathrm{D}_{\mathbb{C}^{\mathrm{x}}}^{b}(E)$$\mathrm{D}^{b}(E)$bethe$\mathbb{C}^{\mathrm{x}}$2.1 Theorem. For any objectofexists the following natural isomorphism:$\mathcal{F}$$\mathrm{D}^{b}(X)$such that$\nu_{\mathrm{Y}}(\mathcal{F})\in \mathrm{O}\mathrm{b}(\mathrm{D}_{\mathbb{C}^{\mathrm{x}}}^{b}(T_{\mathrm{Y}}X))$$f_{\pi}^{-1}\mu_{T_{N}M(\nu_{\mathrm{Y}}(i_{i’}\mathcal{F}))3f_{\pi}^{-1}\tau_{\mathrm{Y}\pi}^{-1}\mu_{N}(\sigma^{-1}\nu_{\mathrm{Y}}(\mathcal{F}))\otimes\omega_{L/X}}$.

2.2 Definition. For any object ofsuch thatby virtue of Kashiwara-Schapira [K-S] and Theorem 2.1:$\mathcal{F}$$\mathrm{D}^{b}(X)$, there$\nu_{\mathrm{Y}}(\mathcal{F})\in \mathrm{O}\mathrm{b}(\mathrm{D}_{\mathbb{C}^{\mathrm{X}}}^{b}(T_{\mathrm{Y}}X))$, we define$\beta:f_{\pi}^{-1}s_{L\pi}^{-1}\mu_{\overline{M}_{N}}(Rj_{L*}\tilde{p}_{L}^{-1}i_{i’}\mathcal{F})arrow f_{\pi}^{-1}\mu_{T_{N}M}(\nu_{\mathrm{Y}}(i_{i’}\mathcal{F}))$$\approx$$f_{\pi}^{-1}\tau_{\mathrm{Y}\pi}^{-1}\mu_{N}(\sigma^{-1}\nu_{\mathrm{Y}}(\mathcal{F}))\otimes\omega_{L/X}$.

of
2.3 Definition (Laurent-Monteiro Fernandes -MF 2]). We say an object
is near-hyperbolic at $x_{0}\in N$ (in -codirection)if there exist positive constants
and such that$\mathcal{F}$$[\mathrm{L}$$\mathrm{D}^{b}(X)$

$dt$

$C$

$\epsilon_{1}$$\mathrm{S}\mathrm{S}(\mathcal{F})\cap\{(z, \tau;z^{*}, \tau^{*})\in T^{*}X;|z-x_{0}|, |\tau|<\epsilon_{1},0<{\rm Re}\tau\}$$\subset\{(z, \tau;z^{*}, \tau^{*})\in T^{*}X;|{\rm Re}\tau^{*}|<C(|{\rm Im} z^{*}|(|{\rm Im} z|+|{\rm Im}\tau|)+|{\rm Re} z^{*}|)\}$

holds by an admissible coordinate system. Here$\mathrm{S}\mathrm{S}(\mathcal{F})$denotes the microsupport of J.

89. Assume that
2.4 Theorem. Let be a object ofis near-hyperbolic at $x_{0}\in N$ . Then, for any$\mathrm{D}^{b}(X)$$\mathcal{F}$$\mathcal{F}$$\nu_{\mathrm{Y}}(\mathcal{F})\in \mathrm{O}\mathrm{b}(\mathrm{D}_{\mathbb{C}^{\mathrm{x}}}^{b}(T_{\mathrm{Y}}X))$and$p^{*}\in T_{T_{N}M^{+}}^{*}T_{\mathrm{Y}}L^{+}$$\beta$:$s_{L\pi}^{-1}\mu_{\overline{M}_{N}}(Rj_{L*}\tilde{p}_{L}^{-1}i_{i’}\mathcal{F})_{P^{*}}$$arrow\mu_{N}(\sigma-1\nu_{\mathrm{Y}}(\mathcal{F}))_{\tau_{Y\pi}(p*)}\otimes\omega_{L/X}$

is an isomorphism.

