数理解析研究所講究録 1261 巻 2002 年 34-45

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BOUNDARY VALUES OF CLASSICAL FORMALSYMBOLS OFPSEUDO-DIFFERENTIAL OPERATORS
東京大・数理片岡 清臣 (Kiyoomi KATAOKA)

Graduate School of Mathematical Sciences
The University of Tokyo
1. INTRODUCTION
Let $X$ be acomplex manifold$\mathbb{C}_{w}\cross \mathbb{C}_{z}^{n}$

and $Z,$ $M$ be its submanifolds$Z=\{(w, z)\in X;{\rm Im} z=0\}\simeq Z^{\mathrm{R}}\supset M=\{{\rm Im} w=0, {\rm Im} z=0\}$Hereis the underlying real manifold of $Z$ . We denote by $(w,$the coordinates of $T^{*}X$ ;.

$Z^{\mathrm{R}}$$w=u+iv\in \mathbb{C},$ $z=x+iy\in \mathbb{C}^{n},$Then the sheaf$C\mathcal{O}_{Z}$$\tau\in \mathbb{C},$$\zeta=\xi+i\eta\in \mathbb{C}^{n}$$z;\tau$, $()$.

on$T_{Z}^{*}X=\{(w, z;\tau, () \in T^{*}X;\tau=0, {\rm Im} z=0, {\rm Re}\zeta=0\}$of microfunctions with aholomorphic parameter$w$

is defined by$C\mathcal{O}_{Z}:=\{f(u,v,x)\in C_{Z^{\mathrm{R};}}\overline{\partial}_{w}f=0\}$.

Hereis the sheaf of usual microfunctions on, and it is welknown that
is identified with the sheafof relative microfunctions as -modules. Then our main theorem is the following:
Theorem. Let $U= \sum_{j=-\infty}^{0}U_{j}(w, z, \zeta)$ be a classical formal symbolof a pseudO-differential operator with orderdefined in an R-conic
open set$Z^{\mathrm{R}}$$C_{Z^{\mathrm{R}}}$$C\mathcal{O}_{Z}$$C_{Z|X}$$\mathcal{E}_{X}^{\mathrm{R}}$$\leq 0$

$W_{f}\equiv\{(w, z;*, \zeta)\in T^{*}X;{\rm Im} w>0,$$|w|<r,$ $|z|<\kappa$ ,$|\zeta j|<\rho|\zeta_{n}|(1\leq\forall j\leq n-1),$ $|{\rm Re}\zeta_{n}|<\delta{\rm Im}\zeta_{n}\}$for some$r,$$\kappa,$$\rho,$and that there
inequalities:. We suppose thatsome constants $C,$ $\mu>0$ satisfying the following$\delta>0(\delta<1)$$e\dot{m}ts$$|U_{-p}(w, z, \zeta)|\leq\alpha_{p!|{\rm Im} w|^{-p-\mu}|\zeta|^{-p}}+1$

Then for any microfunction$f(x)\in C_{N}|(0_{j\dot{l}}dx_{n}),$$U_{j}\in \mathcal{O}(W_{f})(\forall j\leq 0)$on$W_{f}(\forall p\geq 0)$.

a section $U(w, x, D_{x})f(x)\in$
has a microfunction$\Gamma(\{w\in \mathbb{C};{\rm Im} w>0, |w|<r\}\cross\{(0;idx_{tl})\};C\mathcal{O}_{Z})$boundary value at $(0, 0; idx_{n})$ from ${\rm Im} w>0$ .
Further, we show by acounter-example that the growth condition
above is the best possible in some sense.

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2. PRELIMINARIES
We give here the precise meaning concerning boundary values ofwith. Let $K$ be areal analytic submanifold ofsections of$H$passing
incodimension 2, and be areal analytic hypersurface
through $K$ given as follows:$T_{Z}^{*}X$$C\mathcal{O}_{Z}$$T_{Z}^{*}X$$K=\{(w, x;i\eta)\in T_{Z}^{*}X;w=\psi(x, \eta)\}$$\subset H=\{(u+iv, x;i\eta)\in T_{Z}^{*}X;\Phi(u, v, x, \eta)=0\}$.

with h0is acomplex valued analytic function of
mogeneous degree 0with respect to , and $\Phi(u, v, x, \eta)$ is areal-valued
analytic function of $(u, v, x, \eta)$ of homogeneous degree 0with respect
to satisfying the following:Here$(x, \eta)$$\psi(x, \eta)$$\eta$$\eta$$\nabla\Phi\neq 0$

on $1=0$ ,$\Phi\circ\psi=0$.

It is known that we can choose aholomorphic contact transformationsuch thatdefined in aneighborhood$[mathring]_{p}\in K$

$S$

(2.1)

,

$\subset S(H)=\{{\rm Im} w^{*}=0\}\cap T_{Z}^{*}X$$S(K)=\{w^{*}=0\}\cap T_{Z}^{*}X$$\{$$S(T_{Z}^{*}X)$$\subset T_{Z}^{*}X$.

