数理解析研究所講究録 1212 巻 2001 年 144-156

144
ON EXPONENTIAL CALCULUS OF SYMBOLS OFPSEUDODIFFERENTIAL OPERATORS OF MINIMUM
TYPE
東大数理

李

昌勲 (CHANG HOON LEE)GRADUATE SCHOOL OF MATHEMATICAL SCIENCES,
THE UNIVERSITY OF TOKYO

1. INTRODUCTION
Let $X$ andrespectively.

$\mathrm{Y}$be n- and m- dimensional complex manifolds,$S^{*}X:=(T^{*}X-X)/\mathbb{R}^{+}$,$S^{*}\mathrm{Y}:=(T^{*}\mathrm{Y}-\mathrm{Y})/\mathbb{R}^{+}$.

We define the mapping 7as$\gamma$:$T^{*}(X \cross \mathrm{Y})\ni(z, w;\mathrm{o}\mathrm{o} \xi,\eta)-(z; \frac{\xi}{|\xi|})\cross(w;\frac{\eta}{|\eta|})\in S^{*}X\cross S^{*}\mathrm{Y}$

,

where
$T^{*}(X\mathrm{o}\mathrm{o}\cross \mathrm{Y}):=T^{*}(X\cross \mathrm{Y})\backslash \{(T^{*}X\cross \mathrm{Y})\cup(X\cross T^{*}\mathrm{Y})\}$For $d>0$ and an open subset

$U$

of

$S^{*}X\cross S^{*}\mathrm{Y}$.

we denote$\gamma^{-1}(U)\cap\{|\xi|>d, |\eta|>d\}$by $\gamma^{-1}(U;d, d)$ .
Hereafter we write $(z, \xi, w, \eta)$ for coordinates $(z, w;\xi, \eta)$ .

2. SYMBOLS 0F pR0DUCT TYPE
Let K be acompact subset of$S^{*}X\cross S^{*}\mathrm{Y}$.

Definition 2.1. $P(z,\xi, w, \eta)$ is said to be asymbol of product type on$K$ if the following hold:such that(1) There are $d>0$ and $U\supset K$ open in$P(z,\xi, w,\eta)$ is holomorphic in.
such that
there is aconstant(2) For each$S^{*}X\cross S^{*}\mathrm{Y}$$\gamma^{-1}(U;d, d)$$\epsilon>0$(2.1)$|P(z,\xi, w, \eta)|\leq C_{\epsilon}e^{\epsilon(|\xi|+|\eta|)}$$C_{\epsilon}>0$on$\gamma^{-1}(U;d, d)$.

145
We denote by $S(K)$ the set of all such symbols on K.
$S(K)$ becomes acommutative ring with the usual sum and product.

Definition 2.2. We denote by $R(K)$ the set of aU $P(z,\xi, w,\eta)\in S(K)$satisfying the following;, and alocaly
there are $d>0$ , $>0$ , $U\supset K$ open insuch thatonbounded function$S^{*}X\cross S^{*}\mathrm{Y}$$\delta$$(0, \infty)$$C(\cdot)$$|P(z, \xi,w, \eta)|\leq C(|\xi|/|\eta|)e^{-\delta\min\{|\xi|,|\eta|\}}$

on$\gamma^{-1}(U;d, d)$.

We call an element of $R(K)$ asymbol of O-class.
Definition 2.3. Aformal series$\sum_{j,k=0}^{\infty}P_{j,k}(z,\xi,w,\eta)$

is called aformal

symbol of product type on $K$ if the folowing hold:such(1) There are $d>0,0<A<1$ , and $U\supset K$ open in.
that Pjik is holomorphic in $\gamma^{-1}(U;(j+1)d, (k +1)d)$ for each ,such that(2) For each $>0$ , there is$S^{*}X\cross S^{*}\mathrm{Y}$$j$(2.2) $|P_{j,k}(z, \xi,$w,$\eta)|\leq C_{\epsilon}A^{j+k}e^{\epsilon(|\xi|+|\eta|)}$$\geq 0$We denote by$t_{1}$

andon $\gamma^{-1}(U;(j+1)d,$(k$+1)d)$.
$\hat{S}(K)$the set of such formal symbols on $K$.

We often write aformal power seriesminants$\geq 0$$C_{\epsilon}>0$$\epsilon$

for each j, k$k$$t_{2}$

for$\sum_{j,k=0}^{\infty}$$\sum_{j,k=0}^{\infty}t_{1}^{j}t_{2}^{k}P_{j,k}(z,\xi,w, \eta)$, in indeter-

Pjik $(z,\xi, w,\eta)$ .

We can easily obtain the folowing.
becomes a commutative ring with the sum and
Proposition 2.4.
the product as formal power series in and .
$\hat{S}(K)$$t_{1}$$t_{2}$$S(K)$ is identified with asubring of $\hat{S}(K)$ as$S(K)\simeq\hat{S}(K)|t_{1}=0t_{2}=0=${P$= \sum t_{1}^{j}t_{2}^{k}P_{j,k};P_{j,k}\equiv 0$

Definition 2.5. We denote by$\sum_{j,k=0}^{\infty}t_{1}^{j}t_{2}^{k}P_{j,k}(z,\xi,w, \eta)$

and $U\supset K$ open in

in$\hat{S}(K)$$\hat{R}(K)$the set offollows:
for all (j, k) $\neq(0,$ 0)}.

$\mathrm{a}\mathrm{L}$$P(t_{1},t_{2};z,\xi,w,\eta):=$such that there are $d>0,0<A<1$ ,$S^{*}X\cross S^{*}\mathrm{Y}$satisfying the following;

146
for each e $>0$ there is C, $>0$ such that$\rangle$$|\begin{array}{ll}\Sigma P_{j,k}(z,\xi,w,\eta)0<\lrcorner.\leq s0\leq k\leq t \end{array}|\leq C_{\epsilon}A^{\min\{s,t\}}e^{\epsilon(|\xi|+|\eta|)}$on $\gamma^{-1}(U;(s+1)d, (t+1)d)$ for each s,t
We caU an element of$\hat{R}(K)$$\geq 0$.

aformal symbol of zero class.

