数理解析研究所講究録 1211 巻 2001 年 56-65

56

IRREGULARITIES OFMICROHYPERBOLIC OPERATORS(マイクロ双曲形作用素の非正則度)

KEISUKE UCHIKOSHI
Department of Mathematics
National Defense Academy
Yokosuka 239-8686, JAPANe-mail:uchikosh@cc.nda.ac.jp
ABSTRACT. We consider well-posedness of microhyperbolic Cauchy
problems in the category of microfunctions which are the singularity
spectrums of ultradistributions. To obtain aprecise result, we define
the irregularities of microhyperbolic operators, and prove the relation
between irregularities and ultradistribution orders.

1. Introduction.
It is well-known that amicrohyperbolic Cauchy problem is always-posed in the category of microfunctions (See [2]). Let us consider
its well-posedness in the category of microfunctions which are the singularity spectrums of ultradistributions. There is afundamental resultof Kajitani and Wakabayashi for this problem. However, there are somespecial but important cases for which their theory does not give asatisfactory result. Therefore we want to ameliorate it.
$=(x_{1}, x’)--$, and let
Let $(x, \xi)$ be the variables of$(x_{1}, \cdots,x_{n})$ .
(resp.
) beLet
the point defined by $=0,(=(0, \cdots,0, \sqrt{-1})$ (resp. $x’–0,\xi’--$, the sheaves of hyperfunc,0,0,). We denote by , ,tions, microfunctions, microdifferential operators, and holomorphic functions, respectively. For $1<s<\infty$ we denote Gevrey functions with$\mathrm{w}\mathrm{e}\mathbb{I}$$\sqrt{-1}T^{*}\mathrm{R}^{n}$$x$$x^{*}’\in\sqrt{-1}T^{*}\mathrm{R}^{n-1}$$x^{*}\in\sqrt{-1}T^{*}\mathrm{R}^{n}$

$x$

$($

$\cdots$$\sqrt{-1})$$B$$\mathrm{C}$$\mathcal{E}$$\mathcal{O}$

Typese by$\mathrm{A}_{\mathrm{A}}\beta \mathrm{B}$57
KEISUKE UCHIKOSHI

compact supports by$\mathcal{G}_{\mathrm{c}\mathrm{p}\mathrm{t}}^{\{s\}}$and:$\mathcal{G}_{\mathrm{c}\mathrm{p}\mathrm{t}}^{(s)}$is an infinitely differentiable function with, and there exists some $C$ such thatcompact support$\mathcal{G}_{\mathrm{c}\mathrm{p}\mathrm{t}}^{\{s}(\omega)--\{f(x);f$$\subset\omega$$|D^{\alpha}f(x)|\leq C^{|\alpha|+1}\alpha!^{s}\}$,is an infinitely differentiable function with, and for any
there exists somecompact support$\mathcal{G}_{\mathrm{c}\mathrm{p}\mathrm{t}}^{(s)}(\omega)--\{f(x);f$$\epsilon>0$$\subset\omega$$C_{\epsilon}$such thatfor an open subset$\omega$of$|D^{\alpha}f(x)|\leq C_{\epsilon}\epsilon^{|\alpha|}\alpha!^{s}\}$$\mathrm{R}^{n}$. Let$\mathcal{G}_{\mathrm{c}\mathrm{p}\mathrm{t}0}^{\{s\}\prime}--\lim_{arrow}\mathcal{G}_{\mathrm{c}\mathrm{p}\mathrm{t}}^{\{s\}\prime}(\omega)$,$\mathcal{G}_{\mathrm{c}\mathrm{p}\mathrm{t}0}^{(s)\prime}--\lim_{arrow}\mathcal{G}_{\mathrm{c}\mathrm{p}\mathrm{t}}^{(s)\prime}(\omega)$