denotes thesignature of $S^{*}(d{\rm Im} w^{*})/d\Phi$ , where
Set$T^{*}Xarrow X$:
bypull-back of adifferential form by . We denotethe sheaf on $Z$ of
the canonical projection, and by
hyperfunctions with aholomorphic parameter .
$S^{*}(\omega)$$\sigma=\mathrm{t}\mathrm{h}\mathrm{e}$$S$$\omega$$\pi$$B\mathcal{O}_{Z}=C\mathcal{O}_{Z}|_{Z}$$w$

be apoint of $K$ , and $f(w, x)$ be a
Definition 2.1. Let
on $\{\Phi>0\}\cap U$ with an -conic neighborhood
section ofof . Then, $f(w, x)$ is said to have aboundary value at ffom $\Phi>0$
if there exist asmall neighborhood $U’$ of and asection $F(w^{*}, x^{*})\in$$[mathring]_{p}=([mathring]_{w},[mathring]_{x};i[mathring]_{\eta})$$U\subset T_{Z}^{*}X$$\mathbb{R}$$C\mathcal{O}_{Z}$$[mathring]_{p}$$[mathring]_{p}$$[mathring]_{p}$$\Gamma(\{\sigma{\rm Im} w^{*}>0\}\cap\pi(S(U’));B\mathcal{O}_{Z})$satisfying$(\mathcal{T}_{\mathrm{S}}^{-1}f)(w^{*}, x^{*})=[F(w^{*}, x^{*})]$. Here is aquantization
as sections ofof .
Though the boundary value $[F(u^{*}+i\sigma 0, x^{*})]$ itself depends on a
choice of , this definition neither depends on achoice of nor$\Gamma(\{\sigma{\rm Im} w^{*}>0\}\cap S(U’);C\mathcal{O}_{Z})$$\mathcal{T}_{\mathrm{S}}$

$S$

$S$

$\tau_{s}$$\mathcal{T}_{\mathrm{S}}$(shown as below).

. Howis represented by agerm ofRemark 2.2. Agerm ofrepresented
cannot be
ever it is well-known that asection ofIndeed,the cohomological
general.
inglobally by asection ofdefines asecond hyperfunction on
boundary value$\Sigma=\{(w^{*}, x^{*}; i\eta^{*})\in T_{Z}^{*}X;{\rm Im} w^{*}=0\}$ .
On the other hand the sheaf.
of second hyperfunctions is essentially larger than the sheaf$B\mathcal{O}_{Z}$$C\mathcal{O}_{Z}$$C\mathcal{O}_{Z}$$B\mathcal{O}_{Z}$$(\mathcal{T}_{S}^{-1}f)(u^{*}+i\sigma 0, x^{*})$$C_{M}|_{\Sigma}$$B_{\Sigma}^{2}$

36
Here $M=\{(w, z)\in X;{\rm Im} w=0, {\rm Im} z=0\}$ . Hence the definition
above is equivalent to the following:$(\mathcal{T}_{\mathrm{S}}^{-1}f)(u^{*}+i\sigma 0,x^{*})\in C_{M}|_{S([mathring]_{\mathrm{p}})}$.

Further this boundary value is equal to $[F(u^{*}+i\sigma 0, x^{*})]$ as afunction ofat. The uniqueness of this boundary value
for a section
is justified by
Schapira’s $N$-regularity property of -operator. We refer to $[2, 3]$ asfor the second microlocal analysis, and to [8] as for the N-regularityof -operator. Further as for aself-contained proof of the equivalentfact, see Proposition 4.1.11 of [5].
$\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{c}\succ$$(u^{*}, x^{*})$$S([mathring]_{p})$$(\mathcal{T}_{S}^{-1}f)(w^{*},x^{*})$$[F(u^{*}+i\sigma 0, x^{*})]\in C_{M}|_{S\mathrm{C}[mathring]_{p})}$$\overline{\partial}_{w}$$\overline{\partial}_{w}$3. BOUNDARY VALUES AND THE MAIN THEOREM
Lemma 3.I. Let $f(w)$ be a holomorphic function defined in a neighborhood of $G=\{w\in \mathbb{C};r_{0}<{\rm Im} w\leq r_{1}, |{\rm Re}\dot{w}|<r_{1}\}$ satishing an
estimate$|f(w)|\leq C|{\rm Im} w|^{-\mu}(\forall w\in G)$

for some constants
positive integer$p$and $r_{0}(0\leq r_{0}<r_{1}\leq 1)$ . Choose an
as $\mu<p\leq\mu+1$ . Then the $(p+1)$ -times integration$C,$ $\mu,$ $r_{1}>0$$g_{p}(w) \equiv\int_{1f}^{w}.1\frac{(w-w’)^{p}}{p!}f(w’)dw’$

is holomorphic in a neighborhood of $G$ , continuous up to ${\rm Im} w=r_{0}$ ,and satisfies$|g_{p}(w)|\leq C(2^{p}+1)/p!(\forall w\in G)$.

Theorem 3.2. Let
be a classical formal symbolapseudO-differential
operator
with
order
of
defined in an R-conic$U= \sum_{j=-\infty}^{0}U_{j}(w, z, \zeta)$$\leq 0$open set(3.1)$W_{f}\equiv\{(w, z;*, \zeta)\in T^{*}X;{\rm Im} w>0,$ $|w|<r,$ $|z|<\kappa$,$|\zeta_{j}|<\rho|\zeta_{n}|(1\leq\forall j\leq n-1),$ $|{\rm Re}\zeta_{n}|<\delta{\rm Im}\zeta_{n}\}$for some$r,$$\kappa,$$\rho,$and that there
inequalities:(3.2)$\delta>0(\delta<1)$$e\dot{m}ts$. We suppose that
$C,$ $\mu>0$ satisfying thefollowing$|U_{-p}(w, z, \zeta)|\leq C^{p+1}p!|{\rm Im} w|^{-p-\mu}|\zeta|^{-p}$Then, for a sufficiently large number(3.3)$U_{j}\in \mathcal{O}(W_{f})(\forall j\leq 0)$

some constants$\lambda>\mathrm{m}\mathrm{a}s\mathrm{c}\{1,2560C/r\}$on$W_{f}(\forall p\geq 0)$.

