Cauchy Problems for Sheaves and its Applications(Study of Partial Differential Equations by means of Functional Analysis)

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Cauchy

Problems for

Sheaves

and

its

Applications

* _{HIROSHI KOSHIMIZU}

$J^{\mathrm{J}}-/_{\mathrm{H}^{p}\mathrm{K}}^{\mathrm{f}}/_{\urcorner}l^{\backslash }\mathfrak{q}\tau\lceil-$ $/,\cdot*\backslash *\ovalbox{\tt\small REJECT} \mathrm{n}$

** KIYOSHI TAKEUCHI

1 Introduction

–The study of the solvability of partial differential operators has a long history.

When the operator is simple characteristic, Nirenberg-neves [13], Kawai [12],

S-K-$\mathrm{K}[15]$ studied the local solvability very precisely. But if the characteristic variety

ofthe operator has singular points, this problem becomes

more

difficult. One of the

advantage of the employment of the hyperfunction theory is that we can sometimes

treat the operators with multiple characteristics very neatly. For example,

Bony-Schapira [1] showed that the Cauchy problems for general hyperbolic operators are

always solvable in the framework of the hyperfunction theory and such operators

are solvable in the sheaf$B_{M}$ ofSato’shyperfunctions. This result has been extended

by many authors (Kashiwara-Kawai [6], Kashiwara-Schapira [7], Kaneko [4], Oaku

[14]$)$ and

now we

have a general theory for micro-hyperbolic systems ([7] and [8]).

Bony-Schapira also proved the solvability ofpartially elliptic operators in the sheaf

$C_{M}$ of Sato’s microfunctions in another paper [2].

Inthis talk,we provethe solvabilityofaclass ofoperators whichare not (micro-)

hyperbolic nor partially elliptic by making use of the theory of bimicrolocalization

developed in [18] and [20]. Let $M=M’\cross M^{\prime J}$ be a product of two real analytic

manifolds and$X=X’\cross X’’$ acomplexification of$M$. We denote by$D_{X}$ (resp. $D_{X’}$)

the sheaf of ring of holomorphic differential operators on $X$ (resp. $X’$). Then we

prove the following theorem in Section 4.

Theorem: Let $E$ (resp. $Q$) $\in D_{X’}$ be

an

elliptic differential operator (resp. a

hyperbolic differential operator with constant coefficients)

on

$X’$ and set :

$P:=E\cdot Q+$ (lower order terms) (1.1)

by taking arbitrary lower order terms from $D_{X}$. Then

(i) $P:B_{M}arrow B_{M}$ is surjective.

(ii) $P:C_{M}arrow C_{M}$ is surjective at any $p\in\dot{T}_{M}^{*}X$.

Department of Mathematical Sciences, University ofTokyo

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CauchyProbl$\mathrm{e}ms$ forSheaves and its Application$s$

Note that Kashiwara-Kawai [6] showed that $P:=E\cdot Q+(1\mathrm{o}\mathrm{W}\mathrm{e}\mathrm{r})$ issolvable in$B_{M}$

when$X”$ reduces toa point and $Q$ is anarbitrary hyperbolic operator (Theorem 6.5

of [6]$)$. Ourtheorem above canbe considered as a relative(secondmicrolocal) version

of the theorem of [6]. Therefore in Section 2, we give

a

new

and purely algebraic

proofof their theorem generalizing it to systemsofpartialdifferential equations. We

prove the theorem above by combining this reduction with the solution to theCauchy

problem in the sheaves ofbimicrofunctions. Roughly speaking, weperform akind of

“blow-up” along the singular locus of the characteristic variety $\{\sigma(P)=0\}\subset T^{*}X$

of $P$ to separate the partially elliptic factor _{$\{\sigma(E)=0\}$} and the hyperbolic one

$\{\sigma(Q)=0\}$. The theory of [18] and [20] and the flabbiness of the sheaf $C_{ML}$ of

second microfunctions proved in Kataoka-Tose [11] will be essentially used in its

proof.

In Section 5, we also prove the solvability in the sheaf $\hat{B}_{N|\Omega}$ of mild

hyperfunc-tions for microlocally semi-hyperbolic differentialoperators (see Definition 5.7) and

extend a general theorem ofOaku [14] on the solvability of homogeneous boundary

value problems (Theorem 3 of [14]) to inhomogeneous cases. For this purpose, we

generalize in Theorem 5.4 a well-known result $‘\zeta \mathrm{t}\mathrm{h}\mathrm{e}$ hyperfunction solutions to

non-characteristic differential equations are always mild” of Kataoka [9] to systems and

to higher cohomologies at the same time.