OCu$0\in\omega$be the set of germs of ultradistributions at the origin.
:and
Let sp :canonical maps, and let$B_{\mathrm{R}^{n},0}arrow \mathrm{C}_{\mathrm{R}^{n},x^{*}}$,$\mathrm{C}_{\mathrm{R}^{n},x^{*}}^{\{s\}}--\mathrm{s}\mathrm{p}(\mathcal{G}_{\mathrm{c}\mathrm{p}\mathrm{t}\mathrm{R}^{n},0}^{\{s\}}’)$$\mathrm{C}_{\mathrm{R}^{n},x^{*}}^{(s)}--\mathrm{s}\mathrm{p}(\mathcal{G}_{\mathrm{c}\mathrm{p}\mathrm{t}\mathrm{R}^{n},0}^{(s)}’)$,$\mathrm{s}\mathrm{p}’$$B_{\mathrm{R}^{n-1},0}arrow \mathrm{C}_{\mathrm{R}^{n-1},x^{*\prime}}$be the$\mathrm{C}_{\mathrm{R}^{n-1},x^{*\prime}}^{\{s\}}--\mathrm{s}\mathrm{p}’(\mathcal{G}_{\mathrm{c}\mathrm{p}\mathrm{t}\mathrm{R}^{n-1},0}^{\{s\}}’)(1\leq s \leq\infty)$$\mathrm{C}_{\mathrm{R}^{n-1},x^{*\prime}}^{(s)}--\mathrm{s}\mathrm{p}’(\mathcal{G}_{\mathrm{c}\mathrm{p}\mathrm{t}\mathrm{R}^{n-1},0}^{(s)}’)(1<s \leq\infty)$,

which we call microlocal ultradistributions. For the sake of convenience’’andthe set of hyperfunctions, by
we denote by$\mathcal{G}_{\mathrm{c}\mathrm{p}\mathrm{t}\mathrm{R}^{n},\mathrm{O}}^{\{\infty\}}$$\mathcal{G}_{\mathrm{c}\mathrm{p}\mathrm{t}\mathrm{R}^{n},0}^{\{1\}}$the set of distributions. Therefore
microfunctions.
written in the formLet’$\mathcal{G}_{\mathrm{c}\mathrm{p}\mathrm{t}\mathrm{R}^{n},0}^{(\infty)}$$\mathrm{C}_{\mathrm{R}^{n},x^{*}}^{\{1\}}$is the usual set of$P(x, D)\in \mathcal{E}_{x}*\mathrm{b}\mathrm{e}$(1)$\{\begin{array}{l}P(x,D)--D_{1}^{m}+\sum_{\mathrm{o}\leq j\leq m-1}P_{j}(x,D’)D_{1}^{j}\mathrm{o}\mathrm{r}\mathrm{d}P_{j}\leq m-j(0\leq j\leq m-1)\end{array}$Here we define $D$ $–$ afii. We assume that(2)$\{\begin{array}{l}\mathrm{f}\mathrm{o}\mathrm{r}1\leq j\leq m\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{t}\Lambda_{j}(x,\xi)--\xi_{1}-\lambda_{j}(x,\xi’)\in \mathcal{O}_{\mathrm{C}^{2n}x^{*\mathrm{W}}}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{h}\mathrm{o}\mathrm{m}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{n}\xi \mathrm{o}\mathrm{f}\mathrm{d}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}1\mathrm{v}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{a}\mathrm{t}x^{*}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{w}\mathrm{e}\mathrm{h}\mathrm{a}\mathrm{v}\mathrm{e}\sigma_{m}(P)=\prod_{1\leq j\leq m}\lambda_{j}(x,\xi)\end{array}$

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IRREGULARITIES OF MICROHYPERBOLIC OPERATORS

denotes the principal symbol of $P$ . We finally assume that
where
$P$ is microhyperbolic, i.e.,$\sigma_{m}(P)$(3)$(x,\xi’)\in \mathrm{R}^{n}\cross\sqrt{-1}\mathrm{R}^{n-1}$$\lambda_{j}(x,\xi’)\in\sqrt{-1}\mathrm{R}$$\Rightarrow$for $1\leq j\leq m$ . We do not assume any further conditions explicitly
among these characteristic roots.
Let us consider the following Cauchy problem:(4)$P(x, D)u(x)=f(x)$ ,$D_{1}^{j-1}u(0, x’)=v_{j}(x’)(1\leq j\leq m)$.