3Regular-Specializable Systems
In this section, we shall recall the basic results concerning the regular-specializable
Module and its nearby-cycle.
As usual, we denote by
the sheaf on $X$ of holomorphic differential operators, and.
by
the usual order filtration on$\mathcal{D}-$$\mathcal{D}_{X}$$\{\mathcal{D}_{X}^{(m)}\}_{m\in \mathrm{N}_{0}}$$\mathcal{D}_{X}$

3.1 Definition. Denote byinwith aconvention thatthe defining Ideal of(along Y) is afiltration on
for $j\leq 0$ . The filtration
defined by.
$\mathrm{Y}$$O_{X}$$\mathrm{J}_{\mathrm{Y}}$$V$$0_{\mathrm{Y}}^{j}=(9_{X}$$\{V_{\mathrm{Y}}^{k}(\mathcal{D}_{X})\}_{k\in \mathbb{Z}}$$\mathcal{D}_{X}|_{\mathrm{Y}}$$V_{\mathrm{Y}}^{k}( \mathcal{D}_{X}):=\bigcap_{j\in \mathbb{Z}}\{P\in \mathcal{D}_{X}|_{\mathrm{Y}}; P0_{\mathrm{Y}}^{j}\subset \mathrm{J}_{\mathrm{Y}}^{j-k}\}$Let us denote by 19 the Euler operator. Note thatby admissible coordinates.
can be represented by$\theta\in V_{\mathrm{Y}}^{0}(\mathcal{D}_{X})\backslash V_{\mathrm{Y}}^{-1}(\mathcal{D}_{X})$and that$\theta$$\tau\partial_{r}$3.2 Definition. Acoherent-Moduleis said to be regular-specializable (alongY) if there exist locally acoherent-sub-Moduleofand anon-zero polynomial
such that the following conditions are satisfied:; that is,;(1)
generates
over, where $m$ is the degree of.
(2)$\mathrm{M}$$\mathcal{D}_{X}|_{\mathrm{Y}}$$\mathrm{M}_{0}$$\mathrm{t}9_{X}$$\mathrm{M}$$b(\alpha)\in \mathbb{C}[\alpha]$$\mathrm{M}$$\mathrm{M}_{0}$$\mathrm{M}$$\mathcal{D}_{X}$$=\mathcal{D}_{X}\mathrm{M}_{0}$$b(\theta)\mathrm{M}_{0}\subset(\mathcal{D}_{X}^{(m)}\cap V_{\mathrm{Y}}^{-1}(\mathcal{D}_{X}))\mathrm{M}_{0}$$b(\alpha)$

In what follows, we shall omit the phrase “along$\mathrm{Y}$”since$\mathrm{Y}$is fixed.

3.3 Remark. (1) Let
be acoherent-Module for which is non-characteristic.
Then, it is easy to see thatis regular-specializable.
(2) Kashiwara-Kawai [K-K] proved that every regular-holonomic-Module isregular-specializable.
$\mathrm{M}$$\mathrm{Y}$$\mathcal{D}_{X}|_{\mathrm{Y}}$$\mathrm{M}$$\mathcal{D}_{X}|_{\mathrm{Y}}$3.4 Proposition. If M is a regular-specializableandare objects of$\mathrm{D}_{\mathbb{C}^{\mathrm{X}}}^{b}(T_{\mathrm{Y}}^{*}X)$$R\mathcal{H}om_{\mathrm{D}_{X}},(\mathrm{M}, \nu_{\mathrm{Y}}(O_{X}))$

Let$\iota:\mathrm{Y}arrow X$

in the sense of$\mathcal{D}$-Module,and$\mathcal{D}_{X}|_{\mathrm{Y}}$$R\mathcal{H}om_{\mathrm{D}_{X}}(\mathrm{M}, \mu_{\mathrm{Y}}(0_{X}))$$\mathrm{D}_{\mathbb{C}^{\cross}}^{b}(T_{\mathrm{Y}}X)$

respectively.

be the natural inclusion. Then the induced system, or the inverse image-Modules is defined by$D\iota^{*}\mathrm{M}$$:=\mathrm{t}9_{\mathrm{Y}}\otimes^{L}\iota^{-1}\iota^{-1}0_{X}$M.

For any regular-specializable-Module , the nearby-cycleofand thevanishing-cycleofin the theory of -Modules can be defined. For the defini, we refer to Laurent [L], Mebkhout [Me]. We shall recall thetions ofandfollowing two results$\mathcal{D}_{X}$$\Phi_{\mathrm{Y}}(\mathrm{M})$$\Psi_{\mathrm{Y}}(\mathrm{M})$$\mathrm{M}$$\Phi_{\mathrm{Y}}(\mathrm{M})$$\mathrm{M}$$\mathcal{D}$$\Psi_{\mathrm{Y}}(\mathrm{M})$$\mathrm{M}$90
3.5 Proposition (Laurent [L], Mebkhout [Me]). Let M be a regular-specializableand each cohomology of Dc’ M are coherent-Module. Then, ’Modules. Moreover, there exists the following distinguished triangle:$D_{X}|_{\mathrm{Y}}$$\mathrm{Y}(’ \mathrm{C})_{\rangle}I_{\mathrm{Y}}(\ovalbox{\tt\small REJECT} M)$$\ovalbox{\tt\small REJECT} \mathrm{j})_{Y}$Var$/fere$, Var$:=\varphi(\theta)\tau$with$\varphi(\zeta)$$arrow^{+1}$$\Psi_{\mathrm{Y}}(\mathrm{M})arrow D\iota^{*}\mathrm{M}$$\Phi_{\mathrm{Y}}(\mathrm{M})$$:=(e^{2\pi\sqrt{-1}\zeta}-1)/\zeta$.