2 Basic ideas

to

prove the solvability for

$D_{X}$

-modules

–Let $M$ be a real analytic manifold and _{$N=\{x_{1}=0\}$} its closed submanifold of

codimension one. We denoteby $Y\subset X$ acomplexification of$N\subset M,$ $B_{M}$ the sheaf

of Sato’s hyperfunctions on $M,$ $C_{M}$ and $C_{N|X}$ the sheaves of microfunctions ofSato

[15] which are associated to $M$ and $N$ respectively.

We will generalize a theorem on the solvability of single differential equations

(Theorem 6.5 of Kashiwara-Kawai [6]) to the systems ofdifferential equations, that

is, to $D_{X}$-modules.

Theorem 2.1 Let $\mathcal{M}$ \’oe a coherent _{$D_{X}$}-module for which _$Y$ is nonchara_$ct$eris_$tic$.

Assume that $\mathcal{M}$ ismicro-hyperbolic in the $\mathrm{d}i\mathrm{r}ecti_{\mathit{0}}\mathrm{n}s\pm dx_{1}\in\dot{T}_{\mathit{1}\mathrm{v}^{\mathit{1}}}^{*}\iota/I$ on $N\cross_{M}\dot{\tau}_{f_{V}I}^{*x}$.

Then we have the vanishing of cohomologies :

$\mathcal{E}x\rho_{Dx}(\mathcal{M}, B_{M})|_{N}\simeq 0$ for $j>d=proj.\mathrm{d}i\mathrm{m}\mathcal{M}_{Y}$. (2.1)

Proof$\cdot$

First of all, there is a $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma \mathrm{u}\mathrm{i}_{\mathrm{S}}\mathrm{h}\mathrm{e}\mathrm{d}$triangle:

$R\mathcal{H}om_{D_{X}}(\mathcal{M}, \mathcal{O}_{X})|_{\mathrm{J}}\vee I^{arrow R\mathcal{H}}omD_{X}(\mathcal{M}, B_{\mathit{1}\mathrm{v}I})arrow R\dot{\pi}_{l\prime I*}R\mathcal{H}om_{D\mathrm{x}}(\mathcal{M}, C_{M})arrow+1$,

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CauchyProblems for Sheaves and its Applications

where$\dot{\pi}_{M}$ : $\dot{T}_{M}^{*}Xarrow M$is the natural projection. By$\mathrm{C}\mathrm{a}\mathrm{u}\mathrm{c}\mathrm{h}\mathrm{y}-\mathrm{K}\mathrm{o}\mathrm{W}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{v}\mathrm{s}\mathrm{k}\mathrm{i}-\mathrm{K}\mathrm{a}\mathrm{S}\mathrm{h}\mathrm{i}_{\mathrm{W}\mathrm{a}}\mathrm{r}\mathrm{a}’ \mathrm{s}$

theorem, $\mathcal{E}x\rho_{Dx}(\mathcal{M}, \mathcal{O}_{X})|_{N}\simeq 0$ for $j>d$. Hence it is enough to show for any

open subset $U\subset N$

:

$H^{j}(U;R\dot{\pi}_{M*}R\mathcal{H}om_{D_{X}}(\mathcal{M}, C_{M})|_{N})=0$ for $j>d$. (2.3)

Thanks to the micro-hyperbolicity of$\mathcal{M}$ andthe division theorem of

Kashiwara-Kawai [5], the complex $\mathrm{R}\Gamma(U;R\dot{\pi}M*R\mathcal{H}om_{D_{X}}(\mathcal{M}, C_{M})|_{N})$ is a direct summand

of the complex $\mathrm{R}\Gamma(\dot{\tau}_{N}^{*}Y;R\mathcal{H}omD_{Y}(\mathcal{M}_{Y}, C_{N}))$ Hence the assertion follows from

the flabbiness of the sheaf$C_{N}$ of the microfunctions on the initial hypersurface N. $\blacksquare$

Corollary 2.2 (Theorem 6.5of [6]) Let $E$ (resp. $Q$) $\in D_{X}$ bean ellipticdifferenti$\mathrm{a}l$

opera$tor$ (resp. a$hyp$

er

bolic opera$tor$in the $d\mathrm{j}_{T\mathrm{e}Ct}ions\pm dx_{1}\in\dot{T}_{N}^{*}M$) on$X$, and set

$P=E\cdot Q+$ (loweT order $t$erms). Then the coherent $D_{X}$-module $\mathcal{M}=D_{X}/D_{X}P$

satisfies the assumptions of Theorem 2.1 and $P:B_{M}arrow B_{M}$ is surjective.

