数理解析研究所講究録983 巻 1997 年 76-85

76

DIFFERENTIABLE SINGULARITIES OF SOLUTIONSOF MICRODIFFERENTIAL EQUATIONS WITH
DOUBLE INVOLUTIVE CHARACTERISTICSYASUNORI OKADA

1. INTRODUCTION

We study a class of microdifferential equations with double involutive

characteristics in the spaces of micro-distributions.
Let $M$ be a real analytic manifold with complexification $X$ . We
consider the operatorof the form $P=P_{1}P_{2}+lower$ order with
$P\in \mathcal{E}_{X}$simple characteristic operators$\dot{q}\in \mathrm{C}\mathrm{h}\mathrm{a}\mathrm{r}(P1)\mathrm{n}\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{r}(P_{2})\cap T_{M}^{*x}$which are real valued on
canonical 1-form$\omega$$T_{M}^{*}X$$P_{j}’ \mathrm{s}$at a double characteristic point. Herehave principal symbols$P_{j}’ \mathrm{s}$and satisfy$dp_{1}$A$dp_{2}$$p_{j}’ \mathrm{s}$A $\omega\neq 0$ for the.

Note that the structure of the microdifferential equation generated
by $P$ is completely studied outside of Char$(P_{1})\cap \mathrm{C}\mathrm{h}\mathrm{a}\mathrm{r}(P_{2})$et al. [6]. On the double characteristic points Charby M. Sato$(P_{1})\cap \mathrm{C}\mathrm{h}\mathrm{a}\mathrm{r}(P_{2})$,N. Tose $[7, 8]$ treats the analytic singularities by using the method of
second microlocalization, and G. Uhlmann [9] treats the differentiablesingularities by constructing a parametrix.

Under additional assumption of Levi condition, we get(I) the solvability of the equation$Pu=v$ ,in the spaces of micro-distributions.

77(II) the relation between the micro-differentiable singularities of micro-

distribution solutions of$Pu=0$ ,and the Hamiltonian flows ofand$\sigma(P_{1})$$\sigma(P_{2}))$We use Bony-Schapira’s calculus (see [2]), the notion of simple sheet
operators (see [1]), and the relation between microfunctions with holomorphic parameters and their defining holomorphic functions (see [5]and [4] .
$)$2. MAIN THEOREMS
Let $M$ be a real analytic manifold with a complexification $X,$

the projection $T_{M}^{*}Xarrow M=T_{M}^{*}M$ , and$\mathrm{S}\mathrm{p}_{M}$be$\pi_{M}$be the spectrum map: $\pi^{-1}B_{M}arrow C_{M}$ ,$\mathrm{S}_{\mathrm{P}_{M}M}$

where

is the sheaf of Sato’s hyperfunctions defined on $M$ and$B_{M}$

is the sheaf of microfunctions defined on

We introduce the subsheaves$C_{M}^{f}$and$T_{M}^{*}X$$C_{M}^{d}$$D_{M}’$

and$C_{M}^{\infty}$$B_{M}$. Sections of$C_{M}^{f}$(resp.

$C_{M}^{d}$$V$$D_{M}’$$C_{M}^{f}$and$u\in C_{M}^{f}|_{V}$$\mathrm{W}\mathrm{F}_{V}^{2}(u)$

of

$u$

and

$C_{M}^{d}$be a regular involutive submanifold in

Then for any
front set,

$C_{M}^{\infty}$

are subsheaves) are called tempered microfunctions(resp. differentiable microfunctions).

Let

by

are the sheaf of distributions and that of differen-

tiable functions respectively. These sheavesof$C_{M}$,$C_{M}^{d}=\mathrm{s}_{\mathrm{P}}M(\pi_{MM}-1C^{\infty})$

where

.

of

$C_{M}^{f}=\mathrm{s}\mathrm{p}M(\pi_{M}^{-1}D_{M}’)$$C_{M}$are conically soft.

$\dot{T}_{M}^{*}X=T*XM\backslash T_{M}^{*}M$.