Precisely speaking, in order to ascertain that $D_{1}^{j-1}u(0, x’)$ is wel-defined,. For this purpose it suffices to
we must assume that
. However, we are considering in aneighassume
extended as aborhood of , and we may assume that
, whose support does not contain $(0, \pm\sqrt{-1}dx_{1})$ .
global section of
not depend on such an extent
Since the solution
this is well-defined, and we consider (4) in this sense.
,and ,well-posed if for any
We say that $P$ is$(0, \pm\sqrt{-1}dx_{1})\not\in \mathrm{s}\mathrm{p}u$$(0, \pm\sqrt{-1}dx_{1})\not\in \mathrm{s}\mathrm{p}f$$x^{*}$$f\in \mathrm{C}_{\mathrm{R}^{n},x}*\mathrm{i}\mathrm{s}$$\mathrm{C}_{\mathrm{R}^{n}}$$u$$\in \mathrm{C}_{\mathrm{R}^{n},x}*\mathrm{d}\mathrm{o}\mathrm{e}\mathrm{s}$$\{s\}$$f\in \mathrm{C}_{\mathrm{R}^{n},x^{*}}^{\{s\}}$$v_{1}$$\cdots$, there exists
which satisfies (4) (The solution-posedness. Kajitani andis always unique). Similarly we define (s)Wakabayashi [1] proved the folowing$v_{m}\in \mathrm{C}_{\mathrm{R}^{*-1},x^{*}}^{\{s\}}.$$u$$\in \mathrm{C}_{\mathrm{R}^{r\iota},x^{*}}^{\{s\}}$$\mathrm{w}\mathrm{e}\mathrm{U}$Theorem 1. If $1\leq s<m/(m-1)$ , then $P$ is$1<s\leq m/(m-1)$ , then $P$ is (s) well-posed.

$\{s\}$

well-posed.

If

To see that we cannot generally improve this result anymore, let us
consider the folowing
Example 1. Let $P=D_{1}^{m}-D_{n}^{m-1}$ and let us consider$P(x, D)u(x)=0$ ,$D_{1}^{j-1}u(0, x’)=\delta_{j1}v(x’)(1\leq j\leq m)$

It is easy to see that the microfunction solution is given by$u(x)$ $= \frac{1}{m}\sum_{0\leq j\leq m-1}\exp(\frac{2\pi\sqrt{-1}j}{m}x_{1}D_{n}^{(m-1)/m})v(x’)$

If we restrict ourselves to microlocal ultradistributions,$\exp(\frac{2\pi\sqrt{-1}j}{m}x_{1}D_{n}^{(m-1)/m})$:$\mathrm{C}_{\mathrm{R}^{n},x^{*}}^{\{s\}}arrow \mathrm{C}_{\mathrm{R}^{n},x^{*}}^{\{s\}}$

.

.

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KEISUKE UCHIKOSHI

is well-defined if, and only if, $1\leq s<m/(m-1)$ , and Theorem 1is thebest possible result in this sense.

However, this criterion is not satisfactory for the folowing cases:
Example 2(regular involutive operators). Let $n\geq 3$ and let $P=$
$D_{1}(D_{1}+D_{2})+\alpha D_{2}$ ,. The above theorem means that if $1\leq s<2$well-posed (resp. (s) welkposed).
(resp. $1<s\leq 2$ ), then it iswel-posed.
However Okada [5] proved that it is$\alpha\in \mathrm{C}$$\{s\}$$\{\infty\}$

Example 3 (non-involutive operators). Let $P–D_{1}(D_{1}+x_{1}^{q}D_{n})+$-posed(resp. (s) wel-known that $P$ is. It isposed) for any (Among many papers, we refer to [6] ).

$\alpha x_{1}^{q-1}D_{n}$$\mathrm{w}\mathrm{e}\mathrm{l}$$\{s\}\mathrm{w}\mathrm{e}\mathrm{l}$$s$Example 4(constant multiple operators). Assume that $\lambda_{1}--\cdots--$in (1). Komatsu [3] defined the irregularity for this case by$\lambda_{m}--0$$\iota$$\iota$$– \max\{1,0\leq j\leq m-1\max\{\frac{m-j}{m-j-\mathrm{o}\mathrm{r}\mathrm{d}P_{j}}\}\}$

well-posed (resp. (s) well-posed) if
In this case it folows that $P$ is$\leq m$ , and this is$1\leq s<\iota/(\iota-1)$ (resp. $1<s\leq\iota/(\iota-1)$ ). We haveastronger result than the above theorem. Since our theory is strongly
influenced by [3], we briefly sketch the discussions there:$\{s\}$$\iota$(i) Ahyperbolic partial differential operator $P$ with constant multiplicity can be written in aspecial form, which he called De
Paris decomposition.
(ii) Rewriting $P$ in such aform, we can define its irregularity
similarly as above.
well-posed if $1\leq s<\iota/(\iota-1)$ .
(iii) $P$ is$\iota$$\{s\}$

As we shall see in the next section, we can extend this theory to thegeneral case.