3 Cauchy problems

in bimicrofunctions

–In this section, we essentially employ the terminology of [8] and [18]. Let $X\supset$

$L\supset M$ be a sequence of $C^{\infty}$-manifolds. Let us denote it by (X,_$L,$_$M$) and call it a

triplet ofmanifolds. First recall the construction of the binormal deformation of$X$

along $(L, M)$ in [18]. We shall denote itby $\overline{X}_{ML}$ andlet

$t,$ $s\in \mathrm{R}$ be the deformation

parameters. Then we have the commutative diagram below:

(3.1)

By the immersion $s_{X}$ in (3.1), $T_{M}L\cross_{L}T_{L}X$ is identified with $\tilde{X}_{ML}\cap\{t=s=0\}$.

Ifwe choose a local coordinate system $x=(x’, x\prime\prime,\prime J\prime x)$ of$X$ such that $\{$

$L=\{x’=0\}$

$M=\{x’=0, X’’=0\}$, (3.2)

then the morphism $p_{X}\mathrm{i}\mathrm{n}(3.1)$ is described by

$(x’, x”, X^{\prime\prime J}, t, S)-(tsx’, t_{X’’}, X)\prime\prime\prime$ . (3.3)

Let $\mathrm{D}^{\mathrm{b}}(*)$ be the derived category of $\oplus_{-}$ vector spaces on a topological space with

bounded cohomologies. In [18] we defined the functor of bispecialization $\nu_{ML}$ :

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Cauchy Problems for Sheavesan$\mathrm{d}$ its Application

$s$

$\nu_{ML}(F):=\mathit{8}_{X}^{-1}Rj\mathrm{x}*\tilde{p}^{-1}\mathrm{x}F$. (3.4)

We also defined two functors

$\{$

$\nu\mu_{ML}$ : $\mathrm{D}^{\mathrm{b}}(X)arrow \mathrm{D}^{\mathrm{b}}(\tau_{M}L\mathrm{X}_{L}\tau_{L^{*}}X)$

$\mu_{ML}$ : $\mathrm{D}^{\mathrm{b}}(X)arrow \mathrm{D}^{\mathrm{b}}(T_{M}^{*}L\cross_{L}T_{L^{*}}X)$ (3.5)

as

the Fourier-Sato transformations of $\nu_{ML}(*)$. For $F\in \mathrm{D}^{\mathrm{b}}(X),$ _{$\nu_{M}L(F),$} _{$\nu\mu_{ML}(F)$}

and $\mu_{ML}(F)$ are biconic objects. We can give an estimation of the support of the

complex $\mu_{ML}(F)$ (Funakoshi [3]) :

$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\mu_{M}L(F)\subset\tau_{(x}^{*}*)(M\mathrm{X}_{L}T_{L}xT_{L}^{*})\cap C_{T_{L}^{*}\mathrm{x}}(\mathrm{S}\mathrm{S}(F))$ , (3.6)

where we used the natural isomorphism $T_{M}^{*}L\cross_{L}\tau_{L}^{*}X\simeq\tau_{(M\mathrm{x}_{L}}*(\tau_{L}’x)\tau*x)L$ and the

Hamilton $\mathrm{i}_{\mathrm{S}\mathrm{o}\mathrm{m}\mathrm{o}}\mathrm{r}\mathrm{p}\mathrm{h}\mathrm{i}\mathrm{s}\mathrm{m}-H:\tau*(T_{L}^{*}x)\simeq T_{(T_{L}^{*}X})(\tau*x)$ .

Let $g$ : $Marrow M”$be asmoothmorphismof real analytic manifolds and $g\oplus:Xarrow$

$X”$

a

complexification of$g$. Set $L:=g_{\mathrm{G}}^{-1}(M’’)$ and assume $\dim^{\mathrm{R}}M=n$. It follows

from Kashiwara’s abstract edgeof the wedge theorem that the complex$\mu_{ML}(\mathcal{O}_{X})[n]$

on $T_{M}^{*}L\cross_{L}T_{L}^{*}X$ is concentrated in degree $0$. Hence we set :

$c_{ML}:=\mu ML(\mathcal{O}_{\mathrm{x})\mathrm{o}\mathrm{r}_{M}}\otimes[n]$ (3.7)

and call it the sheaf of second microfunctions along $\mathrm{L}$ (Kataoka-Tose [11] and [18]).

Rom now on, to the end of this section, we study the Cauchy problem in the

framework of sheaves of second microfunctions reviewed above. Let $N\subset M=$

$M’\cross M’’=\mathrm{R}^{d}\cross NI’’$ a submanifold of codimension one such that _{$g|_{N}:Narrow M’’$}

is also smooth. We denote by $\mathit{9}\oplus:X=X’\cross X’’=\oplus^{d}\cross X^{\prime/}arrow X’’$ (resp.