.

be the sheaf of real holomorphic microfunctions
3.6 Theorem (Laurent [L]). Let
be a regular-specializable-Module. Then, there exists theon
as usual. Let
following isomorphism of distinguished triangles:$\mathrm{e}_{\mathrm{Y}|X}^{\mathrm{R}}$$\mathrm{M}$$T_{\mathrm{Y}}^{*}X$$\mathcal{D}_{X}|_{\mathrm{Y}}$$R\mathcal{H}om_{\mathrm{D}_{X}}(\mathrm{M}, \mathrm{t}9_{X})|_{\mathrm{Y}}arrow R\}(om_{\mathrm{D}_{X}}(\mathrm{M},\sigma^{-1}\nu_{\mathrm{Y}}(\mathfrak{l}9_{X}))arrow Mom_{\mathrm{D}_{X}}(\mathrm{M},{}^{t}\sigma^{-1}\mathrm{G}_{\mathrm{Y}|X}^{\mathbb{R}})arrow+1$$\downarrow l$$\downarrow[$$\downarrow l$$Rg[om_{\mathrm{D}_{Y}}(D\iota^{*}\mathrm{M}, 19_{\mathrm{Y}})arrow R\sigma(om_{\mathrm{D}_{Y}}(\Psi_{\mathrm{Y}}(\mathrm{M}), 19_{\mathrm{Y}})arrow Mom_{\mathrm{D}_{Y}},(\Phi_{\mathrm{Y}}(\mathrm{M}), \mathrm{t}9_{\mathrm{Y}})arrow+1$

3.7 Remark. (1) The isomorphism (the Cauchy-Kovalevskaja type theorem)$R\mathcal{H}om_{\mathrm{D}_{Y}}(D\iota^{*}\mathrm{M}, \mathrm{t}9_{\mathrm{Y}})\simeq R\mathcal{H}om_{\mathrm{D}_{X}}(\mathrm{M}, \mathrm{t}9_{X})|_{\mathrm{Y}}$holds for Fuchsian systems in the sense of Laurent-Monteiro Fernandes [ -MF 1].
(2) Recently Mandai [Man] extended the definition of boundary values to ageneral
Fuchsian differential equation in the complex domain.
$\mathrm{L}$

4Boundary Values for Regular-Specializable Systemandthe sheaf of holomorphic functions on $X$ , of hyperfunctions
We denote by Ox,
respectively.
on $M$ and of microfunctions on$\mathfrak{B}_{M}$$\mathrm{e}_{M}$$T_{M}^{*}X$

4.1 Definition (Oaku [Oa 2], Oaku-Yamazaki [O-Y]). We set:$\mathrm{G}_{N|M}:=s_{L\pi}^{-1}\mu_{\overline{M}_{N}}(Rj_{L*}\tilde{p}_{L}^{-1}ii’\mathrm{t}9_{X})\otimes or_{M/X}[n+1]$

We can regard$\mathrm{e}_{N|M}$

as amicrolocalization of$\nu_{N}(\mathfrak{B}_{M})$:4.2 Proposition. (1)is concentrated in degree zero; that is,. Furtherholds
sheaf on(2) There exists the following exact sequence on $T_{N}M$ :$\mathrm{e}_{N|M}$.

$\mathrm{e}_{N|M}$

is regarded as$a$$\mathrm{G}_{N|M}|_{T_{N}M}=\nu_{N}(\mathfrak{B}_{M})$$T_{T_{N}M}^{*}T_{\mathrm{Y}}L$.

$0arrow\nu_{\mathrm{Y}}(\mathfrak{B}\mathrm{t}9_{L})|_{T_{N}M}arrow\nu_{N}(\mathfrak{B}_{M})arrow\dot{\pi}_{N|M*}\mathrm{C}_{N|M}arrow 0$

Hereon L. Note thatis the sheaf of hyperfunctions with holomorphic parameters
is concentrated in degree zero;$\mathfrak{B}\mathrm{t}9_{L}:=\mathcal{H}_{L}^{1}(\mathrm{t}9_{X})\otimes or_{L/X}$$\nu_{\mathrm{Y}}(\mathfrak{B}\mathrm{t}9_{L})$91.