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we have 2holomorphic functions $E^{(k)}(w, z, z-z^{*}, s)(k=1,2)$ defined
in

(3.4)

$\mathcal{W}^{(1)}\equiv\{(w, z, z-z^{*}, s)\in \mathbb{C}\cross \mathbb{C}^{n}\cross \mathbb{C}^{n}\cross \mathbb{C};|w|<r/40$,$\max\{0, -80{\rm Im} w/\lambda r\}<{\rm Im} s\leq\lambda^{-1},$ $|{\rm Re} s|<\lambda^{-1}$$|z|<\kappa,$,$|z_{j}-z_{j}^{*}|>\rho^{-1}|z_{n}-z_{n}^{*}|(j=1, \ldots, n-1)\}$

,

and
(3.5)

$\mathcal{W}^{(2)}\equiv\{(w, z, z-z^{*}, s)\in \mathbb{C}\cross \mathbb{C}^{n}\cross \mathbb{C}^{n}\cross \mathbb{C};|{\rm Re} s|<\lambda^{-1}$$|w-ir/80|<r/320,$$|z|<\kappa,$$-(320\lambda)^{-1}<{\rm Im} s\leq\lambda^{-1}$,,$|z_{j}-z_{j}^{*}|>\rho^{-1}|z_{n}-z_{n}^{*}|(j=1, \ldots, n-1)\}$.

respectively satisfying the following:(3.6)$\sum_{k=1}^{2}E^{(k)}(w, z, z-z^{*}, s)=\sum_{|\alpha’|\geq 0,p\geq 0}\frac{\alpha’!}{|\alpha’|!}(\prod_{j=1}^{n-1}\frac{(z_{n}-z_{n}^{*})^{\alpha_{j}}}{(z_{j}^{*}-z_{j})^{\alpha_{j}+1}})$$\cross\int_{i/\lambda}^{s}\frac{(s-s^{*})^{p+\nu+3}}{(p+\nu+3)!}ds^{*}\int_{\lambda}^{\infty}U_{-p,\alpha’}$on$\mathcal{W}^{(1)}\cap \mathcal{W}^{(2)}\equiv \mathcal{W}^{(3)}$(3.7)( $w,$$z,$it)$\cdot(it)^{p-2}e^{:ts^{*}}\frac{dt}{2\pi}$:$\mathcal{W}^{(3)}=\{(w, z, z-z^{*}, s)\in \mathbb{C}\cross \mathbb{C}^{n}\cross \mathbb{C}^{n}\cross \mathbb{C}$$0<{\rm Im} s\leq\lambda^{-1},$$|z|<\kappa,$

Here we expand each$|{\rm Re} s|<\lambda^{-1},$;$|w-ir/80|<r/320$ ,$|z_{j}-z_{j}^{*}|>\rho^{-1}|z_{n}-z_{n}^{*}|(j=1, \ldots, n-1)\}$$U_{-p}$in$W_{r}$

as follows:$U_{-p}(w, z, \zeta)=,\sum_{\alpha\geq 0}U_{-p,\alpha’}(w, z, \zeta_{n})(\zeta’/\zeta_{n})^{\alpha’}$with$\zeta’/\zeta_{n}=(\zeta_{1}/\zeta_{n}, \ldots, \zeta_{n-1}/\zeta_{n})$Proof. Each

$U_{-p,\alpha’}(w, z, \zeta_{n})$.

. Further

$\nu$

,

is some positive integer.

is holomorphic in$W_{r}^{(1)}\equiv$$\{(w, z;\zeta_{n})\in \mathbb{C}^{n+1}\cross \mathbb{C};{\rm Im} w>0, |w|<r, |z|<\kappa, |{\rm Re}\zeta_{n}|<\delta{\rm Im}\zeta_{n}\}$and satisfies
(3.8)

on

$W_{r}^{(1)}$$|U_{-p,\alpha’}(w, z, \zeta_{n})|\leq p!C^{p+1}\rho^{-|\alpha’|}|{\rm Im} w|^{-p-\mu}|\zeta_{n}|^{-p}$. By the preceding lemma, we get holomorphic functions$V_{p,\alpha’}(w, z, \zeta_{n})\in \mathcal{O}(W_{r/2}^{(1)})$(3.9)

satisfying$\partial_{w}^{\mathrm{p}+\nu+1}V_{p,\alpha’}(w, z, \zeta_{n})=U_{-p,\alpha’}(w, z, \zeta_{n})$

,

$\{$

$|V_{p,\alpha’}(w, z, \zeta_{n})|\leq C^{p+1}2^{p+\nu+1}\rho^{-|\alpha’|}|\zeta_{n}|^{-p}$38onffoorr aU. HereTake aconformal mapping$W_{\mathrm{r}/2}^{(1)}$$p,$$\alpha’$$\nu$

is the integer satisfying $\mu<\nu\leq\mu+1$ .
:$\tilde{w}=\varphi(w)$$\varphi:\{w\in \mathbb{C};{\rm Im} w>0, |w|<r/2\}\overline{arrow}\{\tilde{w}\in \mathbb{C};|\tilde{w}|<1\}$such that $\varphi(0)=1$ ;for example,(3.10)$\varphi(w)=\{i(\frac{r+2w}{r-2w})^{2}+1\}/\{(\frac{r+2w}{r-2w})^{2}+i\}$$=\Phi(-2i(\sqrt{2}-1)w/r)\Phi(2i(\sqrt{2}+1)w/r)$ ,where $\Phi(t)=(1+t)(1-t)^{-1}$ . Then by expanding
into apower series of , we have expansions$V_{p,\alpha’}(\varphi^{-1}(\tilde{w}), z, \zeta_{n})$$\tilde{w}$(3.11)

Here$V_{p,\alpha’}(w, z, \zeta_{n})=\sum_{\ell=0}^{\infty}V_{p,\alpha’,\ell}(z, \zeta_{n})\varphi(w)^{\ell}(\forall(w, z, \zeta_{n})\in W_{\mathrm{r}/2}^{(1)})$$V_{p,\alpha’,\ell}(z, \zeta_{n})’ \mathrm{s}$.