, we can define the tempered second wave

along

$V$

, which is a closed biconic subset of

78$\dot{T}_{V}\dot{T}_{M}^{*}X=T_{V}\dot{T}_{M}^{*}X\backslash T_{V}V$

We also define subsheaves. Refer to G. Lebeau [3] for the definition.

and$A_{V}^{2,f}$$A_{V}^{2,d}$of$C_{M}^{f}|_{V}$by,$A_{V}^{2,f}(U)=\{u\in C_{M}^{f2}|_{V}(U);\mathrm{W}\mathrm{F}(Vu)=\emptyset\}$$A_{V}^{2,d}(U)=\{u\in C_{M}^{d}|_{V}(U);\mathrm{W}\mathrm{F}_{V}2(u)=\emptyset\}$,for any open subset $U\subset V$ .
Let$\dot{q}$be a point in$\dot{T}_{M}^{*}X$. We consider a microdifferential equation(2.1)$Pu=v$

and(2.2)$Pu=0$

at , where $P$ is a microdifferential operator satisfying the following$\dot{q}$conditions.
(A.1) There exist two regular involutive submanifolds$T^{*}X$$V_{1}^{\mathbb{C}}$and$V_{2}^{\mathbb{C}}$, which satisfy(A. 1.1) the characteristic variety Char $(P)\subset T^{*}X$ coincides with
$V_{2}^{\mathbb{C}}$(A.1.2)(A.1.3)of,$\mathrm{c}\mathrm{o}\dim_{\mathbb{C}}V_{1}^{\mathbb{C}}=\mathrm{c}\mathrm{o}\dim \mathbb{C}V_{2}\mathbb{C}=1$$V_{1}^{\mathbb{C}}$$V_{1}^{\mathbb{C}}\cup$and$V_{2}^{\mathbb{C}}$,intersect transversally along$V^{\mathbb{C}}=V_{1}^{\mathbb{C}_{\cap V}\mathbb{C}}2$

’

which

is a regular involutive submanifold of complex codimension 2

containing ,$\dot{q}$(A.1.4) $P$ is simple characteristic along Char$(P)\backslash V^{\mathbb{C}}$, i.e., the prin-cipal symbol $p=\sigma(P)$ vanishes exactly up to order 1 along$V_{1}^{\mathbb{C}}\backslash V^{\mathbb{C}}$and$V_{2}^{\mathbb{C}}\backslash V^{\mathbb{C}}$,79(A.1.5)$V_{1}^{\mathbb{C}}$and$V_{1}^{\mathbb{C}}$and$dp_{1}\neq 0$

are real, i.e., we can take defining functions$V_{2}^{\mathbb{C}}$$p_{2}$

of$V_{2}^{\mathbb{C}}$which are real valued onsymbol of (ord $P-1$ )’$V^{\mathbb{C}}$of, and satisfy

and $dp_{2}\neq 0$ .

(A.2) $P$ satisfies the Levi condition along
along$T_{M}^{*}X$$p_{1}$$\mathrm{t}\mathrm{h}$$V^{\mathbb{C}}$. That is, in this case, theorder part of$P$

vanishes up to order 1.

Let us set$V_{1}=V_{1}^{\mathbb{C}}\cap T_{M}^{*x}$,$V_{2}=V_{2}^{\mathbb{C}}\mathrm{n}T_{M}*x$,$V=V^{\mathbb{C}}\cap T_{M}*x$.

Then, from (A.1.2), (A.1.3) and (A.1.5), we findregular involutive submanifolds in$T_{M}^{*}X$$V_{1},$$V_{2}$

and

$V$

are

of real codimension 1, 1 and 2

respectively.

Let

$q$

be a point ofthrough , and by
$q$

$q$

$V$

. We denote by$b_{q}^{\gamma}(j=1,2)$. We also set, for a subset$\hat{K}=${$q\in b_{q}$the bicharacteristic leaf of

the bicharacteristic curve of$K\subset b_{q}$; both$b_{q}$$b_{q}^{1}$$V_{j}$$V$

through,and$b_{q}^{2}$

intersect with $K$ }.

