数理解析研究所講究録 1431 巻 2005 年 165-175

165

Remarks on Local Solvability of Operatorswith Principal Symbol$\xi_{1}^{2}+\cdots+\xi_{n-1}^{2}+x_{n}^{2}\xi_{n}^{2}$

Seiichiro Wakabayashi (若林誠一郎)
University of Tsukuba (筑波大学)

1. Definitions and main results
Many authors have studied local solvability in the spaces of distributions andultradistributions. In the framework of distributions Hormander [6] gave a necessary condition of local solvability, that is, for a differential operator he provedthat the transposed operator of satisfies some estimates if is locally solvable. Conversely, Treves [15] and Yoshikawa [19] proved that the same type of
estimates implies that is locally solvable. In the frameworks of ultradistributions
and hyperfunctions the corresponding treatment is possible (see [4], [1], [3] and[16] .
In this article we shall study local solvability of pseudodifferential operatorsin the spaces of distributions andwith principal symbol
ultradistributions. In [5] Funakoshi proved that these operators are locally solvablein the space of hyperfunctions ( see, also, [16]). Our purpose is to illustrate, withthese examples, how to study local solvability in the spaces of distributions andultrad\‘istributions. For the details we refer to [17].
Let us first define Gevrey classes and symbol classes. Let be a regular
the space ofcompact set in , and let $>1$ and $h>0$ . We denote by
all $f\in C^{\infty}(K)$ satisfying, with some $C>0$ ,$P$${}^{t}P$

$P$

$P$

$P$

$)$

$\xi_{1}^{2}+\cdots+\xi_{n-1}^{2}+x_{n}^{2}\xi_{n}^{2}$$K$$\mathbb{R}^{n}$

(1.1)

these
and

$g^{\{\kappa\},h}(K)$$\kappa$$|D^{\alpha}f(x)|\leq Ch^{|\alpha|}|\alpha|!^{\kappa}$$x=(x_{1}, \cdots,x_{n})\in \mathbb{R}^{n}$$| \alpha|=\sum_{j=1}^{n}\alpha_{j}$space of allfor$\alpha$$f\in C^{\infty}(\mathbb{R}^{n})$,for any $x\in K$ and$D=\mathrm{i}^{-1}\partial=\mathrm{i}^{-1}(\partial/\partial x_{1}, \cdots,\partial/\partial x_{n})$$=(\alpha_{1},\cdots, \iota h)$with$\alpha\in(\mathbb{Z}_{+})^{n}$$\in(\mathbb{Z}_{+})^{n}$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}f\subset K$,, $\mathbb{Z}+=\mathrm{N}\cup\{0\}$. We also denote bysatisfying (1.1).

$\mathit{8}^{\{\kappa\},h}(K)$$\mathscr{D}_{K}^{\{\kappa\},h}$

and

the

$\mathscr{D}_{K}^{\{\kappa\},h}$iee

are Banach spaces under the norm defined by.

$|f; \mathcal{E}^{\{\kappa\},h}(K)|:=\sup_{x\in K,\alpha\in(\mathbb{Z}_{+})^{n}}|D^{\alpha}f(x)|/(h^{|\alpha|}|\alpha|!^{\kappa})$Let$\Omega$be an open subset of$\mathbb{R}^{n}$. We introduce the following locally convex spaces(Gevrey classes):$\mathscr{E}^{(\kappa)}(\Omega):=\underline{]\mathrm{i}\mathrm{m}}\mathscr{E}^{(\kappa)}(K)\backslash$$d^{(\kappa)}(K):=.\mu harrow 0\mathrm{m}g^{\{\kappa\},h}(K)$’,$K\subset\subset\Omega$$\mathscr{E}^{\{\kappa\}}(\Omega):=\underline{1\mathrm{i}\mathrm{m}}\mathscr{E}^{\{\kappa\}}(K)\grave{K}\subset\subset\Omega$$\mathscr{D}^{(\kappa)}(\Omega):=\underline{1\mathrm{i}_{\mathrm{I}}\mathrm{p}}\mathscr{D}_{K}^{(\kappa)}K\subset\subset\acute{\Omega}$’$g^{\{\kappa\}}(K):= \mathscr{E}^{\{\kappa\},h}(K)h\frac{1\mathrm{i}\varphi}{arrow’\infty}$,$\mathscr{D}_{K}^{(\kappa)}:=.\mathscr{D}_{K}^{\{\kappa\},h}\frac{\Phi}{\grave{h}arrow 0}$