Proposition 2.6. Under the previous identification,$S(K)\cap\hat{R}(K)=R(K)$ holds.

Proof. Let$P(_{r}z,\xi,w,\eta)$

be in $S(K)$ . Then $P(z,\xi, w, \eta)\in\hat{R}(K)$ is equiv-

alent to the following;
there exist $d>0,\delta>0$ , and $U\supset K$ open ineach $>0$ there issatisfying$S^{*}X\cross S^{*}\mathrm{Y}$such that for$C_{\epsilon}>0$$\epsilon$$|P(z,\xi,w,\eta)|\leq C_{\epsilon}e^{-\delta\min\{|\xi|,|\eta|\}+\epsilon(|\xi|+|\eta|)}$on$\gamma^{-1}(U;d,d)$.

Using the fact that $(0, \infty)=\{t :=\mathrm{E}\eta ;(\mathrm{z} ,w,\eta)\in\gamma^{-1}(U;d,d)\}$ ,by theobtain the following;$(\subset)$$\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{s},\mathrm{w}\mathrm{e}$$|P(z,\xi,w,\eta)|\leq C_{\epsilon}e^{-\delta\min\{1,\frac{1}{t}\}|\xi|+\epsilon(1+\frac{1}{t})|\xi|}$

for all
We fix any
Then for every

and $(z,\xi,w,\eta)\in\gamma^{-1}(U;d, d)$ .
$<1$ andsuch that.

$t:=\cup\xi|\eta|\in(0, \infty)$$\epsilon>0$$0<\epsilon$$\epsilon$$\leq\frac{\delta}{3}$$t \in[\frac{3}{\delta}\epsilon, 1]$$|P(z,\xi, w,\eta)|\leq C_{\epsilon}e^{-\delta|\zeta|+\epsilon(1+\frac{1}{t})|\xi|}\leq C_{\epsilon}e^{-\delta|\xi|+\epsilon(1+_{\pi}^{\delta})|\xi|}$$=C_{\epsilon}e^{(\epsilon-\frac{2}{\mathrm{s}}\delta)|\xi|}\leq C_{\epsilon}e^{-_{\mathrm{F}}^{1}\delta|\xi|}$.

On the other hand, for any sequence such that
, we define afunction
on $(0, 1]$ as$\epsilon_{n}$$\epsilon_{2}>\cdotsarrow 0$$C(\cdot)$$C(t):=\{\begin{array}{l}C_{S,\tau^{\epsilon_{1}}}C_{S,\mathrm{r}^{\epsilon_{\mathfrak{n}+1}}}\end{array}$Then$C(\cdot)$$\min\{1, \frac{\delta}{3}\}>\epsilon_{1}>$$\epsilon_{n+1}<t\leq\epsilon_{n}\epsilon_{1}<t\leq 1,$

is localy bounded on (0, 1] and$|P(z, \xi,w,\eta)|\leq C(\frac{|\xi|}{|\eta|})e^{-\frac{1}{3}\delta|\xi|}$.

147
on $\gamma^{-1}(U;d, d)\cap\{|\xi|\leq|\eta|\}$ .
In like manners,$|P(z, \xi, w, \eta)|\leq C(\frac{|\eta|}{|\xi|})e^{-\frac{1}{8}\delta|\eta|}$

on $\gamma^{-1}(U;d, d)\cap\{|\xi|\geq|\eta|\}$ .
define afunction
on
Then $C’(t)$ is locally bounded on$C’(\cdot)$$\mathrm{H}\mathrm{e}\mathrm{r}\mathrm{e},\mathrm{w}\mathrm{e}$$(0, \infty)$as$\mathrm{C}’(\mathrm{t})=C(\min\{t, \frac{1}{t}\})$.

$(0, \infty)$

andon $\gamma^{-1}(U;d, d)$ .
That.
0) Let $P(z,\xi, w, \eta)\in R(K)$ .
Then there are $d>0$ , $\delta>0$ , $U\supset K$ open in, and alocallybounded functiononsuch that$|P(z, \xi, w, \eta)|\leq C’(\frac{|\xi|}{|\eta|})e^{-\frac{1}{3}\delta\cdot\min\{|\xi|,|\eta|\}}$$\mathrm{i}\mathrm{s},P(z,\xi, w, \eta)\in \mathrm{R}(\mathrm{K})$$S^{*}X\cross S^{*}\mathrm{Y}$$(0, \infty)$$C(\cdot)$$|P(z, \xi, w, \eta)|\leq C(\frac{|\xi|}{|\eta|})e^{-\delta\min\{|\xi|,|\eta|\}}$

on$\gamma^{-1}(U;d, d)$. We fix any$\epsilon$

such that $0<\epsilon<1$ . Then,$|P(z, \xi, w, \eta)|\leq\max_{\epsilon\leq t\leq 1}C(t)\cdot e^{-\delta\min\{|\xi|,|\eta|\}}$on. We put.
On the other hand, since $P(z, \xi, w, \eta)\in S(K)$ , there exists$\gamma^{-1}(U;d, d)\cap\{\epsilon\leq\frac{|\xi|}{|\eta|}=.

t\leq 1\}$

$C_{\epsilon}’:= \max_{\epsilon\leq t\leq 1}C(t)$$C_{\epsilon}’>0$

such that$|P(z,\xi, w, \eta)|\leq C_{\epsilon}’e^{\epsilon(|\xi|+|\eta|)}$on$\gamma^{-1}(U;d, d)$.