Our aim is to give acriterion which improves Theorem 1, and also
contains all these examples. The main result is the folowing
Theorem 2. If $P$ satisfies (1) $-(3)$ , then we can define Irr $P$ , which is$P\leq m$ . Furthermore,
a rational number satisfying
if $1\leq s$ $<$well-posed, and if, then $P$ is
Irr$P$, then is (s) well-posed.
$1\leq \mathrm{I}\mathrm{r}\mathrm{r}$$P/(\mathrm{I}\mathrm{r}\mathrm{r}P-1)$$1)$$\{s\}$$1<s\leq \mathrm{I}\mathrm{r}\mathrm{r}$ $P/(\mathrm{I}\mathrm{r}\mathrm{r}P-$60
IRRGULARJTIES OF MICROHYPERBOLIC OPERATORS$–1$ , then we define
Remark If. Since, Theorem 2is always stronger than (or equivalent to)
Theorem 1.
In the above examples, it will turn out that$\mathrm{k}\mathrm{r}P$$\mathrm{I}\mathrm{r}\mathrm{r}P/(\mathrm{I}\mathrm{r}\mathrm{r}P-1)=\infty$$1\leq \mathrm{I}\mathrm{r}\mathrm{r}P\leq m$

Irr $P=m$ in Example 1,
Irr $P=1$ in Examples 2,3,
Irr $P=\iota$ (above number)$=\mathrm{t}\mathrm{h}\mathrm{e}$which coincides with thein Example 4,-known results.

$\mathrm{w}\mathrm{e}\mathbb{I}$2. Lascar decomposition.
We ffist want to express $P$ in aspecial form similarly to [3]. If$0\leq q\leq m$ we define
to be the set of all g-tuples,, $\mu_{q}\in\{1,2, \cdots, m\}$ are mutusuch that$\mu=$
ally distinctive. Here we distinguish different arrangements of the same
numbers. Although
does not make sense, we assume that it consistsof only one element, which we denote by . We define,$S_{mq}$$(\mu_{1},\mu_{2}, \cdots,\mu_{q})$$\mu_{1},\mu_{2}$$\cdots$$S_{m0}$$\emptyset$$S= \bigcup_{0\leq q\leq m}S_{mq}$and$S’= \bigcup_{0\leq q\leq m-1}S_{mq}$. If$\mu\in S_{mq}$, then we define $|\mu|=q$ , and$\Lambda^{\mu}(x, D)=\Lambda_{\mu_{q}}(x, D)\cdots$ $\Lambda_{\mu 1}(x, D)$Heresymbol is.

denotes the microdifferential operator whose complete. We also define. We define$\mathcal{E}_{x}*;[P, x_{1}]=0$ ,. By aLascar decomposition we mean an
expression of the following form:$\Lambda_{j}(x, D)$$\Lambda^{\emptyset}=1$$\Lambda_{j}(x,\xi)$$\overline{\mathcal{E}}_{x}*(j)=\{P\in$$\mathrm{o}\mathrm{r}\mathrm{d}P\leq j\}$(5)$\{\begin{array}{l}P(x,D)=\Lambda_{m}(x,D)\cdots\Lambda_{1}(x,D)+\sum_{\mu\in S’}(x_{1}^{-m+|\mu|}a_{\mu}(x,D,)+b_{\mu}(x,D’))\Lambda^{\mu}(x,D)a_{\mu}(x,D,)\in\overline{\mathcal{E}}_{x}*(0),b_{\mu}(x,D’)\in\overline{\mathcal{E}}_{x}*(m-|\mu|-1)\end{array}$Here we consider anegative power of formally. It is easy to see thatan arbitrary operator has an infinitely many Lascar decompositions.
Example 2bis. Let $n\geq 3$ and let$x_{1}$(6)$P=D_{1}(D_{1}+D_{2})+\alpha D_{2}$.