-Module. By Proposition 3.4,
4.3 Definition. Let M be aregular-specializablesatisfies the assumption of Theorem 2.1. Thus, by Definition 2.2 and(M,
Theorem 3.6, we define:$R\ovalbox{\tt\small REJECT} J^{-}Com$$\mathrm{D}_{X}|_{\mathrm{Y}}$$\mathrm{C}9_{X})$$x$.

$\beta:f_{\pi}^{-1}R\mathcal{H}om_{\mathrm{D}_{X}}(\mathrm{M}, \mathrm{G}_{N|M})arrow f_{\pi}^{-1}\tau_{\mathrm{Y}\pi}^{-1}R\mathrm{H}om_{\mathrm{D}_{Y}}(\Psi_{\mathrm{Y}}(\mathrm{M}), \mathrm{G}_{N})$4.4 Theorem. (1) The morphism$\beta^{0}$:$\beta$gives a monomorphism$f_{\pi}^{-1}\mathcal{H}om_{\mathrm{D}_{X}}(\mathrm{M}, \mathrm{C}_{N|M})\mapsto f_{\pi}^{-1}\tau_{\mathrm{Y}\pi}^{-1}\mathcal{H}om_{\mathrm{D}_{Y}}(\Psi_{\mathrm{Y}}(\mathrm{M}), \mathrm{C}_{N})$.

to the zerO-section $T_{N}M^{+}$ coincides with the boundary value(2) The restriction of.
morphism in the sense of Monteiro Fernandes$\beta^{0}$$[\mathrm{M}\mathrm{F}1]$

be acoherent
4.5 Definition. Let
at $x_{0}\in N$ (in -codirection)if
Definition 2.3. Here, we remark that$\mathrm{M}$$dt$$\mathcal{D}_{X}|_{\mathrm{Y}}$is near-hyperbolic-Module. Then we say
is near-hyperbolic in the sense of.
$\mathrm{M}$$R9\{om_{\mathrm{D}_{X}},(\mathrm{M}, 0_{X})$$\mathrm{S}\mathrm{S}(R\mathcal{H}om_{\mathrm{D}_{X}},(\mathrm{M}, \mathrm{t}9_{X}))=\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}(\mathrm{M})$The following theorem is adirect consequence of Theorem 2.4:

be a regular-specializable
4.6 Theorem. Let
hyperbolic at $x_{0}\in N$ . Then, for any$\mathrm{M}$$\mathcal{D}_{X}|_{\mathrm{Y}}$-Module. Assume that$\mathrm{M}$is near-$p^{*}\in T_{T_{N}M^{+}}^{*}T_{\mathrm{Y}}L^{+}$$\beta:R\mathcal{H}om_{\mathrm{D}_{X}},(\mathrm{M}, \mathrm{G}_{N|M})_{p^{*}}arrow R\mathcal{H}om_{\mathrm{D}_{Y}}(\Psi_{\mathrm{Y}}(\mathrm{M}), \mathrm{C}_{N})_{\tau_{Y\pi}(p*)}$is an isomorphism.
, and setbe the sheaf of -mild microfunctions on(see Oaku [Oa 1], [Oa 2], and Oaku-Yamazaki [O-Y]).
M.
-Module. Set
be aregular-specializable

4.7 Remark. Let$F$$\mathrm{C}_{N|M}^{F}$$T_{T_{N}M}^{*}T_{\mathrm{Y}}L$$\overline{\mathrm{e}}_{N|M}^{A}:=\mathcal{H}^{n}(\mu_{N}(\mathit{0}_{X}|_{\mathrm{Y}}))\otimes or_{N/\mathrm{Y}}$Let$\mathrm{M}$$\mathrm{M}_{\mathrm{Y}}:=\mathcal{H}^{0}(D\iota^{*}\mathrm{M})$$\mathcal{D}_{X}|_{\mathrm{Y}}$$=\mathrm{t}9_{\mathrm{Y}}$$\iota^{-1}0_{X}\otimes\iota^{-1}$