Therefore, the following iequalities hold on$\gamma^{-1}(U;d, d)\cap\{\frac{|\xi|}{|\eta|}\leq\epsilon\}$$|P(z,\xi,w, \eta)|\leq C_{\epsilon}’e^{-\delta\min\{(|\xi|,|\eta|)\}+\delta\min\{(|\xi|,|\eta|)\}+\epsilon(|\xi|+|\eta|)}$$\leq C_{\epsilon}’e^{-\delta\min\{(|\xi|,|\eta|)\}+\epsilon(1+\delta)(|\xi|+|\eta|)}$If we put

$C_{\epsilon}:= \max\{C_{\epsilon}’, C_{\epsilon}’\}$.

,$|P(z,\xi, w, \eta)|\leq C_{\epsilon}e^{-\delta\min\{(|\xi|,|\eta|)\}+\epsilon(1+\delta)(|\xi|+|\eta|)}$on $\gamma^{-1}(U;d, d)\cap\{|\xi|\leq|\eta|\}$ .
That is, $P(z, \xi, w, \eta)\in\hat{R}(K)$ .

Proposition 2.7. $R(K)$ is an ideal in $S(K)$ .

Proof. It is clear by the part (C) of the proof of Proposition 2.6.
Proposition 2.8.

$\hat{R}(K)$is an ideal in$\hat{S}(K)$

.

.

148
Proof. Let( , to, )and$U\supset
K$
Then there exist $d>0,0<A<1$ , andopen insatisfying the following:, we have someFor eachsuch that$\sum P_{j,k}(z,\xi,w, \eta)\in\hat{R}(K)$$\epsilon>0$a)b)$\sum Q_{j,k}$$z,\xi$$\eta$$\in\hat{S}(K)$.

$S^{*}X\cross S^{*}\mathrm{Y}$$C_{\epsilon}>0$$|P_{s,t}(z,\xi, w, \eta)|$,$|Q_{s,t}(z,\xi,w,\eta)|\leq C_{\epsilon}A^{s+t}e^{\epsilon(|\xi|+|\eta|)}$w,$\eta)|\leq s\leq C_{\epsilon}A^{\min\{s,t\}}e^{\epsilon(|\xi|+|\eta|)}$$|0 \leq j\sum_{0\leq k\leq t}P_{j,k}(z,\xi,$

on $\gamma^{-1}(U;(s+1)d, (t+1)d)$ for each s,tIt suffices to show that$\sum R_{j,k}\in\hat{R}(K)$$R_{j,k}:=$$\geq 0$.

, where$P_{j_{1},k_{1}}Q_{j_{2},k_{2}}$.

$k_{1}+k_{2}=k \sum_{j_{1}+j_{2}=j},$Since we can easily estimate

for $st=0$ ,$0^{\lrcorner} \leq\sum_{0<\leq s,k\leq t’}.R_{j,k}$.
we suppose $s\geq 1$ andThen we can obtain the folwing inequality,$t\geq 1$$|0 \leq\sum_{0\leq \mathrm{j}\leq s,k\leq t’}R_{j,k}|=|_{0}$$+k_{2}^{2}=k \sum_{+j=j},P_{j_{1},k_{1}}Q_{j_{2},k_{2}}|$$0 \leq k.\leq tk_{1’}\sum_{<\lrcorner\leq sj_{1}}$$\leq|$ $( 0 \leq k_{1}^{1}\leq t\sum_{0\leq j\leq s},P_{j_{1},k_{1}})( 0\leq k_{2}^{2}\leq t\sum_{0\leq j\leq s},Q_{j_{2\prime}k_{2}})|+|s+1\leq\sum_{t+1\leq k\leq 2t}\sum_{k_{1}^{1}+k_{2}=k}P_{j_{1},k_{1}}Q_{j_{2},k_{21}}j\leq 2sj+j_{2}=j$$+|0 \leq j\sum_{t+1\leq k\leq 2t}\sum_{k_{1}^{1}+k_{2}^{2}=k}P_{j_{1},k_{1}}Q_{j_{2},k_{2}}\leq sj+j=j|+|s+1\leq\sum_{0\leq k\leq t}\sum_{k_{1}^{1}+k_{2}^{2}=k}P_{j_{1},k_{1}}Q_{j_{2},k_{2}}j\leq 2sj+j=j|$.

We shaU estimate the four terms in the right side of the inequality,
respectively.
the first term$\leq C_{\epsilon}A^{\min\{s,t\}}e^{\epsilon(|\epsilon|+|\eta|)}$.
$0 \leq k_{2}^{2}\leq t\sum_{0\leq j\leq s},C_{\epsilon}A^{j+k}e^{\epsilon(|\xi|+|\eta|)}$$\leq C_{\epsilon}\cdot C_{\epsilon}\cdot A^{\min\{s,t\}}e^{2\epsilon(|\xi|+|\eta|)}\cdot\frac{1}{1-A}\cdot\frac{1}{1-A}$

149
on $\gamma^{-1}(U;(s+1)d, (t+1)d)$ for each s,t$\geq 1$.

the 2nd term
$C_{\epsilon}A^{j_{1}+k_{1}}e^{\epsilon(|\xi[+|\eta|)}\cdot C_{\epsilon}A^{j_{2}+k_{2}}e^{\epsilon(|\xi|+|\eta|)}$$\leq\sum_{t+1\leq k\leq 2t}s+1\leq j\leq 2s$$k_{1}^{1}+k_{2}^{2}=k \sum_{j+j=j},$$=C_{\epsilon} \cdot C_{\epsilon}\cdot e^{2\epsilon(|\xi|+|\eta|)}\cdot(\sum_{s+1\leq j\leq 2\epsilon}\sum_{j_{1}+j_{2}=j}A^{j})(\sum_{t+1\leq k\leq 2t}\sum_{k_{1}+k_{2}=k}A^{k})$.