$(g|_{N})_{\mathbb{C}}$ : $Yarrow X”$) a complexification of

$g$ : $Marrow M”$ (resp. $g|_{N}:Narrow M’’$).

Finally set $L:=g_{\mathrm{G}}^{-1}(M’’)\subset X$ and $H:=(g|_{N})_{\mathbb{C}}^{-1}(M’’)\subset Y$. Then we have the

canonical injections :

$T_{N}^{*}L\cross_{L}\tau^{*}Lxarrow\delta(N\cross_{M}T_{M}^{*}X)\cross_{L}T_{L}^{*xarrow\tau^{*}L}\varpi \mathrm{o}M\cross_{L}\tau_{L}^{*}x$. (3.8)

The next theorem is a second microlocal version ofa result of Kashiwara-Schapira

(Theorem 6.7.1 of [8]).

Theorem 3.1 Let $\mathcal{M}$ be a coherent _{$D_{X}$}-mod$ul\mathrm{e}$. Suppose there exists a coherent $D_{X’}$-module $\mathcal{M}’$ which satisfies:

(i) $\mathcal{M}’$ is hyperbolic in the $di\mathrm{r}eCtion\mathrm{S}\pm dx_{1}\in\dot{T}_{N}^{*}M$ with constan$\mathrm{t}$ coefficients.

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Cauchy Problems forSheaves and its Applications

Then we $h$

ave

the isomorphism :

$\delta_{*}\varpi_{0}^{-1}R\mathcal{H}om_{D_{X}}(\mathcal{M}, cML)\simeq R\mathcal{H}om_{D_{X}}(\mathcal{M}, C_{NL})[1]$ . (3.9)

Since $V:=X\cross_{x^{l\prime}T^{*}}x\prime\prime$ be a regular involutive submanifold of $T^{*}X$, we can

define the natural injection:

$T_{Y}^{*}Xarrow T_{Y}^{*}X\cross_{X}Varrow T^{*}(X/X’’)\cross_{\mathrm{x}}V\simeq TV(\tau^{*}x)$ (3.10)

by using the zero-section of V. $\mathrm{W}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{S}\mathrm{O}\sim$ use the projection

$\rho_{N}$ : $T_{N}^{*}L\cross_{L}T^{*}XLarrow$

$T_{N}^{*}H\mathrm{x}_{H}T^{*}HY$.

Proposition 3.2 ([21]) Let $\mathcal{M}$ be a coherent $D_{X}$-mod$\mathrm{u}l\mathrm{e}$ which satisfies the

non-microcharacteris$tic$ condition:

$\dot{T}_{Y}^{*}X\cap C_{V}(Ch\mathcal{M})=\emptyset$. (3.11)

Then

we

have the canonical isomorph$\mathrm{i}sm$ :

$R\rho_{N*}R\mathcal{H}omD_{X}(\mathcal{M}, C_{NL})[1]\simeq R\mathcal{H}om_{D_{Y}}(\mathcal{M}_{Y}, C_{NH})$. (3.12)

By using the projection $\rho_{0}:=\rho_{N}\circ\delta$ : $(N\cross_{M}T_{M}^{*}L)\cross_{L}T_{L}^{*}Xarrow T_{N}^{*}H\cross_{H}T_{H}^{*}Y$, the next result follows from Theorem 3.$\mathrm{i}$

and Proposition 3.2 :

Theorem 3.3 Let $\mathcal{M}$ be a $coh$erent $D_{X}$-module which satisfy the conditions $(i)(ii)$

of Theorem 3.1. Then wehave the isomorphism :

$R\rho_{0*0}\varpi-1R\mathcal{H}om_{DX}(\mathcal{M}, C_{ML})\simeq R\mathcal{H}om_{D_{Y}}(\mathcal{M}_{Y}, C_{NH})$ (3.13)

on $T_{N}^{*}H\cross_{H}T_{H}^{*}Y$.

Remark 3.4 If

we

impose the conditions of Theorem 4.1 (in the following section)

on $P$, we can also prove the isomorphism in Theorem 3.1 on $\{\xi_{1}=0\}\cap(\dot{T}_{N}^{*}L\cross_{L}$

$T_{L}^{*}X)=(N\cross_{M}\dot{T}_{M}^{*}L)\cross_{L}\tau_{L}^{*}X$.