Our main results are:
Theorem 2.1. Let

for$C_{M}^{f}$anda solution$u$$C_{M}^{d}$$P$

be as above. Then the equation (2.1) is solvable. That is, if$v$

belongs toof (2.1) belonging to

Theorem 2.2. Let$P$$C_{M,q}^{f}$$C_{M,q}^{f}$(resp.

(resp.
$C_{M,q}^{d}$$C_{M,q}^{d}$), there exists).

be as above. There exists a neighborhood$U\subset b_{q}$of with the following property. Assume that is a continuous map
from the unit square $I\cross I(I=[0,1])$ to satisfying$\dot{q}$$\varphi$

$U$

80

$\bullet$$\varphi(I, t)$

is bicharacteristic curve of$V_{1}$$\bullet$$\varphi(s, I)$is bicharacteristic curve of$V_{2}$and that$u\in C_{M}^{f}(\varphi(I\mathrm{x}I))$

solutions$u_{1}\in A_{V_{1}}^{2,f}(\varphi(I\cross I))$for any ,for any ,$t$$s$

is a solution of (2.2). Then there exist

and$u_{2}\in A_{V_{2}}^{2,f}(\varphi(I\mathrm{X}I))$of (2.2) which

satisfy$u=u_{1}+u_{2}$ .

Moreover if
property$\dot{q}\not\in \mathrm{W}\mathrm{F}(u)$$\dot{q}\not\in \mathrm{W}\mathrm{F}(u_{1})$, the preceding

and$\dot{q}\not\in \mathrm{W}\mathrm{F}(u_{2})$$u_{1}$

and$u_{2}$

can be chosen with the.

Theorem 2.3. Let $P$ be as above. There exists a neighborhood$U\subset b_{q}$of with the following property. Assume that is a continuous map
from the unit square $I\cross I(I=[0,1])$ to satisfying$\dot{q}$$\varphi$$U$$\bullet$$\varphi(I, t)$

is bicharacteristic curve of$V_{1}$$\bullet$$\varphi(s, I)$is bicharacteristic curve of$V_{2}$and that$u\in C_{M}^{f}(\varphi(I\cross I))$$\varphi(\mathrm{o}, 0),$for any ,for any ,$t$$s$

is a solution of (2.2). Then we have$\varphi(1,0),$ $\varphi(0,1)\not\in \mathrm{W}\mathrm{F}(u)\Rightarrow\varphi(1,1)\not\in \mathrm{W}\mathrm{F}(u)$.

Corollary 2.4. Let $P$ be as above. There exists a neighborhoodof$\dot{q}$$U\subset b_{q}$satisfying the following property. Let $K$ be an arbitrary connectedcompact subset of

$U$

with

$\hat{K}\subset U.$

Then, for any solution$u\in C_{M}^{f}(\hat{K})$of (2.2), we have$\mathrm{W}\mathrm{F}(u)\cap K=\emptyset\Rightarrow \mathrm{W}\mathrm{F}(u)\cap\hat{K}=\emptyset$.

3. GOURSAT PROBLEM IN COMPLEX FLAT DOMAINS
We prove the theorems in the previous section by the well-posednessof the Goursat problem of micro-differential equation in suitable flat

81

domains, by using the Bony-Schapira’s actions of micro-differential op-

erators to holomorphic functions. See Bony and Schapira [2].
In this section we assume that

and

$M=\mathbb{R}^{n}$$x=(x_{1}, x2, \ldots, xn)\in M$,$z=(_{Z_{1},Z}2, \ldots, zn)=X+\sqrt{-1}y\in X$$(x, \sqrt{-1}\epsilon\cdot d_{X})\in T_{M}^{*}X$

We take, using these coordinates,with$V_{j},$

with coordinates$X=\mathbb{C}^{n}$,$\xi=(\xi_{1}, \xi_{2}, \ldots, \xi_{n})$

and

$V$

$\dot{q}$as$(j=1_{\gamma}2)$$V_{j}=\{(x, \sqrt{-1}\xi\cdot dx)\in\tau_{M}^{*}x;\xi_{j}=0\}$$V=V_{1}\cap V_{2}=\{\xi 1=\xi_{2}=0\}$$\dot{q}=(0;\sqrt{-1}dX_{n})\in V$

.