If we choose any $B$ and $C$ such that $0<B<1,0<C<1$ , and$BC$, we can get the following inequality:$\geq A$$\sum_{s+1\leq j\leq 2s}\sum_{j_{1}+j_{2}=j}A^{j}\leq C^{s+1}(B^{0}+B^{1}+B^{2}+\cdots)^{2}=C^{s+1}(\frac{1}{1-B})^{2}$

.

Then,

the second term$\leq C_{\epsilon}\cdot C_{\epsilon}\cdot e^{2\epsilon(|\xi|+|\eta|)}\cdot C^{s+1}(\frac{1}{1-B})^{2}\cdot C^{t+1}(\frac{1}{1-B})^{2}$

on $\gamma^{-1}(U;(s+1)d, (t+1)d)$ for each s,t$\geq 1$.

the third term
$C_{\epsilon}A^{j_{1}+k_{1}}e^{\epsilon(|\xi|+|\eta|)}\cdot C_{\epsilon}A^{j_{2}+k_{2}}e^{\epsilon(|\xi|+|\eta|)}$$\leq$$\sum_{0\leq j\leq s,t+1\leq k\leq 2t}$$k_{1}+k_{2}=k \sum_{j_{1}+j_{2}=j},$$=C_{\epsilon} \cdot C_{\epsilon}\cdot e^{2\epsilon(|\xi|+|\eta|)}(\sum_{0\leq j\leq s}\sum_{j_{1}+j_{2}=j}A^{j})(\sum_{t+1\leq k\leq 2t}\sum_{k_{1}+k_{2}=k}A^{k})$$\leq C_{\epsilon}\cdot C_{\epsilon}\cdot e^{2\epsilon(|\xi|+|\eta|)}\cdot(\frac{1}{1-A})^{2}\cdot C^{t+1}(\frac{1}{1-B})^{2}$

on $\gamma^{-1}(U;(s+1)d, (t+1)d)$ for each s, t
In like manners,
the fourth term$\geq 1$.

$\leq C_{\epsilon}\cdot C_{\epsilon}\cdot e^{2\epsilon(|\xi|+|\eta|)}\cdot C^{s+1}\cdot(\frac{1}{1-B})^{2}\cdot(\frac{1}{1-A})^{2}$

on $\gamma^{-1}(U;(s+1)d, (t+1)d)$ for each s, t
Hence, we conclude that.
$\sum R_{j,k}\in\hat{R}(k)$$\geq 1$.

150
becomes acommutative ring by Proposition 2.8. By Propsitions 2.6 and 2.7, the inclusion $S(K)\epsilonarrow\hat{S}(K)$ induces the injective
ring homomorphism$\hat{S}(K)/\hat{R}(K)$$\iota_{K}$: $S(K)/R(K)arrow\hat{S}(K)/\hat{R}(K)$ .

Conversely, we obtain the following.

Theorem 2.9. If $\sum P_{j,k}(z,\xi, w, \eta)\in\hat{S}(K)$ , there exists $P(z, \xi, w, \eta)\in$$S(K)$ such that $P- \sum P_{j,k}\in\hat{R}(K)$ .
Thus, $S(K)/R(K)$ is isomorphic to $\hat{S}(K)/\hat{R}(K)$ in the sense of commutative rings.
apseud0an element in the ring
Definition 2.10. We
$K$
: for
differential operator of the product type on . We write :the associated pseud0-differential operator of the product type on $K$.
inusing an element$\hat{S}(K)/\hat{R}(K)$$\mathrm{c}\mathrm{a}\mathrm{U}$$\sum P_{j,k}$$\sum P_{j,k}$

The mapping$\gamma$$\hat{S}(K)$is the composition of the following$\gamma_{1}$and 72.
,$T^{*}(X \cross \mathrm{Y})\ni(z, w;\xi,\eta)\underline{\gamma 2}(z, w;\frac{\xi}{|(\xi,\eta)|}, \frac{\eta}{|(\xi,\eta)|})\in S^{*}(X\cross \mathrm{Y})\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}$$S^{*}(X \cross \mathrm{Y})\ni(z, w;\frac{\xi}{|(\xi,\eta)|}, \frac{\eta}{|(\xi,\eta)|})\underline{\gamma_{1}}(z, \frac{\xi}{|\xi|})\cross(w, \frac{\eta}{|\eta|})\in S^{*}X\cross S^{*}\mathrm{Y}\mathrm{o}\mathrm{o}$where$S^{*}(X\cross \mathrm{Y}):=S^{*}(X\cross \mathrm{Y})\backslash \{(S^{*}X\cross \mathrm{Y})\mathrm{o}\mathrm{o}\cup(X\cross S^{*}\mathrm{Y})\}$,.

Proposition 2.11. If $P(z, \xi, w, \eta)$ is a symbol of product type on $K$ ,in the sense of AOKI’s symbol
is a symbol on$P$$\gamma_{1}^{-1}(K)$Proof By the hypothesis, there are $d>0$ and$U\supset K$

open in$S^{*}X\cross$satisfying the following:a) $P(z,\xi, w, \eta)$ is holomorphic in $\gamma^{-1}(U;d, d)$ , andsuch that
there iseach$S^{*}\mathrm{Y}$$\mathrm{b})\mathrm{f}\mathrm{o}\mathrm{r}$on$\epsilon>0$$\gamma^{-1}(U;d, d)$$|P(z,\xi,w,\eta)|\leq C_{\epsilon}e^{\epsilon(|\xi|+|\eta|)}$$C_{\epsilon}>0$.

and
Let $K’$ be compact inooThen there exist $d’>0$ and $U’\supset K’$ open in$S^{*}(X\cross \mathrm{Y})\mathrm{o}\mathrm{o}$$\gamma_{1}^{-1}(K)\supset K’$.