4 Solvability

of

operators

with multiple

characteristics

–Inthis section, weshallgive two results whichcan not be covered by Theorem6.5

of Kashiwara-Kawai [6]. We consider the same situation as in Section 3 and inherit

the notations in it. For example, suppose that locally $M=M’\cross M’’=1\mathrm{R}^{d}\cross \mathrm{R}^{n-d}$,

$X=X’\cross X’’=\oplus^{d}\cross\oplus^{n-d}$ _and $N=\{x_{1}=0\}=\mathrm{R}^{d-}1\cross \mathrm{R}^{n-d}\subset M$.

First weconsider the following

case

by making use of the theory of

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Cauchy Problems for Sheaves andits Applications

Theorem 4.1 Let $E,$$Q\in D_{X’}$ be differen$ti\mathrm{a}l$ operators on $X’$. Assum

$\mathrm{e}$ that $E$ is

elliptic and $Q$ is hyper\’oolic in $\pm dx_{1}$-direction with constant _$co$efficients. We set

$P:=E\cdot Q+$ ($l_{ow}\mathrm{e}r$order terms) _{$\in D_{X}$} by taking arbitrary lower order terms

from

the differential operators on the tot$\mathrm{a}l$ space X. Then

we

have:

(i) $P:B_{M}arrow B_{M}$ is $s$urjective.

(ii) $P:C_{M}arrow C_{M}$ is $s$urjective at any point in $\dot{T}_{M}^{*}X$.

Proof$\cdot$

For the sake of the simplicity, set $\dot{\Lambda}_{N}:=N\cross_{L}\dot{\tau}_{L^{*}}X$ and $\dot{\Sigma}:=(N\cross_{M}$ $\dot{T}_{M}^{*}L)\cross_{L}\dot{\tau}_{L^{*}}X$. Considering Sato’s exact sequence, it suffices to show that

$P:\Gamma(N\cross_{M}\dot{T}^{*}x_{;^{c}})MM$

. $arrow\Gamma(N\cross_{M}\dot{T}_{M}*X;c_{M})$ (4.1)

is surjective. But the local solvability in the sheaf$C_{M}$ of$P$ is trivialon $N\cross_{M}\dot{T}_{M}^{*}X-$

$\dot{\Lambda}_{N}$ by

its micro-hyperbolicity there. Hence to prove the theorem, it is enough to

show the surjectivity of the morphism $P:\Gamma(\dot{\Lambda}_{N};c_{M})arrow\Gamma(\dot{\Lambda}_{N};c_{M})$. Now consider

the following commutative diagram with exact rows :

$0_{-}\Gamma(\dot{\Lambda}_{N};C\mathcal{O})-\Gamma(\dot{\Lambda}_{N};C_{M})-\Gamma(\dot{\Sigma}; C_{M}L)-0$

$P\cross\dagger$ _$P\cross|$ $P\cross\dagger$ (4.2)

$0_{-}\Gamma(\dot{\Lambda}_{N;}C\mathcal{O})-\mathrm{r}(\dot{\Lambda}_{N;}CM)-\mathrm{r}(\dot{\Sigma}; c_{ML})-0$_,

where $C\mathcal{O}:=\mu_{L}(\mathcal{O}_{X})\otimes \mathrm{o}\mathrm{r}_{L}[n-d]$ is the sheafof microfunctions with holomorphic

parametersand the exactitude follows from the vanishing of the global cohomology

$H^{1}(\dot{\Lambda}_{N;C\mathcal{O})}$ shown by Theorem 3.1 of Kataoka-Tose [10]. Set _{$z’=(z_{1}, \ldots, z_{d})=$} $(z_{1},\hat{z})$ and $C\mathcal{O}_{\overline{z}}:=\mu_{H}(\mathcal{O}_{Y})\otimes \mathrm{o}\mathrm{r}_{H}[n-d]$. Then by Schapira’s Cauchy-Kowalevski

type theorem of [2] and [16], we have by setting $\mathcal{M}=D_{X}/D_{X}P$ :

$\mathrm{R}\Gamma(\dot{\Lambda}_{N)}.R\mathcal{H}omDX(\mathcal{M}, c\mathcal{O}))$

$\simeq \mathrm{R}\Gamma(\dot{\Lambda}_{\mathit{1}\mathrm{V};}R\mathcal{H}_{\mathit{0}}m_{D_{Y}}(\mathcal{M}Y, c\mathcal{O}_{\overline{z}}))=\mathrm{R}\Gamma(\dot{\Lambda}_{N;}c\mathcal{O}_{\overline{z}})\oplus ordP$. (4.3)