are holomorphic functions in$W^{(2)}\equiv\{\zeta_{n}\in \mathbb{C};|z|<\kappa, |{\rm Re}\zeta_{n}|<\delta{\rm Im}\zeta_{n}\}$satisfying$|V_{p,\alpha’,\ell}(z, \zeta_{n})|\leq 2^{\nu}(2C)^{p+1}\rho^{-|\alpha’|}|\zeta_{n}|^{-p}$on$W^{(2)}$for all(3.12)$p,$$\alpha’,$$\ell$. Therefore we have$U_{-p}= \sum_{\alpha’,\ell}\partial_{w}^{\rho+\nu+1}\{\varphi(w)^{\ell}\}\cdot(\zeta’/\zeta_{n})^{\alpha’}V_{p,\alpha’,\ell}(z, \zeta_{n})$Now for alarge positive constantfor:$\lambda>1$.

we introduce 2kernel functions$V_{p,d,\ell}(z, \zeta_{n})\zeta_{n}^{p-2}$(3.13)$A_{p,\alpha,\ell}^{(1)},(z, s) \equiv\int_{\lambda(\ell+1)}^{\infty}V_{p,\alpha’,\ell}$(3.14)( it)$z,$$\cdot(it)^{p-2}e^{\dot{l}u_{\frac{dt}{2\pi’}}}$( it)$A_{p,\alpha,\ell}^{(2)},(z, s) \equiv\int_{\lambda}^{\lambda(\ell+1)}V_{p,\alpha’,\ell}$$z,$$\cdot(it)^{p-2}e^{\dot{l}ts}\frac{dt}{2\pi}$,which are holomorphic functions defined in$W^{(3)}\equiv\{s\in \mathbb{C};|z|<\kappa, {\rm Im} s>-\delta|{\rm Re} s|\}$with the estimates:(3.15)

on$W^{(3)}$$|A_{p,\alpha,\ell}^{(1)},(z, s)|\leq 2^{\nu}(2C)^{p+1}\rho^{-|\alpha’|}e^{-\lambda(\ell+1){\rm Im} s}/\pi$

for all(3.16)$p,$$\alpha’,$$\ell$. Further$A_{p,\alpha,\ell}^{(2)},(z, s)$are entire functions satisfying$|A_{p,\alpha,\ell}^{(2)},(z, s)|\leq 2^{\nu}(2C)^{p+1}\rho^{-|\alpha’|}e^{\lambda(\ell+1)(-{\rm Im} s)_{+}}/\pi$on. Here $(t)_{+}=t(\forall t\geq 0),$for all
Therefore for each $k=1,2$ and eachthe series$\{s\in \mathbb{C}\}$$p,$$\alpha’,$$\ell$$p,$(3.17)$\alpha’$$A_{p,\alpha}^{(k)},(w, z, s) \equiv\sum_{\ell=0}^{\infty}A_{p,\alpha,\ell}^{(k)},(z, s)\varphi(w)^{\ell}$$=0(\forall t<0)$.

39
converges locally uniformly in(3.18)$\{|\varphi(w)|e^{\lambda(-{\rm Im} s)_{+}}<1\}(\forall s\in \mathbb{C})$We note that
(3.19)

for $k=1$ ,for $k=2$ .

$(\forall s\in W^{(3)})$$\{|\varphi(w)|<e^{\lambda{\rm Im} s}\}$$\{$$\varphi(w)$at (3.10) is holomorphic in $|w|<(\sqrt{2}-1)r/2$ and$\log|\Phi(t)|=\frac{1}{2}\log(\Phi(\frac{2{\rm Re} t}{1+|t|^{2}}))=\sum_{\ell=0}^{\infty}\frac{1}{2\ell+1}(\frac{2{\rm Re} t}{1+|t|^{2}})^{2\ell+1}$$\leq 4({\rm Re} t)_{+}-(-{\rm Re} t)_{+}$

for $\forall t(|t|\leq 1/4)$ . Consequently if $|w|<(\sqrt{2}-1)r/8$ , we have$\log|\varphi(w)|\leq(6\sqrt{2}+10)(-{\rm Im} w)_{+}/r-(10-6\sqrt{2})({\rm Im} w)_{+}/r$

Hence$A_{p,\alpha}^{(1)},(w, z, s)’ \mathrm{s}$(3.20){.

are holomorphic in$(w, z, s)\in \mathbb{C}\cross W^{(3)}$; $|w|<(\sqrt{2}-1)r/8,$ $|z|<\kappa$ ,$(10+6\sqrt{2})(-{\rm Im} w)_{+}<\lambda r{\rm Im} s/2\}$$\supset\{(w, z, s)\in \mathbb{C}^{n+1}\cross \mathbb{C};|w|<r/20,$ $|z|<\kappa$,$(-{\rm Im} w)_{+}< \frac{\lambda r}{40}{\rm Im} s\}\equiv W_{\lambda}^{(4)}$becausehave the following estimates:$|\varphi(w)|e^{-\lambda{\rm Im} s}\leq e^{-\lambda{\rm Im} s/2}<1$on$W_{\lambda}^{(4)}$. At the same time we$|A_{p,\alpha}^{(1)},(w, z, s)| \leq\frac{2^{\nu}(2C)^{p+1}}{\pi\rho^{|\alpha’|}(1-e^{-\lambda{\rm Im} s/2})}\leq\frac{e2^{\nu+1}(2C)^{p+1}}{\pi\lambda\rho^{|\alpha|}{\rm Im} s},\cdot$on$W_{\lambda}^{(4)}\cap\{0<{\rm Im} s\leq 1/\lambda\}$. Consequently we obtain$| \partial_{w}^{\mathrm{p}+\nu+1}A_{p,\alpha}^{(1)},(w, z, s)|\leq(p+\nu+1)!\frac{re2^{\nu+1}(2C)^{p+1}}{80\pi\rho^{|\alpha’|}}(\frac{80}{\lambda r{\rm Im} s})^{p+\nu+2}$