61
KEISUKE UCHIKOSHI, and by aLascar decomposition we mean
Here $\Lambda_{1}--D_{1}+D_{2}$ ,an expression of the following form:$\Lambda_{2}=D_{1}$$\{\begin{array}{l}P--\Lambda_{2}\Lambda_{1}+(x^{-1}a_{1}+b_{1})\Lambda_{1}+(x^{-1}a_{2}+b_{2})\Lambda_{2}+(x^{-2}a_{\emptyset}+b_{\emptyset})\mathrm{o}\mathrm{r}\mathrm{d}a_{\mu}\leq 0,\mathrm{o}\mathrm{r}\mathrm{d}b_{j}\leq 0(j=1,2),\mathrm{o}\mathrm{r}\mathrm{d}b_{\emptyset}\leq 1\end{array}$

Note that (6) is aLascar decomposition as it stands. In fact we may, andthe other coefficient operators to be 0. We also
takehave another expression:$b_{\emptyset}--\alpha D_{2}$$\mathrm{a}1$(7)$P=\Lambda_{2}\Lambda_{1}+\alpha\Lambda_{1}-\alpha\Lambda_{2}$.

This means $b_{1}---b_{2}--\alpha$ , and all the other coefficient operators are 0.
We have still other expressions, but they are not important. We shallsee that some expressions are heavy, and some expressions are light.

Example 3bis. Let(8)$P–D_{1}(D_{1}+x_{1}^{q}D_{n})+\alpha x_{1}^{q-1}D_{n}$.

. Again this is aLascar decomposition,Here
as it stands. We also have another expression:$\Lambda_{1}--D_{1}+x_{1}^{q}D_{n}$(9)$\Lambda_{2}--D_{1}$$P–\Lambda_{2}\Lambda_{1}+x_{1}^{-1}\alpha\Lambda_{1}-x_{1}^{-1}\alpha\Lambda_{2}$.

denotes
In (5), $P$ is decomposed into three parts. Firstly,
the principal part. The lower order terms are formaly written in aform, $\mu\in S’$ . For the-module generated bylike an element of somethe generator part, andsake of convenience, let us
the coefficient part. Roughly speaking we have$\Lambda_{m}\cdots$$\Lambda^{\mu}$$\mathcal{E}_{x}*$$\mathrm{c}\mathrm{a}\mathrm{l}$$P(x, D)-$
-principal– principal$\Lambda_{1}$$x_{1}^{-m+|\mu|}a_{\mu}+b_{\mu}$$\Lambda^{\mu}$order part(coefficient part$\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}+\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}$

part

$+$

$\cross \mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}$part).

coefficient partIf we calculate the amount of the lower order part
generator part), we can prove Theorem 1. To the contrary, if we calculate the amount of the coefficient part alone, we can prove Theorem
2. Of course less amount gives abetter result, so the latter calculatio$(_{-}^{-}$$\cross$$\mathrm{n}$62IRREGULARITIES OF MICROHYPERBOLIC OPERATORS

is preferable. However, this amount depends on Lascar decompositions,and we determine the best one as follows.
For each Lascar decomposition (5) we define$\kappa$.

$= \mathrm{m}\mathrm{a}\mathrm{s}\mathrm{c}\{1, \max_{\mu\in S},\{\frac{m-|\mu|}{m-|\mu|-\mathrm{o}\mathrm{r}\mathrm{d}b_{\mu}}\}\}$$\leq m$ .
This number depends on the expression and we
We haveas the minimum value of among all the Lascar decomposidefine
tions. Although there are infinitely many decompositions, the minimum
value is wel-defined.
$=1$ ,.
Example 2tris. In (6) we have $m=2$ , andTherefore we have
$1\leq\kappa$$\mathrm{i}\mathrm{r}\mathrm{r}P$$\kappa$$|\emptyset|--0$$\mathrm{o}\mathrm{r}\mathrm{d}b0$$\kappa=\max\{1, (2-0)/(2-0-1)\}=2$

for this decomposition. On the other hand, in (7) we have
ord $b_{2}=0$ , $|1|=|2|=1$ . Therefore we have$\kappa$$\mathrm{o}\mathrm{r}\mathrm{d}b_{1}$–$= \max\{1, (2-1)/(2-1-0)\}=1$

for this decomposition. This means that (7) is abetter expression than.
(6). We obtain$\mathrm{i}\mathrm{r}\mathrm{r}P=1$, 1, . for Examples 1,3,4, respecWe can similarly prove
tively.
Remark. Although we have infinitely many Lascar decompositions,
to construct the fundamental solution we can choose the best decomposition, and forget all the other expressions. This means that we only
use the minimum value of , and we may neglect all the other values.
Therefore we define irrP $= \min$ { ;Lascar decompositions}.
$\mathrm{i}\mathrm{r}\mathrm{r}P=m$$\iota$$\kappa$$\kappa$