,

,

,

,

$\mathscr{D}_{K}^{\{\kappa\}}:=\lim_{\vec{harrow\infty}}\mathscr{D}_{K}^{\{\kappa\},h}$$\mathscr{D}^{\{\kappa\}}(\Omega):=\mathscr{D}_{K}^{\{\kappa\}}K\frac{1\mathrm{i}\mathit{1}\mathrm{p}}{\subset \mathrm{e}\acute{\Omega}}$means that the closure of is compact and included in the interior
where
and
the strong dual spaces ofandof . We denote byarespacesofthese. Elements, respectively, where denotes (k) or,called ultradistributions (see, $e.g.$ , [11]). We also write $g*$ , , instead of
R. We say that a symbol, where ,classes. Let us define symbol.
and for any there$p(x,\xi)$ belongs to) if( resp.
is $C\equiv C_{A}>0$ (resp. there are $A>0$ and $C>0$ ) such that$A$$\overline{A}$$A\subset\subset B$$B\mathrm{o}$$B$$\mathscr{D}^{*}(\Omega)$$\mathscr{E}^{*/}(\Omega)$$\mathscr{D}^{*/}(\Omega)$$\{\kappa\}$$*$$g^{*}(\Omega)$$d^{*}(\mathbb{R}^{n})$$\cdots$$S_{*}^{m;\delta}$$m$$\ldots$$S_{(\kappa)}^{m,\delta}$(1.2)$\delta\in$$A$$p(x, \xi)\in C^{\infty}(\mathbb{R}^{n}\mathrm{x} \mathbb{R}^{n})$$S_{\{\kappa\}}^{m,\delta}.$$|p_{(\beta)}^{(\alpha)}(x,\xi)|\leq CA^{|\alpha|+|\beta|}(|\alpha|+|\beta|)!^{\kappa}\langle\xi\rangle^{m-|\alpha|}e^{\delta\{\xi\rangle}$

for any $x,\xi$$\in \mathbb{R}^{n}$$(1+|\xi|^{2})^{1/2}$

and$\alpha,\beta\in(\mathbb{Z}_{+})^{n}$, where$p_{(\beta)}^{(\alpha)}(x, \xi)=\partial_{\xi}^{\alpha}D_{X}^{\beta}p(x, \xi)$and$\langle\xi\rangle=$. We define$S_{(\kappa)}^{0,\infty}.:=\cup S_{(\kappa)}^{0,\delta}\delta>0^{\cdot}$

We also use the usual symbol classes
if $p(x,\xi)$that $p(x,\xi)$such that$\in S_{\rho,\delta}^{m}$’$S_{\{\kappa\}}^{+}:= \bigcap_{\delta>0}S_{\{\kappa\}}^{0_{j}\delta}$$S_{\rho,\delta}^{m}$. We sayandand there are positive constants, where $0\leq p$ ,$\in C^{\infty}(\mathbb{R}^{n}\cross \mathbb{R}^{n})$.

$\delta\leq 1$$m\in \mathbb{R}$$C_{\alpha,\beta}$$(\alpha,\beta\in(\mathbb{Z}_{+})^{n})$$|p_{(\beta)}^{(\alpha)}(x,\xi)|\leq C_{\alpha,\beta}\langle\xi\rangle^{m-\rho|\alpha|+\delta|\beta|}$

for any$x,\xi$$\in \mathbb{R}^{n}$and$\alpha,\beta\in(\mathbb{Z}_{+})^{n}$.