,

,

.

Let $P=D_{1}D_{2^{-}}D_{1}A-D_{2}B-C$ be a microdifferential operator definedin a neighborhood of $\dot{q}=(0;\sqrt{-1}dx_{n})\in T_{M}^{*}X$ , where $A,$ $B$ and $C$ aremicrodifferential operators of order$\leq 0$.

We consider the Goursat problem$(D_{1}D_{2^{-}}D_{1}A_{\Sigma}-D_{2}B_{\Sigma}-C_{\Sigma})f=g$

(3.1)

$\{$

$f|_{z_{1}=0}=0$

for a given function
where$B_{\Sigma}$$\Sigma$and$g\in \mathcal{O}(\Omega)$

is a hyperplane$C_{\Sigma}$,$f|_{z_{2}=0=}\mathrm{o}$and an unknown function$\{z_{n}=\dot{\sigma}\},$$\Omega$

is a suitable flat domain, andare the Bony-Schapira’s action of $A,$

Moreover, for a holomorphic function$\ell$$f\in \mathcal{O}(\Omega)$$f\in \mathcal{O}(\Omega)$, and an arbitrary positive valued function$\lambda$

$B$

and

$C$

$A_{\Sigma}$,,.

, non-negative integer

defined in$\Omega$depending82

only on$z_{n}$, we set$d’(z)=,$$\inf_{)zn\not\in\Omega}\sup_{=}(\tilde{z};_{z}’|\tilde{z}_{i}-Z_{i}|\}i1,2$”$d”(Z)=,, \inf_{\prime,\prime\Omega 3\leq\leq n}\sup_{i}|(z\overline{z}zn)\not\in-1\tilde{z}_{i}-Z_{i}|$$\lambda(z_{n})*=\sup_{0\leq t\leq 1}\lambda(tZn+(1-t)\dot{\sigma})$,$||f||_{\lambda}= \sup_{z\in\Omega}\frac{|f(z)|}{\hat{\lambda}(z_{n})d’(_{Z})-1d’’(z)-1}$$||f||_{\lambda,l}= \sup_{+|\alpha|\alpha_{n}\leq\ell}||D^{\alpha}f||_{\lambda}$

Theorem 3.1. Let $P$ be as above,$\Omega$,,.

be sufficiently small, andarbitrary positive valued function defined in$\Omega$$G_{\lambda}(\Omega)$

and$G_{\lambda}^{\infty}(\Omega)${$G_{\lambda}$and$z_{n}$$G_{\lambda}^{\infty}$.
.

are vector spaces defined by$G_{\lambda}(\Omega)=\{f\in O(\Omega);||f||\lambda<\infty\}$$G_{\lambda}^{\infty}(\Omega)=$be andepending only on

Then the Goursat problem (3.1) is well-posed for classHere$\lambda$$f\in \mathcal{O}(\Omega);||f||_{\lambda,\ell}<\infty$,

for any }.
$l$In the proof of this theorem, we use the notion of simple sheet operators due to Bony [1]. We omit the details.

4.

$\mathrm{P}\mathrm{R}\mathrm{o}\mathrm{O}\mathrm{F}$OF THE RESULTS

In this section, we give a sketch of proof of our results.

By a suitable real (quantized) contact transformation, we can reduce

our problem to the case
$M=\mathbb{R}^{n}\mapsto X=\mathbb{C}^{n}$,$V_{1}=\{\xi_{1}=0\},$ $V_{2}=\{\xi 2=0\},$ $V=\{\xi_{1}=\xi_{2}=0\}$$\dot{q}=(0;\sqrt{-1}dx_{n})$

.

,

83

Moreover from the assumption of Levi condition, we deduce$P=Q(D_{1}D_{2}-D1A-D_{2}B-^{c)}$with an elliptic operator$Q$and operators $A,$ $B,$$C$of orderaccount that an elliptic operator operates bijectively on$A_{\#}^{2,*}$$\leq 0$$C_{M}^{f},$. Taking$C_{M}^{d}$and, we may assume, from the beginning, that
$P=D_{1}D_{2^{-}}D_{1}A-D_{2}B-C$ .