We next consider permutations in the principal part. Let
and let us consider the following expression:(10)$\sigma\in S_{mm}$$\{\begin{array}{l}P(x,D)-+\sum_{\mu\in S’}^{-}(x_{1}^{-m+|\mu|}a_{\mu}’(x,D’)+b_{\mu}’(x,D’))\Lambda^{\mu}(x,D)\Lambda^{\sigma}(x,D)a_{\mu},(x,D,)\in\overline{\mathcal{E}}_{x}*(0),b_{\mu},(x,D’)\in\overline{\mathcal{E}}_{x}*(m-|\mu|-1)\end{array}$,

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KEISUKE UCHIKOSHI

We call (10) aLascar decomposition subordinate to . We have infinitely
many expressions again, and for each expression we define$\sigma$$\kappa’--\max\{1, \max_{\mu\in S},\{\frac{m-|\mu|}{m-|\mu|-\mathrm{o}\mathrm{r}\mathrm{d}b_{\mu}’}\}\}$.

We define$\mathrm{i}\mathrm{r}\mathrm{r}_{\sigma}P--\min${ ;Lascar decompositions subordinate to }.
$\kappa’$$\sigma$

Finaly we define the irregularity
Irr$I\mathrm{r}\mathrm{r}$$P$

of $P$ by$P– \max\{\mathrm{i}\mathrm{r}\mathrm{r}_{\sigma}P;\sigma\in S_{mm}\}$.

Irr $P$ .
In all the above examples we have
Remark. R. Lascar considered an expression of the form (5) in [4].
In his paper he assumed that the characteristic variety of $P$ is regu. Under theselarly involutive, and he assumed that $a_{\mu}--0$ ,assumptions he proved that the wave front set of the distribution soluthan of Pu –0 propagates along the integral manifold defined by thecharacteristic variety.
$\mathrm{i}\mathrm{r}\mathrm{r}P--\mathrm{i}\mathrm{r}\mathrm{r}_{\sigma}P--$$\mathrm{o}\mathrm{r}\mathrm{d}b_{\mu}\leq 0$

3.

.

and Irr $P$
In the previous section we defined the irregularity in three steps. We, and finaly Irr $P$ . One may think this unfirst calculate , next
comfortable, and it may be preferable if we can omit the last step. Thisis possible in two special cases. The first case is the folowing$\mathrm{i}\mathrm{r}\mathrm{r}P$$\kappa$$\mathrm{i}\mathrm{r}\mathrm{r}P$

Lemma 1. Assume that(11)for each$\{\Lambda_{i}(x, \xi), \Lambda_{j}(x,\xi)\}\in x_{1}^{-1}\Lambda_{i}(x, \xi)\mathcal{O}_{x}*+x_{1}^{-1}\Lambda_{j}(x, \xi)\mathcal{O}_{x}*$$i$and . Then we have$j$$\mathrm{i}\mathrm{r}\mathrm{r}_{\sigma}P--\mathrm{i}\mathrm{r}\mathrm{r}_{\tau}P--$

for each$\sigma$

Irr $P$, $\tau\in S_{mm}$ .

denotes the Poisson bracket. Regularly
Herelutive operators and non-involutive operators satisfy (11). In such cases
instead of Irr $P$ . We want to emphasize
we only need to calculatethat the former number is more easy to calculate than the latter one.
The second case is the folowing$\{\Lambda_{i}(x,\xi), \Lambda_{j}(x, \xi)\}$$\mathrm{i}\mathrm{r}\mathrm{r}P$$\mathrm{i}\mathrm{n}\mathrm{v}\epsilon\succ$

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IRREGULARITIES OF MICROHYPERBOLIC OPERATORS

Lemma 2. If$\sigma,\tau\in S_{mm}$, then we have$\mathrm{i}\mathrm{r}\mathrm{r}_{\tau}P\leq\max(2, \mathrm{i}\mathrm{r}\mathrm{r}_{\sigma}P)$, Irr$P \leq\max(2, \mathrm{i}\mathrm{r}\mathrm{r}_{\sigma}P)$.

This result is very interesting. Sometimes we are interested in microlocal ultradistributions of some special order . Theorem 2meansthat $P$ iswell-posed if$s_{0}$$\{s_{0}\}$

Irr

(12)

$P( \leq\max(2, \mathrm{i}\mathrm{r}\mathrm{r}P))<s_{0}/(s_{0}-1)$.