Next we shall define the Fourier transformation and pseudodifferential opera, and define
tors in the space of ultradistributions. Let $>1$ and$\kappa$$\epsilon\in \mathbb{R}$,$\overline{\mathscr{S}_{\kappa,\epsilon}}:=\{v(\xi)\in C^{\infty}(\mathbb{R}^{n});\exp[\epsilon\langle\xi\rangle^{1/\kappa}]v(\xi)\in \mathscr{S}\}$167

so thatdenotes the Schwartz space. We introduce the topology in
where
is a homeomorphism. Since
the mappingofis identified with, the dual spaceis dense in, and define. Let$\overline{\mathscr{S}_{\kappa,\epsilon}}$$\mathscr{S}$$\overline{\mathscr{S}_{\kappa,\epsilon}}\ni v(\xi)\vdash+\exp[\epsilon\langle\xi\rangle^{1/\kappa}]v(\xi)\in \mathscr{S}$$\mathscr{D}$$\overline{\mathscr{S}_{\kappa,\epsilon}}$$(=C_{0}^{\infty}(\mathbb{R}^{n}))$$\overline{\mathscr{S}_{\kappa,\epsilon}}$$\mathscr{S}_{\kappa,\epsilon}^{\overline{\prime}}$$\{\exp[\epsilon\langle\xi\rangle^{1/\kappa}]v(\xi)\in \mathscr{D}’;v\in \mathscr{S}’\}$$\epsilon\geq 0$,$\mathscr{S}_{\kappa,\epsilon}:=\mathscr{F}^{-1}[\overline{\mathscr{S}_{\kappa,\epsilon}}](=\mathscr{F}[\overline{\mathscr{S}_{\kappa,\epsilon}}]=\{u\in \mathscr{S};\exp[\epsilon\langle\xi\rangle^{1/\kappa}]\hat{u}(\xi)\in \mathscr{S}\})$