$S^{*}(X\cross \mathrm{Y})$such that$\gamma^{-1}(U)\cap\{|\xi|>d, |\eta|>d\}\supset\gamma_{2}^{-1}(U’)\cap\{|\xi|+|\eta|>d’\}$In fact, for each ( ,$z\circ$$w;\circ$$(, \eta)\circ\in\gamma_{1}^{-1}(K)$.

we can choose $d’>0$ such that$d’> \frac{d}{\min\{|\xi|,|\eta|\}\circ\circ}$.

151
Then there exists aneighborhood $U’$ ofsuch that$(\mathrm{z}, w;^{\mathrm{O}}\xi, \eta)\mathrm{o}\mathrm{o}\in\gamma_{1}^{-1}(K)$in$S^{*}(X\cross \mathrm{o}\mathrm{o}$$\mathrm{Y})$$\gamma^{-1}(U)\cap\{|\xi|>d, |\eta|>d\}\supset\gamma_{2}^{-1}(U’)\cap\{|\xi|+|\eta|>d’\}$.

By the compactness of $K’$ , the proof is completed.

Proposition 2.12. If $P(z,\xi, w,\eta)$ is a symbol of product type of 0class on $K$ , that is, $P\in R(K)$ , $P$ is a zero symbol onin the$OKI$
sense of A’s symbol.
$\gamma_{1}^{-1}(K)$Proof Letbe compact inand.
It suffices to show that is azero symbol on $K’$ in the sense of AOKI’s
symbol. By the hypothesis, there exist $d>0$ , $\delta>0$ , $U\supset K$ open in, and alocally bounded functiononsuch that$K’$$S^{*}(X\mathrm{o}\mathrm{o}\cross \mathrm{Y})$$\gamma_{1}^{-1}(K)\supset K’$$P$$S^{*}X\cross S^{*}\mathrm{Y}$$C(\cdot)$$(0, \infty)$$|P(z, \xi, w, \eta)|\leq C(\frac{|\xi|}{|\eta|})e^{-\delta\min\{|\xi|,|\eta|\}}$

on $\gamma^{-1}(U)\cap\{|\xi|>d, |\eta|>d\}$ .
Let$(z, w;\xi, \eta)\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}$be any point of$\gamma_{1}^{-1}(K)$. By Proposition 2.11,

there exist $d’>0$ and aneighborhood $U’$ ofsuch thatin$(z, w\mathrm{o}\mathrm{o};\xi, \eta)\circ\circ$$S^{*}(X\mathrm{o}\mathrm{o}\cross \mathrm{Y})$$\gamma^{-1}(U)\cap\{|\xi|>d, |\eta|>d\}\supset\gamma_{2}^{-1}(U’)\cap\{|\xi|+|\eta|>d’\}$,and that there exists $\delta’>0$ satisfying$\min\{\frac{|\xi|}{|\xi|+|\eta|}, \frac{|\eta|}{|\xi|+|\eta|}\}>\delta’$on$\gamma_{2}^{-1}(U’)$.

Hence,$|P(z, \xi, w, \eta)|\leq C(\frac{|\xi|}{|\eta|})e^{-\delta\delta’(|\xi|+|\eta|)}$on $\gamma_{2}^{-1}(U’)\cap\{|\xi|+|\eta|>d’\}$ .
Since $K$ is compact,
is azero symbol on
AOKI’s symbol.
$P$

Definition 2.13. Tne canonical mapping$H_{K}$$\gamma_{1}^{-1}(K)$in the sense ofis defined as follows;$S(K)/R(K)\ni:P:^{\underline{H_{K}}}[P]\in 1\dot{E}_{1}\mathcal{E}^{\mathrm{R}}(U)U\supset\gamma_{1}^{-}(K)$

Proposition 2.14. Suppose
andrespectively, and. Then,for all $P\in S(K)/R(K)$ .
$K_{1}$$K_{1}\supset K_{2}$$K_{2}$are compact in.
$S^{*}X\cross S^{*}\mathrm{Y}$$H_{K_{1}}$ $(: P:)|_{\gamma_{1}^{-1}(K_{2})}=H_{K_{2}}($:$P|_{K_{2}}$

:

,

$)$

152
Definition 2.15. We define the productas follows:w,$( \sum_{j,k=0}^{\infty}P_{j,k}(z,\xi,w, \eta))*(\sum_{j,k=0}^{\infty}Q_{j,k}(z,\xi,$$*\mathrm{o}\mathrm{f}$two elements of$\hat{S}(K)$w,$\eta)$$\eta))=\sum_{j,k=0}^{\infty}R_{j,k}(z, \xi,$

.

,

where
$\sum_{j,k=0}^{\infty}t_{1}^{j}t_{2}^{k}R_{j,k}(z,\xi,$w,w,$\eta):=e^{t_{1}\langle\partial_{\xi},\partial_{z}*\rangle+t_{2}\langle\partial_{\eta},\partial_{w}*\rangle}((\sum_{j,k=0}^{\infty}P_{j,k}(z, \xi,$$\eta))$$\cross(\sum_{j,k=0}^{\infty}Q_{j,k}(z^{*},\xi^{*},w^{*}, \eta^{*})))|_{w^{*}=w,\eta^{*}=\eta}z^{*}=z,\xi^{*}=\xi$That is,$R_{j,k}(z,\xi, w,\eta):=$$\frac{1}{\alpha!\beta!}\partial_{\xi}^{\alpha}ff_{\eta}iP_{j_{1}k_{1}}(z,\xi,w, \eta)$$k_{1}+k_{2}+| \beta|=k\sum_{j_{1}+j_{2}+|\alpha|=j},$$\cross\partial_{z}^{\alpha}\partial_{w}^{\beta}Q_{j_{2}k_{2}}(z,\xi, w, \eta)$.