Byvirtueof Theorem 3.1 of Kataoka-Tose [10] again, it implies that the firstvertical

arrow

$P:\Gamma(\Lambda_{N;c\mathcal{O}}|_{\Lambda_{N}})arrow\Gamma(\Lambda_{\mathit{1}}\mathrm{V};C\mathcal{O}|_{\Lambda_{N}})$ (4.4)

of the diagram (4.2) is surjective. We

can

also show the surjectivity of the third

vertical arrow $P:\Gamma(\dot{\Sigma}; C_{ML})arrow\Gamma(\dot{\Sigma}; C_{ML})$ by applying the

same

arguments as in

the proof of Theorem 2.1 to (the proofs of) Theorem 3.1 and Proposition 3.2. In

this case, we require also the flabbiness of the sheaf $C_{NH}$ of second microfunctions

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CauchyProblems forSheaves and its Application$s$

Next take anelliptic operator $E$

on

$X’=\oplus_{z’}^{d}$ (that is, $\mathrm{C}\mathrm{h}(D_{X’}/D_{X’}E)\cap\dot{T}_{\mathrm{A}I’}^{*},X^{J}=$ $\emptyset)$ and a hyperbolic operator $Q\mathrm{i}\mathrm{n}\pm dx_{1}$-direction

on

the total space $x_{--}X’\cross X^{\prime/}$.

We set :

$P=E\cdot Q+$ (lower order terms). (4.5)

Ifwe consider $E$

as

a differential operator

on

$X$,

we

have ;

$\mathrm{C}\mathrm{h}(D_{X}/D_{X}E)\cap T_{M}^{*}X=M’\cross T_{M}^{*},,X’’=M\cross_{L}\tau_{L}^{*}x$. (4.6)

Theorem 4.2 Assume the separation condition :

$Ch(D_{X}/D_{X}E)\cap Ch(D_{X}/D_{X}Q)\mathrm{n}\dot{\tau}_{M}^{*}x=Ch(D_{X}/D_{X}Q)\cap(M\cross_{L}\dot{T}_{L}^{*x)=\emptyset}$. $(4.7)$

Then $P:B_{M}arrow B_{M}$ is $s$urjective.

Proof.$\cdot$ By virtue

ofSato’sexact sequence and Cauchy-Kowalevski-Kashiwara’s

the-orem, it suffices to show the surjectivity of the morphism :

$P$: $\Gamma(N\cross_{M}\dot{\tau}_{M}^{*}X;C_{M})arrow\Gamma(N\cross_{M}\dot{\tau}_{M}^{*}X;c_{M})$. (4.8)

Weknowfrom theproofof Theorem 2.1 that $P$is globallysolvableinaneighborhood

of $\mathrm{C}\mathrm{h}(D_{X}/D_{X}Q)\cap\dot{T}_{M}^{*}X$ in $\dot{T}_{M}^{*}X$. The problem is to show the surjectivity of the

morphism:

$P:\Gamma(\dot{\Lambda}N;cM)arrow\Gamma(\dot{\Lambda}_{N;}c_{M})$ (4.9)

for $\dot{\Lambda}_{N}:=N\cross_{L}\dot{\tau}_{L^{*}}X$. This is equivalent to the vanishing for the $D_{X}$-module

$\mathcal{M}:=D_{X}/D_{X}P$ :

$H^{1}\mathrm{R}\mathrm{r}(\dot{\Lambda}N;R\mathcal{H}_{om}D_{X}(\mathcal{M}, C_{M}))\simeq 0$ . (4.10)

Since the $D_{X}$-module $\mathcal{M}$ is partially elliptic along $\dot{V}=X’\cross\dot{\tau}*x\prime\prime\subset T^{*}X$ in the

sense of Bony-Schapira [2], we have the chain of isomorphisms :

$\mathrm{R}\Gamma(\dot{\Lambda}_{N};R\mathcal{H}omDX(\mathcal{M}, C_{M}))$

$arrow^{\sim}\mathrm{R}\Gamma(\dot{\Lambda}N;R\mathcal{H}om_{D_{X}}(\mathcal{M}, C\mathcal{O}))\simeq \mathrm{R}\Gamma(\dot{\Lambda}_{N;R}\mathcal{H}om_{\mathcal{E}Y}(\mathcal{E}^{\oplus d}YorE, c\mathcal{O}_{\hat{z}}))$ (4.11) $=\mathrm{R}\Gamma(\dot{\Lambda}_{N};^{c}\mathcal{O}_{\hat{z}})^{\oplus\sigma}\Gamma dE$,

where we used Schapira’s Cauchy-Kowalevski type theorem of [16] to show the

sec-ond isomorphism. The first cohomology group $H^{1}(\dot{\Lambda}_{N;}CO_{\hat{z}}\oplus ordE)$ of the last term

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Cauchy Problems for Sheaves and its Applications

5 Solvability

of

boundary value

problems

–In thissection, we apply the methods in Theorem2.1 to boundaryvalueproblems.