on(3.21)$W_{\lambda}^{(5)}\equiv\{(w, z, s)\in \mathbb{C}^{n+1}\cross \mathbb{C};|w|<r/40,$$|z|<\kappa$,$\max\{0, -80{\rm Im} w/(\lambda r)\}<{\rm Im} s\leq 1/\lambda,$ $|{\rm Re} s|<\lambda^{-1}\}$Further(3.22)$A_{p,\alpha}^{(2)},(w, z, s)’ \mathrm{s}$are holomorphic in$\{(w, z, s)\in \mathbb{C}^{n+1}\cross \mathbb{C};|w|<(\sqrt{2}-1)r/8,$ $|z|<\kappa$,$\lambda(-{\rm Im} s)_{+}-(10-6\sqrt{2}){\rm Im} w/(2r)<0\}$$\supset\{(w, z, s)\in \mathbb{C}^{n+1}\cross \mathbb{C};|w|<r/20,$ $|z|<\kappa$${\rm Im} w>0,$,$-{\rm Im} w/(2\lambda r)<{\rm Im} s\}\equiv W_{\lambda}^{(6)}$and satisfy the following estimates,$|A_{p,\alpha}^{(2)},(w, z, s)| \leq\frac{2^{\nu}(2C)^{p+1}}{\pi\rho^{|\alpha’|}(1-e^{-{\rm Im} w/(2r)})}\leq\frac{re2^{\nu+1}(2C)^{p+1}}{\pi\rho^{|\alpha|}{\rm Im} w}$

.

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on

because$W_{\lambda}^{(6)}$on$|\varphi(w)|e^{\lambda(-{\rm Im} s)_{+}}\leq e^{-{\rm Im} w/(2r)}<1$

quently, setting(3.23)$W_{\lambda}^{(6)}\supset W_{\lambda}^{(7)}\equiv\{(w, z, s)\in \mathbb{C}^{n+1}\cross \mathbb{C};|z|<\kappa$$|w-ir/80|<r/320,$

Conse-$W_{\lambda}^{(6)}.$,${\rm Im} s>-(320\lambda)^{-1}\}$,

we get the following estimates:(3.24)$| \Psi_{w}^{\nu+1}A_{p,d}^{(2)}(w, z, s)|\leq(p+\nu+1)!,\frac{re2^{\nu+1}(2C)^{\mathrm{p}+1}}{\pi\rho^{|\alpha|}(r/320)^{p+\nu+2}}$

on
Fixing the initial point $s=i/\lambda,$ we apply again the preceding lemma to
for $k=1,2$ . That is, we have
holomorphic functions
satisfying$W_{\lambda}^{(7)}.$$\partial_{w}^{p+\nu+1}A_{p,d}^{(k)}(w, z, s)$$E_{p,d}^{(1)}(w, z, s)\in \mathcal{O}(W_{\lambda}^{(5)})$(3.25)$\{\begin{array}{l}\theta_{s}^{+\nu+4}E_{\mathrm{p},d}^{(1)}(w,z,s)=\theta^{+\nu+1}A_{p,\alpha}^{(1)},(ww,z,s)|E_{p,a}^{(1)},(w,z,s)|\leq\frac{2^{p+\nu+4}re2^{\nu+1}(2C)^{\mathrm{p}+1}}{80\pi\rho^{|\alpha|}},(\frac{80}{\lambda r})^{p+\nu+2}\end{array}$on
for allsatisfying$W_{\lambda}^{(5)}$(3.26)

on$p,$$\alpha’$, and $E_{p,d}^{(2)}(w, z, s)\in \mathcal{O}(W_{\lambda}^{(7)}\cap\{|s-i/\lambda|<2/\lambda\})$$\{\begin{array}{l}\Psi_{s}^{+\nu+4}E_{p,\alpha’}^{(2)}(w,z,s)=\partial_{w}^{\rho+\nu+1}A_{p,d}^{(2)}(w,z,s)|E_{p,\alpha}^{(2)},(w,z,s)|\leq,\frac{re2^{\nu+1}(2C)^{p+1}}{\pi\rho^{|a|}(r/320)^{\mathrm{p}+\nu+2}}(2/\lambda)^{p+\nu+4}\end{array}$. Choose A $(>1)$ as(3.27)$1280C/(\lambda r)\leq 1/2$ .
Then we can introduce 2kernel functions $E^{(k)}(w, z, z-z^{*}, s)$ for $k=1,2$$W_{\lambda}^{(7)}\cap\{|s-i/\lambda|<2/\lambda\})$for all$p,$$\alpha’$as follows:(3.28).

$E^{(k)} \equiv\sum_{|d|\geq 0,p\geq 0}\frac{\alpha’!}{|\alpha|!},E_{p,d}^{(k)}(w, z, s)\prod_{j=1}^{n-1}\frac{(z_{n}-z_{n}^{*})^{\alpha_{g}}}{(z_{j}^{*}-z_{j})^{\alpha_{\mathrm{j}}+1}}$

are holomorphic in
at (3.4), (3.5), respectively for$k=1,2$ . On the other hand, ffom (3.13), (3.14) and (3.11) we obtain
Here,$E^{(k)}$$\mathcal{W}^{(k)}$the folowing:(3.29),$\sum_{k=1}^{2}\partial_{w}^{\mathrm{p}+\nu+1}A_{p,\alpha}^{(k)},(w, z, s)=\partial_{w}^{\mathrm{p}+\nu+1(\sum_{\ell=0}^{\infty}\sum_{k=1}^{2}A_{p,\alpha,\ell(z,s)\varphi(w)^{\ell)}}^{(k)}}$$= \Psi_{w}^{+\nu+1}(\int_{\lambda}^{\infty}V_{p,\alpha},$$= \int_{\lambda}^{\infty}U_{-p,\alpha’}$( $w,$ $z,$ it)( $w,$ $z,$ it)$\cdot(it)^{p-2}e^{\dot{*}u}\frac{dt}{2\pi})$$\cdot(it)^{p-2}e^{ts}.\cdot\frac{dt}{2\pi}$41
for any$(w, z, s)\in W_{\lambda}^{(5)}\cap W_{\lambda}^{(7)}\cap\{|s-i\lambda^{-1}|<2\lambda^{-1}\}$$=\{(w, z, s)\in \mathbb{C}^{n+1}\cross \mathbb{C};|w-ir/80|<r/320$$|z|<\kappa,$$0<{\rm Im} s\leq\lambda^{-1},$$|{\rm Re} s|<\lambda^{-1}\}$,.