Then we obtain the folowing.

Lemma 2.16. If
andare formal symbols of product type
on $K$ ,theriis also a formal symbol of product type on $K$ .
$\sum P_{j,k}$$\sum Q_{j,k}$$\sum R_{j,k}$

Proposition 2.17. If $\sum P_{j,k}\in\hat{S}(K)$ and $\sum Q_{j,k}\in\hat{R}(K)$ , otherwise$\sum P_{j,k}\in\hat{R}(K)$and $\sum Q_{j,k}\in\hat{S}(K)$ ,is also in.
$\hat{R}(K)$$\sum R_{j,k}$

By Lemma 2.JL6 and Proposition 2.17, the following composition of
two elements
is weU defined$\mathrm{i}\mathrm{n}\hat{S}(K)/\hat{R}(K)$:$\sum P_{j,k}$: 0 : $\sum Q_{j,k}::=:(\sum P_{j,k})*(\sum Q_{j,k})$ :.

We can easily verify the associativity about the operation . That is,
becomes an associative algebra. Hence the mapping
is ahomomorphism about the operation, and where$\circ$$\hat{S}(K)/\hat{R}(K)$.,$\mathbb{C}$$H_{K}$$\circ,$$+$$\mathcal{E}_{X\mathrm{x}\mathrm{Y}}\mathrm{R},p\mathrm{r}\mu(K)\equiv\hat{S}(K)/\hat{R}(K)\mathcal{E}_{X\mathrm{x}\mathrm{Y}}^{\mathrm{R}}(\gamma^{-1}(K))\underline{\underline{H_{K_{1}}}}$

Definition 2.18. The reverse of$\sum P_{j,k}$in$\hat{S}(K)$.

is defined as$( \sum t_{1}^{j}t_{2}^{k}P_{j,k})^{R}:=e^{t_{1}(\partial_{\xi},\partial_{z})+t_{2}\langle\partial_{\eta},\partial_{w})}(\sum\dot{\theta}_{1}t_{2}^{k}P_{j,k}(z,\xi, w, \eta))$

We can verify that if$\hat{S}(K)(\hat{R}(K))$.respectively$\sum P_{j,k}$

is in $\hat{S}(K)(\hat{R}(K))$ then.

$( \sum P_{j,k})^{R}$

is in

153
3. EXPONENTIAL calculus OF symbols OF MINIMUM TYPE
Definition 3.1. Afunction
if the following hold;$\Lambda:\mathrm{R}_{>0}arrow \mathrm{R}_{>0}$

is said to be infra-linear(1) Ais continuous,(2) for each $\alpha>1$ ,$\Lambda(\alpha t)\leq\alpha\Lambda(t)$on (0,$\infty)$,(3) Ais increasing,(4)$\lim_{tarrow\infty}\frac{\Lambda(t)}{t}=0$.

Definition 3.2. $P(z,\xi, w, \eta)\in S(K)$ is called asymbol of minimum
on $K$ if there exist constants $C>0$ ,type of growth ordersatisfying the following;$d>0$ , and $U\supset K$ open ind, d), and(1) $P(z, \xi,$w, is holomorphic in$(\Lambda_{1}, \Lambda_{2})$$S^{*}X\cross S^{*}\mathrm{Y}$$\gamma^{-1}(U;$$\eta)$(2) $|P(z, \xi,$w,$\eta)|\leq C$ .

$\min\{\Lambda_{1}(|\xi|), \Lambda_{2}(|\eta|)\}$

on$\gamma^{-1}(U;$Example 3.3. (by K. Kataoka)$\Omega=\Omega’:=\mathbb{C}\cross\{\xi\in \mathbb{C};|\arg\xi|<\delta, \xi\neq 0\}(0<\delta<\frac{\pi}{2})$Let $K$ be any compact subset of$\Omega\cross\Omega’$.

such that$\gamma^{-1}(K)\subset$.

$P(z,\xi, w, \eta):=(\xi\eta)^{(1+\sigma)/2}/(\xi+\eta)$

Ai (t)$S^{*}\mathbb{C}_{z}\cross S^{*}\mathbb{C}_{w}$d, d).

$=\mathrm{A}2(\mathrm{t}):=t^{\sigma}$,with $0<\sigma<1$ .

Remark 3.4. If P is asymbol of minimum type on K,of product type on K.

$e^{P}$

is asymbol

is called aformal symbol of minimuminDefinition 3.5.
on $K$ if there exist constants $C>0$ ,type of growth ordersatisfying the$d>0,0<A<1$ , and $U\supset K$ open in$\sum P_{j,k}$$\hat{S}(K)$$(\Lambda_{1}, \Lambda_{2})$$S^{*}X\cross S^{*}\mathrm{Y}$following;(1)(2)$P_{j,k}$is holomorphic in $\gamma^{-1}(U;(j+1)d, (k+1)d)$ for each , $k\geq 0$ ,$j$$|P_{j,k}(z, \xi, w, \eta)|\leq C\cdot\min\{\Lambda_{1}(|\xi|), \Lambda_{2}(|\eta|)\}\cdot A^{j+k}$

on $\gamma^{-1}(U;(j+1)d, (k+1)d)$ for each , $k\geq 0$ .
$j$

is aformal symbol of minimum type on K,
Remark 3.6. If
is aformal symbol of product type on K.
$\sum P_{j,k}$$e^{\Sigma P_{j,k}}$