By the argument in Oaku-Yamazaki [O-Y] we have the following commutative diagram:$f_{\pi}^{-1} \mathcal{H}om_{\mathrm{D}_{X}}(\mathrm{M}, \mathrm{G}_{N|M}^{F})\approx f_{\pi}^{-1}\tau_{\mathrm{Y}\pi}^{-1}\mathcal{H}om_{\mathrm{q})_{X}}(\mathrm{M},\tilde{\mathrm{C}}_{N|M}^{A})\int 0\downarrow\backslash -\vec{\mathrm{o}}\iota’f_{\pi}^{-1}\tau_{\mathrm{Y}\pi}^{-1}\mathrm{H}om_{\mathrm{D}_{Y}}(\mathrm{M}_{\mathrm{Y}}, \mathrm{C}_{N})$,$f_{\pi}^{-1}\mathcal{H}om_{\mathrm{D}_{X}},(\mathrm{M}, \mathrm{G}_{N|M})>arrow f_{\pi}^{-1}\mathcal{H}om_{\mathrm{D}_{X}}(\mathrm{M},\overline{\mathrm{G}}_{N|M})arrow-f_{\pi}^{-1}\tau_{\mathrm{Y}\pi}^{-1}f\mathrm{f}om_{\mathrm{D}_{Y}}(\Psi_{\mathrm{Y}}(\mathrm{M}), \mathrm{C}_{N})$that is, the boundary value morphism$\gamma^{F}$

and
that

$\beta^{0}$:$f_{\pi}^{-1}\mathrm{H}om_{\mathrm{D}_{X}}(\mathrm{M}, \mathrm{G}_{N|M}^{F})\mapsto f_{\pi}^{-1}\tau_{\mathrm{Y}\pi}^{-1}\mathrm{f}\{om_{\mathrm{D}_{Y}}(\mathrm{M}_{\mathrm{Y}}, \mathrm{C}_{N})$are compatible. In particular, if is non-characteristic for , then it is known
and by Oaku [Oa2] (cf. Oaku-Yamazaki [O-Y]) we have$\Psi_{\mathrm{Y}}(\mathrm{M})\approx$$\mathrm{Y}$$D\iota^{*}\mathrm{M}$$\overline{\gamma}_{N|M}$:$\mathrm{M}$$\simeq \mathrm{M}_{\mathrm{Y}}$$R\mathcal{H}om_{\mathrm{D}_{X}}(\mathrm{M},\overline{\mathrm{C}}_{N|M})\approx$.

$\tau_{\mathrm{Y}\pi}^{-1}R\mathcal{H}om_{\mathrm{D}_{Y}}(\mathrm{M}_{\mathrm{Y}}, \mathrm{G}_{N})$92
In this case we see thatis equivalent to the non-characteristic boundary value morphism(see Kataoka [Kat] and Oaku [Oa 2]). In particular, the restriction of to the zer0-section$T_{N}M^{+}$ is equivalent toKomatsu-Kawai [KO-K] and Schapira [Sc 1]. Further, if is noncharacteristic for
and $\pm dt\in T_{N}^{*}M$ is hyperbolic for , then the nearly-hyperbolic
condition is satisfied and is an isomorphism.
$\beta^{0}$$\beta^{0}$$\mathrm{Y}$$\mathrm{M}$$\mathrm{M}$$\beta$4.8 Example. Assume thatand so on by an admissible coordinate system.
(1) Letbe anon-zero polynomial with degree $m$ , and.
Set. Thenis regular-specializable. Assume that
$X=\mathbb{C}^{n+1}$$b(\alpha)$$\mathrm{M}$$Q\in \mathcal{D}_{X}^{(m)}\cap V_{\mathrm{Y}}^{-1}(\mathcal{D}_{X})$$:=\mathcal{D}_{X}/\mathcal{D}_{X}(b(\theta)+Q)$$\prod_{j=1}^{\mu}(\alpha-\alpha_{j})^{\nu_{\mathrm{j}}}$($\mathrm{M}$$\mathrm{b}(\mathrm{a})=$for $1\leq i\neq j\leq\mu$ , note that$\alpha_{i}-\alpha_{j}\not\in \mathbb{Z}$$\sum_{j=1}^{\mu}\nu_{j}=m$). Then adirect

calculation shows that, andis equivalent to 7in Oaku [Oa2]: Let $p^{*}=$
be apoint of, and $f(x,t)$ agerm of
at . Then, we can see that $f(x, t)$ has adefining function$\Psi_{\mathrm{Y}}(\mathrm{M})\simeq \mathcal{D}_{\mathrm{Y}}^{\oplus m}$$(x_{0}, t_{0};\sqrt{-1}\langle\xi_{0}, dx\rangle)$$\beta^{0}$$T_{T_{N}M}^{*}+T_{\mathrm{Y}}L^{+}$$\mathcal{H}om_{\mathrm{D}_{X}},(\mathrm{M}, \mathrm{C}_{N|M})$$p^{*}$$F(z, \tau)=\sum_{j=1}^{\mu}\sum_{k=1}^{\nu_{j}}F_{jk}(z, \tau)\tau^{\alpha_{\mathrm{j}}}(\log\tau)^{k-1}$.