In particular, we extend a result of Oaku [14] to inhomogeneous boundary value

problems.

Firstwe shall recall

some

basic notions concerning boundary valueproblems. Let

$N=\{x_{1}=0\}\subset M$ be areal analytic submanifold ofcodimension

one as

before and

$\Omega=\{x_{1}>0\}\subset M$ an open subset such that $N=\partial\Omega$. We take

a

complexification

$Y\subset X$ of $N\subset M$ as usual.

Definition 5.1 $(\mathrm{S}\mathrm{c}\mathrm{h}\mathrm{a}_{\mathrm{P}^{\mathrm{i}}}\mathrm{r}\mathrm{a}[17])$ We define the complex

$C_{\Omega|X}$ by $C_{\Omega|X}:=\mu hom(\oplus\Omega$,

$\mathcal{O}_{X})\otimes \mathrm{o}\mathrm{r}_{M}[n]$, where $\mu hom(\cdot, \cdot)$ : $\mathrm{D}^{\mathrm{b}}(X)^{op}\cross \mathrm{D}^{\mathrm{b}}(X$

. $)arrow \mathrm{D}^{\mathrm{b}}(\tau^{*},x)$ is a bifunctor

introduced in [8] and $n=\dim^{\mathrm{R}}M$.

We set $M_{+}:=\{x_{1}\underline{>}0\}=\overline{\Omega}$ and we can also define Kataoka’s sheaf

$C_{M|X}+$ by

replacing $\oplus_{\Omega}$ with $\oplus_{M}+\mathrm{i}\mathrm{n}$the definition above. Recall the following results proved

by Schapira-Zampieri.

Theorem 5.2 (Schapira-Zampieri [19]) Let $T^{*}Xarrow Y\rho\cross x\tau*x\varpi-\tau*x$ _be

nat-ural $\mathrm{m}$orphisms. Then the complex$R\rho_{!}\varpi^{-1}C\Omega|\mathrm{x}$ of sheaves on $T^{*}Y$ is $co\mathrm{n}$centrated

in degree $0$ and coincides with Kataoka’s shea$f\hat{C}_{N|\Omega}$ of$\mathrm{m}il\mathrm{d}$ microfunctions ([91).

Remark 5.3 The sheaf $\hat{C}_{N|\Omega}$ is supported by

$T_{N}^{*}Y$ and $\hat{B}_{N|\Omega}:=\hat{C}_{N|\Omega}|_{T_{Y}^{*}Y}$ was

called the sheaf of mild hyperfunctions in Kataoka [9]. Since $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\hat{B}_{N|\Omega}\subset N$, we

sometimes consider it a sheaf on $N$.

The sheaf $\hat{B}_{N|\Omega}$ of mild hyperfunctions is a subsheaf of

$\Gamma_{\Omega}B_{M}|_{N}$ and we

can

explicitly take the boundary values to $N$ of mild hyperfunctions by restricting their

defining holomorphic functions in the complex domain. Kataoka [9] found that the

hyperfunction solutions $u\in\Gamma_{\Omega}B_{M}|_{N}$ of the single differential equations for which

$Y$ is noncharacteristic are always mild and made the defintion of the boundary value

more explicit. The next proposition extends this classical result to systems.

Theorem 5.4 Let $\mathcal{M}$ be a coherent _{$D_{X}$}-modulefor which _$Y$ is noncharacteristic.

Then wehave the isomorphism:

$R\mathcal{H}om_{D_{X}}(\mathcal{M}, \Gamma_{\Omega}BM)|_{N^{arrow\sim}}R\mathcal{H}mDx(\mathcal{M},\hat{B}N|\Omega)$ . (5.1)

To define the semi-hyperbolicity of differential operators, we take

a

coordinate

system $(z, \zeta),$$Z=x+iy,$_{$\zeta=\xi+i\eta$} of$T^{*}X$ such that _{$T_{M}^{*}X=\{y=0, \xi=0\}$}, and

$N=\{x_{1}=0\},$ $\Omega=\{x_{1}>0\}\subset M$. Ifwe take a point $p=(x_{0}’;i\eta’0)\in T_{N}^{*}Y\subset T^{*}Y$,

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CauchyProblems for Sheavesandits Applications

operator or

a

pseudo-differential operator defined

on

a neighborhood of $\rho^{-1}(p)\simeq$

$\oplus_{\zeta_{1}}^{1}\subset T^{*}X$.

..