Hence we have our conclusion (3.6).

$\square$By using the expressions of the kernel functions obtained in the preceding theorem, we prove that $U(w, x, D_{x})f(x)$ has aboundary value
for any microfunction $f(x)$ . To do so, we
fromat
introduce actions of $E^{(*)}(w, z, z-z^{*}, s)$ on holomorphic functions $F(z)$similar to the Bony-Schapira actions of microdifferential operators on
holomorphic functions [3] (also see [4] concerning the action of ).
${\rm Im} w>\mathrm{O}$$w=\mathrm{O}$$\mathcal{E}_{X}^{\mathrm{R}}$Definition 3.3. We inherit the notation from the preceding theorem.
Let $F(z)$ be aholomorphic function defined in(3.30)$\Omega\equiv\{z\in \mathbb{C}^{n};|z’|<r’, |{\rm Re} z_{n}|<(3\lambda)^{-1}+r’, r’>{\rm Im} z_{n}>k|{\rm Im} z’|\}$$\cup\bigcup_{\sigma=\pm 1}\{z\in \mathbb{C}^{n};|z’|<r’, |z_{n}-\sigma/(3\lambda)|<r’\}$for positive small constants $r’(r’<1/(3\lambda))$ , and $k(<\rho/(2n))$ . Letat$E(w, z, z-z^{*}, s)$ be aholomorphic kernel function defined in, we define aholomorphic function(3.4). For asufficiently smallbydepending on$\mathcal{W}^{(1)}$$\epsilon>0$$(E*F)_{\lambda,\epsilon}(w, z)$(3.31)$\lambda,$$\epsilon$.

$\int_{i\epsilon}^{z_{n}}dz_{n}^{*}\int_{\gamma}dz^{*\prime}\int_{-1/(3\lambda)}^{1/(3\lambda)}E(w, z, z-z^{*}, z_{n}^{*}-z_{n}^{**})F(z^{*\prime}, z_{n}^{**})dz_{n}^{**}$

Here the path for$z_{n}^{*}$

is the line segment$z_{n}^{*}(t)=z_{n}+t(i\epsilon-z_{n})(0\leq t\leq 1)$

with iff, $7^{=}\{z_{j}^{*}=zj+R(z_{n}, z_{n}^{*}(t))e^{i\theta_{j}}(0\leq\theta_{j}\leq 2\pi);j=$$n-1\}$ with some $R(z_{n}, z_{n}^{*}(t))>\rho^{-1}|z_{n}-z_{n}^{*}(t)|=t\rho^{-1}|z_{n}-i\in|$ .
is the line graph passing throughFurther the path for

combining$1,$$z_{n}$$\ldots,$$z_{n}^{**}$$-(3\lambda)^{-1},$$-(3\lambda)^{-1}+ih{\rm Im} z_{n}^{*}(t),$$(3\lambda)^{-1}+ih{\rm Im} z_{n}^{*}(t),$$(3\lambda)^{-1}$for aconstant $h(1/2<h<1)$ . That is,$z_{n}^{**}(\theta;t)=\{\begin{array}{l}-1/(3\lambda)+3ih\theta \mathrm{I}\mathrm{m}z_{n}^{*}(\mathrm{t})(0\leq\theta\leq 1/3)(2\theta-1)/\lambda+ih\mathrm{I}\mathrm{m}z_{n}^{*}(\mathrm{t})(\mathrm{l}/3\leq\theta\leq 2/3)1/(3\lambda)+3ih(1-\theta)\mathrm{I}\mathrm{m}z_{n}^{*}(t)(2/3\leq\theta\leq 1)\end{array}$Indeed this integral is well-defined if ${\rm Im} w>0,$ $|w|<r/40,$ $|z|<$$|{\rm Re} z_{n}|<(3\lambda)^{-1},0<{\rm Im} z_{n}\leq\epsilon<r’$ and the following sets arecontained in :$\kappa,$$\Omega$$\{(z_{1}+t\omega_{1}, \ldots, z_{n-1}+t\omega_{n-1}, q+ih({\rm Im} z_{n}+t(\in-{\rm Im} z_{n})));0\leq t\leq 1$$|q|\leq(3\lambda)^{-1}(q\in \mathbb{R}),$ $|\omega_{1}|\leq\rho^{-1}|z_{n}-i\epsilon|,$,$|\omega_{n-1}|\leq\rho^{-1}|z_{n}-i\epsilon|\}$$\ldots,$,42$\{(z_{1}+t\omega_{1}, \ldots, z_{n-1}+\mathrm{t}\omega_{n-1}, \pm(3\lambda)^{-1}+iq({\rm Im} z_{n}+t(\epsilon-{\rm Im} z_{n})))$$0\leq t\leq 1,0\leq q\leq h,$ $|\omega_{1}|\leq\rho^{-1}|z_{n}-i\epsilon|,$$\ldots,$

The former set is contained in$\Omega$

if$\epsilon<r’$$|{\rm Re} z_{n}|<(2h-1)(\epsilon-{\rm Im} z_{n}),$$|z’|+(n/\rho)|z_{n}-i\epsilon|<r’$

The latter set is contained in$\Omega$;$|\omega_{n-1}|\leq\rho^{-1}|z_{n}-i\epsilon|\}$.

and${\rm Im} z_{n}>(k/h)|{\rm Im} z’|$

,

.

if

$|{\rm Im} z_{n}|\leq\epsilon<r’$$|z’|+(n/\rho)|z_{n}-i\epsilon|<r’$and.