Here eachis holomorphic on aneighborhood of $\{(z, \mathrm{O})\in X;|x_{0}-z|<\epsilon$ ,with apositive constant and an open convex cone such that(the interior of the dual coneof ). Then,is equivalent to$\leq j\leq\mu$$k\leq\nu_{j}$ , 1}. Moreover, if the principal symbol of $b(\theta)+Q$ is written as
for ahyperbolic polynomial
at -codirection, the nearly-hyperbolic
condition is satisfied. Note that this operator is aspecial case of Fuchsian hyperbolic
operators due to Tahara [T].
(2) Take an operator
at the origin and set
and $A^{(j)}:=$
for $j\geq 1$ . Let
be apoint of
and$F_{jk}(z, \tau)$${\rm Im} z\in$$\Gamma\}$$\Gamma$$\epsilon$$\Gamma^{\mathrm{o}}$$\beta^{0}(f)$$\Gamma$$\{\mathrm{s}\mathrm{p}_{N}(F_{jk}(x+\sqrt{-1}\Gamma 0,0));1\leq$$P$$\tau^{m}P(z, \tau;z^{*}, \tau^{*})$$dt$$A(z;\partial_{z})\in \mathcal{D}_{\mathrm{Y}}^{(1)}$set$p_{0}:=(0;\sqrt{-1}\langle\xi, dx\rangle)\in T_{N}^{*}\mathrm{Y}$$(\alpha_{1}, \alpha_{2})\in \mathbb{C}^{\oplus 2}$

agerm of(i) If. Consider$A^{0}:=\mathrm{i}\mathrm{d}$$p^{*}=(0,1;\sqrt{-1}\langle\xi, dx\rangle)$$\frac{1}{j!}A\circ A^{(j-1)}\in \mathcal{D}_{\mathrm{Y}}^{(j)}$$\mathrm{M}$. Set$\xi_{0}\in \mathrm{I}\mathrm{n}\mathrm{t}(\mathrm{I}^{\mathrm{o}})$$T_{T_{N}M^{+}}^{*}T_{\mathrm{Y}}L^{+}$, where. Let $f(x, t)$ be$P:=(\theta-\alpha_{1})(\theta-\alpha_{2})-\tau A(z;\partial_{z})\theta\in \mathcal{D}_{X}|_{\mathrm{Y}}$$:=\mathcal{D}_{X}/\mathcal{D}_{X}P=\mathcal{D}_{X}u$, where$u:=1\mathrm{m}\mathrm{o}\mathrm{d} P$

at . Then:$(\alpha_{1}, \alpha_{2})=(-1,0)$ , then$\mathcal{H}om_{\mathrm{D}_{X}}(\mathrm{M}, \mathrm{C}_{N|M})$$p^{*}$$\Phi_{\mathrm{Y}}(\mathrm{M})=\frac{V_{\mathrm{Y}}^{0}(\mathcal{D}_{X})u+V_{\mathrm{Y}}^{1}(\mathcal{D}_{X})(\theta+1)u}{V_{\mathrm{Y}}^{-1}(\mathcal{D}_{X})u+V_{\mathrm{Y}}^{0}(\mathcal{D}_{X})(\theta+1)u}=\mathcal{D}_{\mathrm{Y}}[u]+\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}(\theta+1)u]\simeq \mathcal{D}_{\mathrm{Y}}^{\oplus 2}$,$\Psi_{\mathrm{Y}}(\mathrm{M})=\frac{V_{\mathrm{Y}}^{-1}(\mathcal{D}_{X})u+V_{\mathrm{Y}}^{0}(\mathcal{D}_{X})(\theta+1)u}{V_{\mathrm{Y}}^{-2}(\mathcal{D}_{X})u+V_{\mathrm{Y}}^{-1}(\mathcal{D}_{X})(\theta+1)u}=\mathcal{D}_{\mathrm{Y}}[\tau u]+\mathcal{D}_{\mathrm{Y}}[(\theta+1)u]\simeq \mathcal{D}_{\mathrm{Y}}^{\oplus 2}$,

and Var: $([u], [\partial_{\tau}(\theta-1)u])-+([\tau u], 0)$ . Hence$f(x,t)$ has the following defining function$\mathrm{M}_{\mathrm{Y}}\simeq \mathcal{D}_{\mathrm{Y}}[(\theta+1)u]\simeq \mathcal{D}_{\mathrm{Y}}$. In this case$F(z, \tau)=U_{0}(z)+\frac{U_{-1}(z)}{\tau}-\sum_{j=1}^{\infty}\frac{A^{(j)}U_{-1}(z)}{j-1}\tau^{j-1}-AU_{-1}(z)\log\tau$