Definition 5.5 (Kaneko [4], Kataoka [9]) We say $P$ is sehyperbolic (resp.

mi-crolocally semi-hyperbolic) $\mathrm{i}\mathrm{n}+dx_{1}$-direction at _$p$, if there exists a constant $\epsilon>0$

such that $p$.

$\sigma(P)(x_{1}, x’;\xi_{1}+i\eta_{1}, i\eta’)\neq 0$ (5.2)

for $0\leq x_{1}<\epsilon$, $|x’-X_{0}’|<\epsilon$, $0<\xi_{1}$ (resp. $0<\xi_{1}<\epsilon$), $\eta_{1}\in \mathrm{R}$, $|\eta’-\eta_{0}’|<$

$\epsilon$.

Remark 5.6 Asumme $p\in N\subset T_{N}^{*}Y$ and $P$ is a differential operator defined on

a neighborhood of$p\in N\subset X$. Then the definition above coincides with that of

Kaneko [4].

’

For example, the operator $Q=D_{1}^{2}-x_{1}^{k}D_{x}^{2}$,($k\in$ IN) is semi-hyperbolic $\mathrm{i}\mathrm{n}+dx_{1^{-}}$

direction at any$p\in T_{N}^{*}Y$. Ifwe take an elliptic operator $\mathrm{E}$ and set :

$P=E\cdot Q+$ (lower order terms), (5.3)

then $P$ is microlocally semi-hyperbolic at any$p\in\dot{T}_{N}^{*}Y$. As we will show later, such

operators $P$

are

solvable in the sheaf $\hat{B}_{N|\Omega}$ of mild hyperfunctions. Hence we define

a family of differential operators to include these examples.

Definition 5.7 We say $P\in D_{X}$ is microlocally semi-hyperbolic $\mathrm{i}\mathrm{n}+dx_{1}$-direction,

if $P$ is

so

at any$p\in\dot{T}_{N}^{*}Y$.

$\mathrm{R}\mathrm{e}.\mathrm{m}$ark 5.8 A similar but

stron..g

er condition

“-microloCally..

$\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{b}\vee \mathrm{o}\mathrm{l}\mathrm{i}_{\mathrm{C}}$

” was

intro-duced in [21].

Theorem 5.9 Let $P$ be a differen$\mathrm{t}i\mathrm{a}l$ operator for which $Y$ is noncharacteris$tic$.

$Ass\mathrm{u}meP$ is microl$0$callysemi-hyperbolic$in+dx_{1}$-direction. Then :

(i) $P:\Gamma_{\Omega}B_{M}|_{N}arrow\Gamma_{\Omega}B_{M}|_{N}$ is surjective.

(ii) $P:\hat{B}_{N|\Omega}arrow\hat{B}_{N|\Omega}$ is surjecti$\mathrm{r}^{r}\mathrm{e}$.

(iii) $P:\hat{C}_{N|\Omega}arrow\hat{C}_{N|\Omega}$ is surjective at any$p\in\hat{T}_{N}^{*}Y$.

Remark 5.10 The proofof (i) is reduced to show :

(10)

on $N\cross_{M}\dot{T}_{M}^{*}X$ by using the

same

arguments as in the proof ofTheorem 2.1. (5.4)

follows fromKataoka’s resultonthe solvability of semi-hyperbolic pseudo-differential

operators (Corollary 1.9 of $[9]$)

$.\cdot$ The part (ii) is a direct consequence from

$(\mathrm{i})$ ,

and

Theorem 5.4.

As an application of Theorem 5.9 (ii), we get a result which extends Theorem 3

of Oaku [14] to inhomogeneous boundary value problems.

Corollary 5.11 Let $P$ be a differenti$\mathrm{a}l$ operator for which _$Y$ is noncharacteristic.

$Ass\mathrm{u}m\mathrm{e}P$ is microlocally semi-hyperbolic _{$in+dx_{1}$}-direction and

$\#$ [$\{q\in\rho^{-}1(p)$;a$(P)(q)=0\}\mathrm{n}\{\xi_{1}\leq 0\}$] $\geq m’$ (5.5)

holds for any$p\in T_{N}^{*}Y$, where $0\leq m’\leq m=ordP$. Then there always exists amild

hyperfunction solution $u\in\hat{B}_{N|\Omega}$ to the inhomogeneous bound$\mathrm{a}ry\mathrm{b}^{r}\mathrm{a}l\mathrm{u}e$ problem:

$\{$ $Pu=f$,

$\dot{\sigma}_{1}u|_{x_{1^{arrow}+0}}=v_{j}$ $(j=0, \ldots, m’-1)$ (5.6)

for any $f\in B_{N|\Omega}$ an$d$ any$v_{j}\in B_{N}$ $(j=0, \ldots, m’-1)$.

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(11)

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