The solvability (Theorem 2.1) and the decomposition of solutions ofhomogeneous equations (the first half of Theorem 2.2) is a easy conse-

quence of solvability of the Goursat problem in the previous section.
The last half of Theorem 2.2 can be proved by using the uniquenessof the Goursat problem.
Now we give the proof of Theorem 2.3. From Theorem 2.2, we cantake $u_{j}’ \mathrm{s}(j=1,2)$ with
$u=u_{1}+u_{2}$$u_{j}\in A_{V_{j}}^{2,f}(L)$,$(j=1,2)$ ,$\varphi(0,0)=(0;\sqrt{-1}dx_{n})\not\in \mathrm{W}\mathrm{F}(u_{j})$We give a remark that each$u_{j}$$(j=1,2)$ .

is a restriction of a tempered microfunc-tion with a holomorphic parameter$z_{j}$

to $\{{\rm Im} z_{j}=0\}$ . Thus, from the

propagation theorem of differentiable singularities along holomorphic
parameters (refer to [1]. See also [4]), we have$\varphi(1,0)\not\in \mathrm{W}\mathrm{F}(u_{1})$,$\varphi(0,1)\not\in \mathrm{W}\mathrm{F}(u_{2})$.

84

Since these two points are also outside of$\mathrm{W}\mathrm{F}(u)$, the estimates$\varphi(1, \mathrm{O})\not\in \mathrm{W}\mathrm{F}(u-u_{1})=\mathrm{W}\mathrm{F}(u_{2})$,$\varphi(0,1)\not\in \mathrm{W}\mathrm{F}(u-u_{2})=\mathrm{W}\mathrm{F}(u_{1})$,andfollow. Applying the propagation theorem again, we get$\varphi(1,1)\not\in \mathrm{W}\mathrm{F}(u_{2})$,$\varphi(1,1)\not\in \mathrm{W}\mathrm{F}(u_{1})$,which concludes the desired result.

REFERENCES[1] Bony, J.M., Propagation des singularit\’es diff\’erentiables pour une classed’op\’erateurs diff\’erentiels \‘a coefficients analytiques, Ast\’erisque, Soc. Math. deFrance, 34-35 (1976) 43-92.
[2] Bony, J.M. and P. Schapira, Propagation des singularit\’es analytiques pour lessolutions des \’equations aux d\’eriv\’ees partielles, Ann. Inst. Fourier, Grenoble,

26 (1) (1976) 81-I40.
[3] Lebeau G. Deuxi\‘eme microlocalisation \‘a croissance, S\’eminaire Goulaouic-

Meyer-Schwartz 1982-83, Expos\’e 15.
[4] Okada, Y. Second microlocal singularities of tempered and Gevrey classes, J.

Fac. Sci., Univ. Tokyo, Sect. IA, Math., 39 (1992), 475-505.
[5] Okada, Y., and N. Tose, FBI transformation and second microlocalization-

Equivalence between second analytic wave front sets and second singular spec-

trums, J. Math. Pures Appl. 70 (1991), 427-453.

[6] Sato, M., T. Kawai, and M. Kashiwara, Microfunctions and pseudo-differential
equations Lect. Notes in Math., 287 Springer, (1973) 265-529.

[7] Tose, N. On a class of microdifferential equations with involutory double characteristics, J. Fac. Sci., Univ. Tokyo, Sect. IA, Math., 33 (1986), 619-634.

85[8] –The 2-microlocal canonical form for a class of microdifferential equations and propagation of singularities, Publ. of RIMS, Kyoto Univ., 23 (1987),

101-116.
[9] Uhlmann, G. Pseudodifferential operators with double involutive characteristics, Comm. in P.D.E. 2 (1977), 713-779.

DEPARTMENT OF MATHEMATICS AND INFORMATICS, FACULTY OF SCIENCE,CHIBA UNIVERSITY, 1-33 YAYOI-CHO, INAGE-KU,$E$-mail address: okada@math. s. chiba-u. ac. jp$\mathrm{c}_{\mathrm{H}\mathrm{l}\mathrm{B}\mathrm{A}},$

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