,

93
and !

is given by$(f^{\ovalbox{\tt\small REJECT}}(\ovalbox{\tt\small REJECT} \mathrm{m}\ovalbox{\tt\small REJECT}, t))$$U.(z)\ovalbox{\tt\small REJECT}$

0 and$\{\mathrm{s}\mathrm{p}_{N}(U_{\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}})(\mathrm{r})\}_{*\ovalbox{\tt\small REJECT}}\mathrm{i},0$$\mathrm{t}^{F}(f(x, t))\ovalbox{\tt\small REJECT}$(ii) If C’r:’2)$\ovalbox{\tt\small REJECT}$$(0_{\ovalbox{\tt\small REJECT}}1)_{\ovalbox{\tt\small REJECT}}$$\{f(\mathrm{r}, +\mathrm{O})\}\ovalbox{\tt\small REJECT}$at$p_{\mathit{0}}$. If$f(\mathrm{m}\ovalbox{\tt\small REJECT}, [])$

is $F$-mild at Po\rangle then$\{\mathrm{s}\mathrm{p}_{N}(U_{0})(\mathrm{z})\}_{\ovalbox{\tt\small REJECT}}$then$\Phi_{\mathrm{Y}}(\mathrm{M})$$= \frac{V_{\mathrm{Y}}^{1}(\mathcal{D}_{X})u+V_{\mathrm{Y}}^{2}(\mathcal{D}_{X})\theta u}{V_{\mathrm{Y}}^{0}(\mathcal{D}_{X})u+V_{\mathrm{Y}}^{1}(\mathcal{D}_{X})\theta u}=\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}u]+\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}^{2}\theta u]\simeq \mathcal{D}_{\mathrm{Y}}^{\oplus 2}$$\Psi_{\mathrm{Y}}(\mathrm{M})$$= \frac{V_{\mathrm{Y}}^{0}(\mathcal{D}_{X})u+V_{\mathrm{Y}}^{1}(\mathcal{D}_{X})\theta u}{V_{\mathrm{Y}}^{-1}(\mathcal{D}_{X})u+V_{\mathrm{Y}}^{0}(\mathcal{D}_{X})\theta u}=\mathcal{D}_{\mathrm{Y}}[u]+\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}\theta u]\simeq \mathcal{D}_{\mathrm{Y}}^{\oplus 2}$,

and Var. Hence$f(x, t)$ has the following defining function:$[\partial_{\tau}u]=\mathrm{V}\mathrm{a}\mathrm{r}$$[\partial_{\tau}^{2}\theta u]=0$$\mathrm{M}_{\mathrm{Y}}\simeq \mathcal{D}_{\mathrm{Y}}[u]+\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}\theta u]\simeq \mathcal{D}_{\mathrm{Y}}^{\oplus 2}$$F(z, \tau)=U_{0}(z)+\sum_{j=0}^{\infty}\frac{A^{(j)}U_{1}(z)}{j+1}\tau^{j+1}$

and $f(x, t)$ is always $F$-mild.

(iii) If

for which$\mathrm{Y}$In this case,

Hence $\beta^{0}(f(x, t))$ at
coincides with $\gamma^{F}(f(x, t))=$(if $\tau\neq 0$ ,is isomorphic to$p_{0}$$\{\partial_{t}^{i}f(x, +0)\}_{i=0,1}=\{\mathrm{s}\mathrm{p}_{N}(U_{i})(x)\}_{i=0,1}$$A(z;\partial_{z})\partial_{\tau})$,$\mathrm{M}$$\mathcal{D}_{X}/\mathcal{D}_{X}(\partial_{\tau}^{2}$

is non-characteristic).
, then$(\alpha_{1}, \alpha_{2})=(1,1)$,$\Phi_{\mathrm{Y}}(\mathrm{M})$$= \frac{V_{\mathrm{Y}}^{2}(\mathcal{D}_{X})u}{V_{\mathrm{Y}}^{1}(\mathcal{D}_{X})u}=\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}^{2}u]+\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}^{2}(\theta-1)u]\simeq \mathcal{D}_{\mathrm{Y}}^{\oplus 2}$$\Psi_{\mathrm{Y}}(\mathrm{M})$$= \frac{V_{\mathrm{Y}}^{1}(\mathcal{D}_{X})u}{V_{\mathrm{Y}}^{0}(\mathcal{D}_{X})u}=\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}u]+\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}(\theta-1)u]\simeq \mathcal{D}_{\mathrm{Y}}^{\oplus 2}$