Hence we obtain the folowing lemma:
Lemma 3.4. Let
Then$\epsilon(>0)$$(E*F)_{\lambda,\epsilon}(w, z)$(3.32)

be smaller than $\min\{\kappa/2, (1+2n/\rho)^{-1}r’\}$ .

is holomorphic in$\{{\rm Im} w>0,$ $|w|<r/40,$$|z’|<\epsilon,$ $|z_{n}|<\epsilon$,${\rm Im} z_{n}>k|{\rm Im} z’|,$ $|{\rm Re} z_{n}|<\epsilon-{\rm Im} z_{n}\}$let $E’(w, z, z-z^{*}, s)$ be a holomorphic kernelfunction defined in
at (3.5). Thenis holomorphic in a neighborhood$R\iota\hslash her$$\mathcal{W}^{(2)}$of$(E’*F)_{\lambda,\epsilon}(w, z)$$\{|w-ir/80|<r/320,$ $z=0\}$ .

Proof. We have only to prove the latter statement. In this case we

modify the paths of integrations as follows:$z_{n}^{*}(t)=z_{n}+t(i\epsilon-z_{n})$$(0\leq t\leq 1)$,$z_{n}^{**}(\theta;t)=\{\begin{array}{l}-1/(3\lambda)+3i\theta\psi(t)(0\leq\theta\leq 1/3)(2\theta-1)/\lambda+i\psi(t)(1/3\leq\theta\leq 2/3)1/(3\lambda)+3i(1-\theta)\psi(t)(2/3\leq\theta\leq 1)\end{array}$where$\psi(t)=\max\{{\rm Im} z_{n}^{*}(t), \epsilon\}$$\epsilon<\min\{(640\lambda)^{-1}, \epsilon\}$have, for any

with some smaU

. If we choose
and $t\in[0,1]$ we$\epsilon>0$$z_{n}=iy_{n}(y_{n}\in(-\epsilon, \epsilon))$$\psi(t)=\max\{y_{n}+t(\epsilon-y_{n}), \epsilon\}>\frac{t}{2}(\epsilon-y_{n})\geq\frac{nk}{\rho}t(\epsilon-y_{n})$Therefore$(E’*F)_{\lambda,\epsilon}(w, z)$.

is holomorphic in aneighborhood of$\{|w-ir/80|<r/320,$ $z’=0,$ ${\rm Re} z_{n}=0,$ $-\epsilon<{\rm Im} z_{n}<\epsilon\}$ .

This completes the proof.

$\square$43
The following is our main result. K. Uchikoshi [9] used asimilar
method (Bronshtein’s method) of considering boundary values of hol0morphic pseudodifferential operators for constructing fundamental s0lutions of weakly hyperbolic microdifferential operators. However thesituations are different from each other, and the proofs and results arecompletely independent.
The proof of this theorem is alittle long and the most part of theproof is devoted to the proof of the compatibility of actions of pseud0differential operators. So we omit the proof, which will be given inanother paper.
Theorem 3.5. Let $U= \sum_{j=-\infty}^{0}U_{j}(w,$ , be the classical formal sym$bol$of the pseudO-differential operator treated in Theorem 3.2. Thenfor any microfunction $f(x)\in C_{N}|(0;idx_{n}),$ a section $U(w, x, D_{x})f(x)\in$$\Gamma(\{w\in \mathbb{C};{\rm Im} w>0, |w|<r\}\cross\{(0;idx_{n})\};C\mathcal{O}_{Z})$ has a boundary value
in the sense of Definition 2.1.
at $(0, 0; idx_{n})$ from$z$$\langle$${\rm Im} w>\mathrm{O}$Remark 3.6. The growth order condition (3.2) for the lower orderis the best possible in the following sense:
terms ofFor any constant $k(1<k<2)$ there exists aclassical formal symbol
satisfying the following (1) $-(3)$ :
.
(1)
$C,$
$\mu>0$ we have(2) For some constants$\sum_{j=-\infty}^{0}U_{j}(w, z, \zeta)$$U= \sum_{j=-\infty}^{0}U_{j}(w, z, \langle)$$U_{j}\in \mathcal{O}(W_{r})(\forall j\leq 0)$$|U_{-p}(w, z, \zeta)|\leq C^{p+1}p!|{\rm Im} w|^{-kp-\mu}|\zeta|^{-p}$on$W_{r}(\forall p\geq 0)$.

(3) $U(w, x, D_{x})\delta(x_{n})$ does not have aboundary value at $(0, 0; idx_{n})$in microfunctions of $({\rm Re} w, x)$ from ${\rm Im} w>0$ .
Indeed, we can give an explicit example as follows:$U_{-p}(w, z, \zeta)\equiv p!\{-i(w/i)^{k}\}^{-p-1}\zeta_{n}^{-p-1}$ ,(3.33)

and $|\arg(w/i)|<\pi/2$ . It is easy to see that thewhere $p=0,1,2,$
above conditions 1, 2are satisfied and that$\ldots$$U(w, x, D_{x}) \delta(x_{n})=[\sum_{p=0}^{\infty}-\frac{1}{2\pi i}\{-i(w/i)^{k}\}^{-p-1}z_{n}^{p}\log z_{n}]$$=[ \frac{1}{2\pi i}\cdot\frac{1\mathrm{o}\mathrm{g}z_{n}}{z_{n}+i(w/i)^{k}}]$.