Assume that $1\leq s_{0}<2$ . (12) is equivalent to in $P<s_{0}/(s_{0}-1)$ , whichmeans that we can use
instead of Irr $P$ , and otherwise we must
calculate Irr $P$ . The author thinks that it coincides with historical experience: The well-posedness is an easy problem in hyperfiinction theory$\mathrm{i}\mathrm{r}\mathrm{r}P$(where $s=1$ ), and is adifficult problem in distribution theory (where$s=\infty)$ . Even in the case $2\leq s_{0}\leq\infty$ , the situation is not so bad ifeither we can use Lemma 1or $m$ is not large. In distribution theory it
is usual to assume such an assumption. Otherwise we need to calculate
for $\sigma\in S_{mm}$ , which contains $m!$ elements. Then the criterion may$\mathrm{i}\mathrm{n}_{\sigma}$

be very complicated.
At the end we consider the case of $m=2$ as an example. In this case
Irr $P\in\{1,2\}$ , and we have
Irr

$P=1\Leftrightarrow \mathrm{i}\mathrm{r}\mathrm{r}_{(1},{}_{2)}P=\mathrm{i}\mathrm{r}\mathrm{r}_{(2},{}_{1)}P=1$$\Leftrightarrow\{\begin{array}{l}P\in\Lambda_{2}\Lambda_{1}+x_{1}^{-1}\overline{\mathcal{E}}_{x}*(0)\Lambda_{1}+x_{1}^{-1}\overline{\mathcal{E}}_{x}*(0)\Lambda_{2}+x_{1}^{-2}\overline{\mathcal{E}}_{x}*(0)P\in\Lambda_{1}\Lambda_{2}+x_{1}^{-1}\overline{\mathcal{E}}_{x}*(0)\Lambda_{1}+x_{1}^{-1}\overline{\mathcal{E}}_{x}*(0)\Lambda_{2}+x_{1}^{-2}\overline{\mathcal{E}}_{x}*(0)\end{array}$$\Leftrightarrow\{\begin{array}{l}P\in\Lambda_{2}\Lambda_{1}+x_{1}^{-1}\overline{\mathcal{E}}_{x}*(0)\Lambda_{1}+x_{1}^{-1}\overline{\mathcal{E}}_{x}*(0)\Lambda_{2}+x_{1}^{-2}\overline{\mathcal{E}}_{x}*(0)[\Lambda_{1},\Lambda_{2}]\in x_{1}^{-1}\overline{\mathcal{E}}_{x}*(0)\Lambda_{1}+x_{1}^{-1}\overline{\mathcal{E}}_{x}*(0)\Lambda_{2}+x_{1}^{-2}\overline{\mathcal{E}}_{x}*(0)\end{array}$This is equivalent to(13),$P\in\Lambda_{2}\Lambda_{1}+x_{1}^{-1}\overline{\mathcal{E}}_{x}*(0)\Lambda_{1}+x_{1}^{-1}\overline{\mathcal{E}}_{x}*(0)\Lambda_{2}+x_{1}^{-2}\overline{\mathcal{E}}_{x}*(0)$

and
(14)

$\Lambda_{1}$and$\Lambda_{2}$satisfy (11)65
KEISUKE UCHIKOSHI

well-posed for any
If (13) and (14) axe true, then Irr $P$ $=1$ and $P$ iswell-posed for $1\leq s<2$ . In other. Otherwise Irr $P$ $=2$ and $P$ iswords, according to our result we must assume (13) and (14) for the case$2\leq s$. (13) means that the lower order terms must vanish according
to some rule, and is not surprising. However as far as our theory applies,
we must also assume condition (14) for the principal symbol.
$\{s\}$$\{s\}$$s$$\leq\infty$

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Math. Pures et Appl, 59(1980), 145-185.
[4] R. Lascar, Propagation des singularit\’es pour des op\’erateurs pseud\infty diff\’erentiels \‘a caracteristiques de multiplicit\’e variable, Lecture Notes in Math.,
856(1980), 1-26, Springer.
[5] Y. Okada, On distribution solutions of microdifferential equations withdouble involutive characteristics, Comm. in Partial Differential Equations, 24(1999), 1419-1444.
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on Pure and Appl. Math., 23(1970), 569-586${\rm Res}$