denote the Fourier transformation and the inverse Fourier
and
where
), respectively, and( ortransformation on
so that :. We introduce the topology infor. Then we can dethe dual space ofis a homeomorphism. Denote byandwhich mapof andandfine the transposed operators, we can de, respectively. Since
and
onto
:
, and in troduce the co pology so thatfine
. Then we
of
space
the
dual
denotes
is a homeomorphism.
. From the definitions it folows that (i)haveis a dense
on , (iii), (ii)forandand, (iv)
subspace of, and (vi)
and
, (v)
( see,
1f
as . Let $p(\xi,y, \eta)$ be a symbol satisfying$e.g.$ , [10] . So we write$\mathscr{F}^{-1}$$\mathscr{F}$$\hat{u}(\xi)\equiv \mathscr{F}[u](\xi):=f$ $e^{-ix\cdot\xi}u(x)$$\mathscr{S}’$$\mathscr{S}$$\mathscr{S}_{\kappa,\epsilon}$$\mathscr{S}_{\kappa,\epsilon}$$\mathscr{S}_{\kappa,\epsilon}^{t}$$t\mathscr{F}\mathscr{F}^{-1}$$t\mathscr{F}$$\mathscr{F}^{-1}$$\mathscr{F}$$\mathscr{S}_{\kappa,\epsilon}’$$\overline{\mathscr{S}_{\kappa,-\epsilon}}\subset \mathscr{S}_{\kappa,\epsilon}^{\overline{/}}$$\mathscr{S}_{\kappa,\epsilon}^{\overline{\prime}}$$\mathscr{S}_{\kappa,\epsilon}^{\overline{/}}$$\overline{\mathscr{S}_{\kappa,\epsilon}}arrow \mathscr{S}_{\kappa,\epsilon}$$\mathscr{F}$$u\in \mathscr{S}$$\mathrm{x}dx$$(\subset \mathscr{D}^{l})$$\mathscr{S}_{\kappa,\epsilon}’$$\mathscr{F}^{-1}$$\mathscr{S}_{\kappa,-\epsilon}:=^{t}\mathscr{F}^{-1}[\overline{\mathscr{S}_{\kappa,-\epsilon}}\rfloor\sim$$\mathscr{S}_{\kappa,-\epsilon}$$\mathscr{S}_{\kappa,-\epsilon}’$$\mathscr{S}_{\kappa,-\epsilon}$$\mathscr{S}_{\kappa,-\epsilon}^{\overline{\prime}}\subset \mathscr{S}’\subset$$\mathscr{S}_{\kappa,-\epsilon}’=\mathscr{F}[\mathscr{S}_{\kappa,-\epsilon}^{\overline{l}}]$$\epsilon$$\mathscr{S}_{\kappa,-\epsilon}’\subset \mathscr{S}’\subset \mathscr{S}_{\kappa,\epsilon}’$$\mathscr{S}_{\kappa_{?}\epsilon}^{\overline{\prime}}$$\mathscr{F}=^{t}\mathscr{F}$$\geq 0$$\mathscr{D}^{\{\kappa\}}\subset \mathscr{S}_{\kappa,+}:=\bigcup_{\epsilon>0}\mathscr{S}_{\kappa,\epsilon}$$\mathscr{S}_{\kappa}|\epsilon,$$\mathscr{D}\{\kappa\};$$\mathscr{F}^{(\kappa\rangle/}\subset g_{\kappa,-}$$\mathscr{D}(\kappa)\subset \mathscr{S}_{\kappa,\epsilon}\subset \mathscr{S}_{\kappa,\epsilon},$$\overline{\mathscr{S}_{\kappa,-\epsilon}}arrow$$\mathscr{D}^{(\kappa)}$$\mathscr{S}’$$:= \bigcap_{\epsilon>0}\mathscr{S}_{\kappa,\epsilon}’\subset$$\mathscr{F}\{\kappa\}$$\mathscr{E}^{\{\kappa\}\prime}\subset \mathscr{E}_{\kappa,0}:=\bigcap_{\epsilon>0}\mathscr{S}_{\kappa,-\epsilon}$$:= \bigcup_{\epsilon>0}\mathscr{S}_{\kappa,-\epsilon}$$\epsilon\geq\epsilon^{t}\geq\epsilon’$$\subset \mathscr{F}_{(\kappa)}:=\bigcup_{\epsilon>0}\mathscr{S}_{\kappa,\epsilon}’\subset \mathscr{D}^{(\kappa)\prime}$$\subset \mathscr{S}_{\kappa,-\epsilon^{t}}’,$$t\mathscr{F}$$)$$\mathscr{F}$$|\partial_{\xi}^{\alpha}D_{y}^{\beta}\partial_{\eta}^{\gamma}p(\xi,y,\eta)|\leq C_{\alpha,\gamma}A^{|\beta|}|\beta|!^{\kappa}\exp[\delta_{1}\langle\xi\rangle^{1/\kappa}+\ \langle\eta\rangle^{1/\kappa}]$, $h$, where A $>0$ ,
,
and
we
paper
are independent of . Throughout this, ,and , ,$(A,B, \cdots)$ constants depending on
and, respectively. Define

for any
and the positive constants$(\xi,y,\eta)\in \mathbb{R}^{n}\cross$denote by, $A$ , $B$ ,$\ldots$$C_{a_{1}b},\cdots$$\cdots$$\mathbb{R}^{n}\mathrm{x}$$\mathbb{R}^{n}$$\alpha,\beta$$\delta_{1}$$\gamma\in(\mathbb{Z}_{+})^{n}$$\in \mathbb{R}$$\beta$$C_{\alpha,\gamma}$$a$$C_{a,b},\cdots$$b$$\cdots$$a$$b$$p(D_{X},y,D_{y})u(x):=(2 \pi)^{-n}\mathscr{F}_{\xi}^{-1}[\int e^{-iy\cdot\xi}(\int e^{iy\cdot\eta}p(\xi,y, \eta)\hat{u}(\eta)d\eta)dy](x)$for.