and Var:In this case $f(x, t)$ has the following defining function:$([\partial_{\tau}^{2}u], [\partial_{\tau}^{2}(\theta-1)u])-t(2\pi\sqrt{-1}[\partial_{\tau}(\theta-1)u], 0)$. Hence$\mathrm{M}_{\mathrm{Y}}\simeq \mathcal{D}_{\mathrm{Y}}[\partial_{\tau}u]\simeq \mathcal{D}_{\mathrm{Y}}$and $\beta^{0}(f(x, t))$ is given by. If $f(x, t)$ is $F$-mild at
at$U_{0}(z)=0$ and $\gamma^{F}(f(x, t))=\{\partial_{t}f(x, +0)\}=\{\mathrm{s}\mathrm{p}_{N}(U_{1})(x)\}$ .
, then:(iv) If$\{\mathrm{s}\mathrm{p}_{N}(U_{i})(x)\}_{i=0,1}$$p_{0}$$p_{0}$, then$(\alpha_{1}, \alpha_{2})=(1,2)$,$\Phi_{\mathrm{Y}}(\mathrm{M})$$= \frac{V_{\mathrm{Y}}^{2}(\mathcal{D}_{X})u+V_{\mathrm{Y}}^{3}(\mathcal{D}_{X})(\theta-1)u}{V_{\mathrm{Y}}^{1}(2)_{X})u+V_{\mathrm{Y}}^{2}(\mathcal{D}_{X})(\theta-1)u}=\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}^{2}u]+\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}^{3}(\theta-1)u]\simeq \mathcal{D}_{\mathrm{Y}}^{\oplus 2}$$\Psi_{\mathrm{Y}}(\mathrm{M})$$= \frac{V_{\mathrm{Y}}^{1}(\mathcal{D}_{X})u+V_{\mathrm{Y}}^{2}(\mathcal{D}_{X})(\theta-1)u}{V_{\mathrm{Y}}^{0}(\mathcal{D}_{X})u+V_{\mathrm{Y}}^{1}(\mathcal{D}_{X})(\theta-1)u}=\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}u]+\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}^{2}(\theta-1)u]\simeq \mathcal{D}_{\mathrm{Y}}^{\oplus 2}$$([\partial_{\tau}^{2}u], [\partial_{\tau}^{3}(\theta-1)u])\vdash\Rightarrow(0,2A[\partial_{\tau}u])$. Hence$\mathrm{M}_{\mathrm{Y}}\simeq\frac{\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}u]+\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}^{2}(\theta-1)u]}{\mathcal{D}_{\mathrm{Y}}A[\partial_{\tau}u]}$

.

.

,

$F(z, \tau)=\sum_{j=0}^{\infty}A^{(j)}U_{0}(z)\tau^{j+1}-\sum_{j=1}^{\infty}\sum_{k=1}^{j}\frac{A^{(j)}U_{1}(z)}{k}\tau^{j+1}+\sum_{j=0}^{\infty}A^{(j)}U_{1}(z)\tau^{j+1}\log\tau$

and Var:-,94
In this case $f(x, t)$ has the following defining function:$F(z, \tau)=\sum_{j=0}^{\infty}A^{(j)}U_{2}(z)\tau^{j+2}+U_{1}(z)\tau-\sum_{j=2}^{\infty}\sum_{k=1}^{j-1}\frac{jA^{(j)}U_{1}(z)}{k}\tau^{j+1}$$+( \sum_{j=0}^{\infty}(j+1)A^{(j+1)}U_{1}(z)\tau^{j})\tau^{2}\log\tau$.
and $\beta^{0}(f(x, t))$ is given by
dition that $AU_{1}(z)=0$ , and in this case$\{\mathrm{s}\mathrm{p}_{N}(U_{\dot{1}})(x)\}_{*=1,2}$at$p_{0}$.

$\gamma^{F}(f(x, t))$$\{\partial_{t}^{:}f(x, +0)\}_{:=1,2}=\{\mathrm{s}\mathrm{p}_{N}(U_{1})(x), 2\mathrm{s}\mathrm{p}_{N}(U_{2})(x)\}$,$f(x, t)$at$p_{0}$

is -mild under the conis given by $\gamma^{F}(f_{3}(x, t))=$$F$

with $A\partial_{t}f(x, +0)=A\mathrm{s}\mathrm{p}_{N}(U_{1})(x)=0$.

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