154
Proposition 3.7. If $P$ and
(3.1)

$Q$

are in $S(K)$ , thenw,$\eta)*(Q(z,\xi,$w,$P(z, \xi,$$\eta))^{R}$$=e^{t_{1}(\partial_{\xi},\partial_{z}*\rangle+t_{2}\langle\partial_{\eta},\partial_{w}*\rangle}P(, \xi,$w,$\eta)Q(z^{*},\xi,w^{*}, \eta)|_{w^{*}=w}z^{*}=z$

Theorem 3.8. If $P$ andare symbols of minimum type of growthorder
on $K$, there exists a formal symbol, $\sum Rjik$ , of minimum
type on $K$ satisfying.
$Q$$(\Lambda_{1}, \Lambda_{2})$$e^{P}*e^{Q}=e^{\Sigma t_{1}^{\mathrm{j}}t_{2}^{k}R_{\mathrm{j},k}}$Proof. $W(s,$t;$z,\xi,$w,$\eta, z^{*},\xi^{*},w^{*},\eta^{*})$

is the uniqueformal series solution to the following system of partial differential
equations :$:=e^{s\langle\partial_{\xi},\partial_{z}*\rangle+t(\partial_{\eta},\partial_{w}*\rangle}\exp(P(z,\xi, w, \eta)+Q(z^{*},\xi^{*}, w^{*},\eta^{*}))$(3.2)$\{\begin{array}{l}\partial_{s}W=\langle\partial_{\xi},\partial_{z}*\rangle W,\partial_{t}W=\langle\partial_{\eta},\partial_{w}\cdot\rangle WW_{s=t=0}=\mathrm{e}\mathrm{x}\mathrm{p}(P(z,\xi,w,\eta)+Q(z^{*},\xi^{*},w^{*},\eta^{*}))\end{array}$If we putsystem, we obtain the folowing recursive formulas
about:$W= \exp(\sum_{j,k}^{\infty}s^{j}t^{k}W_{j,k}(z,\xi,w, \eta, z^{*},\xi^{*}, w^{*}, \eta^{*}))$, by the above$\{W_{j,k}\}_{j,k\geq 0}$

(3.3)

Then

$\{W_{j+1,k}W_{j,k+1}W_{\mathrm{o},\mathrm{o}}=P(,\xi,w,\eta)+Q(z^{*}=\frac{1}{k+1}\{\langle\partial_{\eta},\partial_{w}*\rangle W_{j,k}’+=\frac{z_{1}}{j+1}\{\langle\partial_{\xi},\partial_{z}*\rangle W_{j,k}+\xi^{*},w^{*}k_{1}^{1}+k_{2}^{2}’=kj+j=jk_{1}+k_{2}^{2}kj_{1}+j=j\Sigma\langle\partial’ W\partial_{z^{*}}W_{j_{2},k_{2}}\rangle\}\sum_{=}^{\xi j_{1\prime}k_{1}}\eta^{*})\langle\partial_{\eta}W_{\mathrm{j}_{1},k_{1}}’,\partial_{w^{l}}W_{j_{2},k_{2}}\rangle\}’$$Rjik=W_{j,k}(z,\xi, w,\eta, z^{*}, \xi^{*}, w^{*}, \eta^{*})|z^{*}w^{*}=w,*\eta^{*}=\eta=z,\xi=\xi$

Suppose there exist $C_{P}(=C_{Q})>0$ , $d>0$ , and an open subset $U(\supset K)$ofsatisfying the following:$S^{*}X\cross S^{*}\mathrm{Y}$(1) $P$ and are holomorphic in $\gamma^{-1}(U;d, d)$ ,(2) $|P(z,\xi, w, \eta)|$ , and, where.
$Q$$|Q(z,\xi,w,\eta)|\leq C_{P}\cdot\tilde{\Lambda}(|\xi|, |\eta|)$on$\gamma^{-1}(U;d, d)$$\tilde{\Lambda}(|\xi|, |\eta|):=\min\{\Lambda_{1}(|\xi|), \Lambda_{2}(|\eta|)\}$$V:=\gamma^{-1}(U;d, d)\cross\gamma^{-1}(U;d, d)$,$V^{\epsilon_{1},\epsilon_{2}}:=\{(z, w,\xi,\eta, z^{*},w^{*},\xi^{*},\eta^{*})\in V;|\xi’-\xi|\leq\epsilon_{1}|\xi|$,Then we obtain the following lemma$|\eta’-\eta|\leq\epsilon_{2}|\eta|$,$|z^{*}-\prime z^{*}|\leq\epsilon_{1}$$|w^{*}-\prime w^{*}|\leq\epsilon_{2}\Rightarrow(z, w, \xi’,\eta’, z^{*}w^{*}\xi^{*}’,’,,\eta^{*})\in V\}$

.

,

155
Lemma 3.9. Suppose$\{C_{j,k}^{(\mu,\nu)}\}_{j,k,\mu,\nu\geq 0}$(1)

satisfy the following conditions:$C_{j+1,k}^{(\mu,\nu)}$,$\geq\frac{9ne^{10}}{j+1}\{C_{j,k}^{(\mu,\nu)}(j+1)^{2}+\sum^{*}(j_{1}+1)(j_{2}+1)C_{j_{1},k_{1}}^{(\mu_{1},\nu_{1})}C_{j_{2},k_{2}}^{(\mu_{2},\eta)}\}$(2)$C_{j,k+1}^{(\mu,\nu)}$$\geq\frac{9me^{10}}{k+1}\{C_{j,k}^{(\mu,\nu)}(k+1)^{2}+\sum^{**}(k_{1}+1)(k_{2}+1)C_{j_{1},k_{1}}^{(\mu_{1},\nu_{1})}C_{j_{2},k_{2}}^{(\mu_{2},\eta)}\}$(3) $C_{0,0}^{0,0}\leq C_{P}+C_{Q}$ ,(4) $C_{j,k}^{(\mu,\nu)}\geq 0(j, k\geq 0, 0\leq\mu\leq j, 0\leq\nu \leq k)$ ,(5)(otherwise).
Here, the sum,
mean $\sum j_{1}+j_{2}=j$, ’,$C_{j,k}^{(\mu,\nu)}=0$$\sum^{*}$$\sum^{**}$$\sum j_{1}+j_{2}=j$$k_{1}+k_{2}=k$,$\mu_{1}+\mu_{2}=\mu-1$

Then for eachfollowing hold:(3.4)$\epsilon_{1}$and$\nu_{1}+\nu_{2}=\nu$$\epsilon_{2}$

,

, ’
,
,

respectively.