over
Here the equalities are valid for sections of$\{(0;idx_{n})\}$ with any small positive. Then by the following lemma we
get the condition (3) above for .
The following example is avariant of the example in [7] of second
hyperfunctions:be a constant satisfying 1 $<k<2$ . Then theLemma 3.7. Let$C\mathcal{O}z$$\{{\rm Im} w>\epsilon\}\cross$$\epsilon$

$U$

$k$

microfunction$f(w, x)= \frac{1\mathrm{o}\mathrm{g}(x+i0)}{x+i(w/i)^{k}}$estends to$\{(w, x;i\eta dx)\in \mathbb{C}\cross$44
as a microfunction with holomorphic parame. However $f(w, x)$ never has a microfunction boundary value at$(0, 0; idx)$ from ${\rm Im} w>0$ .
$\eta>0\}$$T_{\mathrm{R}}^{*}\mathbb{C};{\rm Im} w>0,$

ter

$w$

Proof. Consider aholomorphic function$F_{1}(w, z)= \frac{\log z-1\mathrm{o}\mathrm{g}\{(w/i)^{k}/i\}}{z+i(w/i)^{k}}$defined in $\{(w, z)\in \mathbb{C}^{2};{\rm Im} z>0,\pi(1-k^{-1})/2<\mathrm{a}r\mathrm{g}w<\pi\}$, where$-\pi<\arg\{(w/i)^{k}/i\}<\pi(k-1)/2<\pi/2$ . Further set$F_{2}(w, z)=F_{1}(w, z)+ \frac{-2\pi i}{z+i(w/i)^{k}}$.

Then $F_{2}(w, z)\in \mathcal{O}(\{(w, z)\in \mathbb{C}^{2};{\rm Im} z>0,0<\arg w<\pi(1+k^{-1})/2\})$ .
Hence the extension of $f(w, x)$ is given by$f(w,x)=\{$$[F_{1}(w,x+i\mathrm{O})](\pi(1-k^{-1})/2<\arg w<\pi)$ ,$[F_{2}(w,x+i\mathrm{O})](0<\arg \mathrm{w}<\pi(1+\mathrm{k}^{-1})/2)$

as microfunctions with holomorphic parameter . We suppose herethat $f(w, x)$ has amicrofunction boundary value at $(0, 0; idx)$ from${\rm Im} w>0$ . Therefore, we have aholomorphic
function $G(w, z)$ defined
in $\{{\rm Im} w>0, {\rm Im} z>0, |w|+|z|<\epsilon\}$ with somesatisfying$w$$\epsilon>0$$f(w, x)=[G(w, x+i0)]$

as sections of microfunctions with holomorphic parameter{($w,$ $x$;iqdx)$\in \mathbb{C}\cross T_{\mathrm{R}}^{*}\mathbb{C};|w|+|x|<\epsilon,{\rm Im} w>0,$

$w$

in

$\eta>0$}.

Since$0=\{x+i(w/i)^{k}\}(f(w, x)-[G(w, x+i0)])$$=[\log(x+i0)-\{x+i(w/i)^{k}\}G(w, x+i\mathrm{O})]$ ,we conclude that
$A(w, z)\equiv\log z-\{z+i(w/i)^{k}\}G(w, z)$$\in \mathcal{O}(\{{\rm Im} w>0, {\rm Im} z>0, |w|+|z|<\epsilon\})$extends holomorphically to $\{|w|+|z|<\epsilon’, {\rm Im} z=0, {\rm Im} w>0\}$ withsome smaller. Therefore by Kashiwara’s theorem on the local
version of Bochner’s tube theorem we can extend $A(w, z)$ to ahol0morphic functionin$\epsilon’>0$$\tilde{A}(w, z)$$\Omega=\{|w|+|z|<\epsilon’’, |{\rm Im} z|<\epsilon’’{\rm Im} w\}$

for some smaller$\epsilon’’>0$. Set$P(r, \theta)=(re^{\dot{l}\theta}, r^{k}e^{:(k\theta-\pi(k+1)/2)})\in\{(w, z)\in \mathbb{C}^{2};z+i(w/i)^{k}=0\}$

45
for $r>0,0<\theta<\pi$ . We note that $P(r, \theta)\in\Omega$ for any fl $\in(0, \pi)$with any sufficiently small
because $k>1$ . Therefore $H(w)\equiv$$\tilde{A}(w, -i(w/i)^{k})$ is aholomorphic function in$r>\mathrm{O}$$W=\{w\in \mathbb{C};0<\arg w<\pi, 0<|w|<\varphi(\arg w)\}$with apositive valued continuous function
hand, we have that$\varphi(\theta)$

on $(0, \pi)$ . On the other$H(re^{i\theta})=A(re^{i\theta}, r^{k}e^{ik\{\theta-\pi(1+k^{-1})/2\}})=k\log$ $r+ik\{\theta-\pi(1+k^{-1})/2\}$

for $\pi(1+k^{-1})/2<\theta<\pi$ , and that
$H(re^{i\theta})=A(re^{i\theta}, r^{k}e^{ik\{\theta+\pi(3k^{-1}-1)/2\}})=k\log r+ik\{\theta+\pi(3k^{-1}-1)/2\}$

coincides
for $0<\theta<\pi(1-k^{-1})/2$ . That is, $H(w)$ -klog
in the above
with 2different
domains, respectively. This contradicts with the connectedness of $W$ .
Thus $f(w, x)$ never have amicrofunction boundary value at $(0, 0; idx)$
from ${\rm Im} w>0$ .
$w\in \mathcal{O}(W)$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{s}-\pi(k+1)i/2\mathrm{a}\mathrm{n}\mathrm{d}-\pi(k-3)i/2$$\square$

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$d\acute{e}r\dot{\eta}v\acute{e}es$${\rm Res}$