$u \in \mathscr{S}_{\kappa,\infty}:=\bigcap_{\epsilon>0}\mathscr{S}_{\kappa,\epsilon}$

Proposition 1.1 (Proposition 2.3 of [10]), $p(D_{X},y,D_{y})$ maps continuously$\mathscr{S}_{\kappa}|\epsilon_{2}$

and

,

to

$\mathscr{S}_{\kappa,\epsilon_{1}}$aann$d\mathscr{S}_{\kappa,-\epsilon_{2}}’$$\epsilon_{1}<\kappa(nA)^{-1/\kappa}-\delta_{1}$Let$p(x,\xi)\in S_{(\kappa)}^{0,\infty}.$ttoo$\mathscr{S}_{\kappa,-\epsilon_{1}}’$

if,$\delta_{2}-\kappa(nA)^{-1/\kappa}<\epsilon_{2}$$\epsilon_{1}\leq\epsilon_{2}-\delta_{1}-h$.

. From Proposition 1.1 we can define $p(x,D)$ and

by
(1.3)

$p(x,D)=p(D_{X},y,D_{y})$ ,${}^{t}p(x,D)=q(D_{X},y,D_{y})$,$\mathrm{r}p(x,D)$168

where $p(\xi,y,\eta)=p(y, \eta)$ and $q(\xi,y,\eta)=p(y, -\xi)$ . It follows from Proposito
andtotion 1.1 that $p(x,D)$ and ${}^{t}p(x,D)$ map continuously, and that $p(x,D)$ and ${}^{t}p(x,D)$ map
if
for any. Similarly, we can define $p(x,D)$
. Let
to
to
and
.
to
to
and
,
to
and ${}^{t}p(x,D)$ by (1.3), which map
adoptedsolvability
localof
In order to state our main results we give definitions
here.
$\mathscr{F}_{(\kappa)}$$\mathscr{S}_{\kappa,\infty}$$p(x,\xi)\in S_{(\kappa)}^{0,\delta}.$$\epsilon\in \mathbb{R}$$\mathscr{S}_{\kappa,\epsilon+\delta}’$$\mathscr{S}_{\kappa_{\{}\infty}$$\mathscr{S}_{\kappa,\epsilon}’$$\mathscr{S}_{\kappa,\epsilon-\delta}$$\mathscr{S}_{\kappa,\epsilon}$$p(x,\xi)\in S_{\{\kappa\}}^{+}$$\mathscr{F}_{(\kappa)}$$\mathscr{S}_{\kappa,+}$$\mathscr{S}_{\kappa,+}$$\mathscr{F}_{\{\kappa\}}$$\mathscr{F}\{\kappa\}$$\mathscr{E}_{\kappa,0}$$\mathscr{F}_{\kappa,0}$) we say that( resp.
Rn. (i) For
Definition 1.2. Let
if there is an open neighborhood ofin$p(x,D)$ is locally solvable at),satisfying $p(x, D)u=f$ in ( inthere issuch that for any$p(x,D)$
solvable
is locally). Moreover, we say that
where $*=(\kappa)$ ( resp.
neighborhood
open
are
an
there
any
for
if
in a germ sense
at in). (ii) For $p(x, \xi)$satisfying $p(x,D)u=f$ in ( inof andwe say that $p(x_{7}D)$ is locally solvable at in ?’ if there is an open neighborhood
satisfying $p(x,D)u=f$ in ( inthere isof such that for any. Similarly, we define local solvability at in in a germ sense.
$x^{0}\in$$p(_{\backslash }x,\xi)\in S_{(\kappa)}^{0_{j}\infty}$$x^{0}$$S_{\{\kappa\}}^{+}$$U$$u\in \mathscr{F}_{*}$$f\in \mathscr{D}^{*\prime}$$x^{0}$$U$$\mathscr{D}^{*\prime}$$\mathscr{D}^{*/}(U)$$*=\{\kappa\}$$x^{0}$$x^{0}$$U$$f\in \mathscr{D}^{*\prime}$$\mathscr{D}^{*/}$$U$$u\in \mathscr{F}_{*}$$\mathscr{D}^{*/}(U)$$\in S_{1,0}^{m}$$x^{0}$$U$$x^{0}$$U$$u\in \mathscr{S}’$$f\in \mathscr{D}’$$x^{0}$$\mathscr{D}’$$\mathscr{D}’(U))$

