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AN ELEMENTARY APPROACH TO THE MICROSUPPORT-THEORY OF HOLOMORPHIC SOLUTION COMPLEXES FOR $\mathcal{E}_X$- AND $\mathcal{E}^{\Bbb R}_X$-MODULES (Asymptotic Analysis and Microlocal Analysis of PDE)

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AN ELEMENTARY APPROACH TO THE

MICROSUPPORT-THEORY OF HOLOMORPHIC

SOLUTION COMPLEXES FOR $\mathcal{E}_{X}-$ AND $\mathcal{E}_{X}^{\mathbb{R}}$-MODULES

KIYOOMI KATAOKA

Department of Mathematical Sciences, University of Tokyo

3-8-1 Komaba, Meguro-Ku, Tokyo 153 JAPAN

Kaehiwara-Schapira originated the micr0-support theory for complexes of

ar-bitrary sheaves

on

$C^{1}$-differentiable manifolds in 1980. They applied successfully

this theory to the microlocal analysis of $D_{X}$-modules. For example, let $\mathcal{M}$ be a

coherent $D_{X}$-module

on a

complex manifold $X$, and set the holomorphic solution

complex $\mathcal{F}$ by

$F=\mathbb{R}\mathcal{H}om_{D_{X}}(\mathcal{M};O_{X})$

.

Then, the microfunction solution complex for $\mathcal{M}$ is expressed

ae

$\mathbb{R}\mathcal{H}om_{D_{X}}(\mathcal{M};C_{M})=\mu_{M}(F)$

.

Here $\mu_{M}$ is the microlocalization functor along $M$, which operates

on

complexes

of any sheaves

on

$X$

.

Many results

on

the microfunction solution complex are

de-rived from the estimation of the microsupport of $\mathbb{R}\mathcal{H}om_{D_{X}}(\mathcal{M};C_{M})$ from above ;

for example, the vanishing

or

the propagation results

on

microfunction solutions.

The

essence

of the microsupport theory is the following: Find the microsupport of

the most fundamentsal complex like $F$

.

Then, give

an

estimation of the complex derived from $F$ from above by using

some

estimation formulas concerning

micr0-supports for several functors. Indeed, Kaehiwara proved that the microsupport of

$F$ coincides with the classical characteristic variety of $\mathcal{M}$ (we call this result as the generalization of Cauchy-Kovalevski theorem). Hence many results

on

PDE reduce to functorial and geometric arguments

on

the estimation of microsupports;

for example, the solvability and propagation results in hyperfunction theory for

hyperbolic

or

microhyperbolic equations.

数理解析研究所講究録 1211 巻 2001 年 120-121

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On the other hand, this beautiful and strong theory has

a

weak point

con-cerning treating $\mathcal{E}_{X}-$

o

$\mathrm{r}$ $\mathcal{E}_{X^{-}}^{\mathrm{R}}$ equations. This is because the sheaf $O_{X}$ is neither $\mathcal{E}_{X}-$

no$\mathrm{r}$ $\mathcal{E}_{X}^{\mathbb{R}}$-modules. Kashiwara-Schapira gave

an

approach to

overcome

this difficulty

by using more abstract theories; the microlocalization ofcategories

or

the theory of

Ind-sheaves. Here, we introduce another approach by using

more

explicit methods.

That is, we construct explicitly

$F=\mathbb{R}\mathcal{H}om_{\mathcal{E}_{X}}(\mathcal{M};\mathcal{O}_{X})$

as a complex of abelian groups, where

we

abandon the usual sheaf theory. At

the same time we define

a

generalization of microsupports for $F$

as a

conic closed

subset of $T^{*}X$. Such

an

approach is not unusual because

we

often

use

Martineau’s

definition ofhyperfunctions by using formal

sums

ofdefining holomorphic functions

in the elementary lecture of microlocal analysis. There, only abelian groups of

formal

sums

of holomorphic functions appear. Owing to

a

development of

FBI-theory,

we can

prove almost all basic theorems concerning microlocal analysis only

by this naive method. We will show here that we

can

extend this method to the

theory of microsupports for $\mathcal{E}_{X}-$

o

$\mathrm{r}$ $\mathcal{E}_{X}^{\mathbb{R}}arrow \mathrm{m}\mathrm{o}\mathrm{d}\mathrm{u}1\mathrm{e}\mathrm{s}$.

参照

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