$k_{1}+k_{2}=k$$\mu_{1}+\mu_{2}=\mu$$\nu_{1}+\iota \mathrm{q}=\nu-1$

such that $0<\epsilon_{1}\ll 1$ and $0<\epsilon_{2}\ll 1$ , the$\frac{C_{j,k}^{(\mu,\nu)}}{\epsilon_{1}^{2j}\epsilon_{2}^{2k}|\xi|^{j}|\eta|^{k}}(\overline{\Lambda}(|\xi|, |\eta|)+\tilde{\Lambda}(|\xi^{*}|, |\eta^{*}|))$$|W_{j,k}|\leq 0$$\leq\mu\leq j\sum_{0\leq\nu\leq k},$,$\cross(\Lambda_{1}(|\xi|)+\Lambda_{1}(|\xi^{*}|))^{\mu}(\Lambda_{2}(|\eta|)+\Lambda_{2}(|\eta^{*}|))^{\nu}$on

for all ,$V^{\epsilon_{1},\epsilon_{2}}$$j$$k\geq 0$.

The following lemma guarantees the existence ofsatisfying theconditions ffom (1) through (5) of the previous lemma.
$C_{j,k}^{(\mu,\nu)}$

Lemma 3.10. The following sequence
ditions from (1) through (5) of the previous lemma.

$\{C_{j,k}^{(\mu,\nu)}\}_{j,k,\mu,\nu\geq 0}$

satisfies the con$0\leq\nu$ $\leq k)$$C_{j,k}^{(\mu,\nu)}:=\{\begin{array}{l}lB^{j+k}(j+\mathrm{l})^{j-\mu-3}(k+\mathrm{l})^{k-\nu-3},(0\leq\mu\leq j0,(otherwise)\end{array}$,where 1and $B$ are constants satisfying $\geq\max\{C_{P}+C_{Q}, 1\}$ and$B\geq 72l$ $\max\{m, n\}\cdot$$(c^{2}+1)$ andis a constant satisfying thefollowing T. Aoki ’s iequality;$l$$\cdot$$e^{10}\cdot$$c$(3.5) $\frac{1}{j+1}\sum_{\mu=0}^{\nu-1j-}\sum_{k=\mu}^{\nu+\mu+1}(k+1)^{k-\mu-2}(j-k+1)^{j-k-\nu+\mu-1}\leq c(j+1)^{j-\nu-2}$

156
for all j,$\nu$such that $0\leq\nu$ $-1\leq j$ .

(continued) We can prove the theorem using the above two lemmas.

REFERENCES[A1] Aoki, T., Calcul exponentiel des opirateurs microdifferentiels d ’ordre infini. I,
Ann. Inst. Fourier, Grenoble 33-4 (1983), 227-250.
[A2] –, Symbols and formal symbols of pseudodifferential operators, Advanced
Syudies in Pure Math. 4(K.Okamoto, ed.), Group Representation and Systemsof Differential Equations, Proceedings Tokyo 1982, Kinokuniya, Tokyo; NorthHolland, Amsterdam-New York-Oxford, 1984, pp.181-208.
[A3] –, Exponential calculus of pseudodifferential operators. (Japanese)
S\^ugaku 35 (1983), n0.4, 302-315.
$.[\mathrm{A}4]$–, The exponential calculus of microdifferential operators of infinite order III. Proc. Japan Acad. Ser. Amath Sci. 59 (1983), n0.3, 79-82.

[A5] –, The erponential calculus of microdifferential operators of infinite order II. Proc. Japan Acad. Ser. AMath. Sci. 58 (1982), no.4,154-157.
[A6] –, The espnential calculus of microdifferential operators of infinite order I. Proc. Japan Acad. Ser. AMath. Sci. 58 (1982), n0.4,58-61.
[A7] –, The theory of symbols of pseudodifferential operators with infinite
order. Lectures in Mathematical Sciences (Japanese), Univ. Tokyo, 14 (1997).
[A8] –, Invertibility for microdifferential operators of Infinite Order., Publ.
RIMS, Kyoto. Univ. 18 (1982) 1-29.

[K1] Kataoka, K., On the theory of Radon transformations of hyperfunctions andits applications, Master’s thesis in Univ. Tokyo, 1976 (Japanese).
[K2] –, On the theory of hyperfunctions, J. Fac. Sci. Univ. Tokyo Sect. IA28
(1981),331-412.
[K3] –, Microlocal energy methods and pseudO-differential operators, Invent.
math. 81 (1985), 305-340.

[S-K-K] Sato, M., T.Kawai and M.Kashiwara, Microfunctions and PseudOdifferential Equations, Hyperfunctions and PseudO-Differential Equations(H.Komatsu, ed.), Proceeding, Katata 1971, Lecture Notes in Math. 287,
Springer, Berlin-Heidelberg-New York, 1973, pp.265-529.
[K-K] Kashiwara, M. and Kawai, T., On holonomic systems of micrO-differential
equations III, publ. RIMS, Kyoto Univ., 17 (1981), 813979.
[U] Uchikoshi, K., Microlocal analysis partial differential operators with irregular
singularities, J. Fac. Sci., Univ. Tokyo, Sect. IA, 30 (1983), 299-332