Remark, (i) We remark that the above definitions of local solvability are slightly
local solvabiiity in a germ sense implies
different from usual ones, (ii) In$\mathscr{D}^{\{\kappa\};}$

local solvability” for properly supported pseudodifferential operators ( see [17]).
, and letby . Let. We denote orLet$\kappa>1$$(\kappa)$$\{\kappa\}$$\alpha(x,\xi)\in S_{*}^{1,0}.$$*$$L(x,\xi)=|\xi’|^{2}+x_{n}^{2}\xi_{n}^{2}+\alpha(x, \xi)$where$\xi’=(\xi_{1}, \cdots, \xi_{n-1})$for$\xi=(\xi_{1},\cdots,\xi_{n})\in \mathbb{R}^{n}$,. Then we have the following$when*=(\kappa)$ , and if $\kappa<2$ when $*=\{\kappa\}$ , thenTheorem 1,3. (i) If
can be. (\"u) Assume that
$L(x,D)$ is locally solvable at the origin in
written as$\kappa\leq 2$$\alpha(x,\xi)$$\mathscr{D}^{*/}$$\alpha(x, \xi)=\overline{\sum_{k=1}^{n1}}\alpha_{k}(x,\xi)\xi_{k}+x_{n}\mathrm{o}\mathrm{e}_{\iota}(x, \xi)+\infty(x,\xi)$

where $\alpha_{j}(x,\xi)\in S_{*}^{0;0}(0\leq j\leq n-1)$ and.
solvable at the origin in$\alpha_{n}(x,\xi)$$\in S_{*}^{1,0}.$,. Then $L(x,D)$ is locally$\mathscr{D}^{*\prime}$

Remark It was shown that $L(x,D)$ is locally solvable at the origin in the space
is an analytic symbol ( see, $e.g.$ , Chapter of [16]).
of hyperfunctions if
Let $P(x,D)$ be a differential operator of the form$\alpha(x,\xi)$$\mathrm{V}$189$P(x,D)=D_{1}^{2}+\cdots+D_{n-1}^{2}+x_{n}^{2}D_{n}^{2}-x_{n}a(x)D_{n}$$-(1+2ix_{1}+x_{1}^{2}b(x))D_{n}-\overline{\sum_{k=1}^{n1}}c_{k}(x)D_{k}+d(x)$ ,Then we have the following theorem which gives necessary conditions of local
solvability.

and $d(x)$ are analytic near
Theorem 1.4. (i) Assume that $a(x)$ , $b(x)$ , the$\kappa>2$. (ii)$P(x,D)$is not locally solvable at the origin in
the origin. Then
. Then $P(x,D)$ is not locally solvable
Assume thatat the origin in .
$c_{k}(x)$$\mathscr{D}^{*/}if$$a(x),b(x),c_{k}(x),d(x)\in \mathrm{C}^{\infty}(\mathbb{R}^{n})$$\mathscr{D}’$Remark From Hormander [7] and Olejnik and Radkevic [12] it follows that theoperator$D_{1}^{2}+\cdots+D_{n-1}^{2}+x_{n}^{2}D_{n}^{2}+(i\alpha(x)+x_{n}a(x))D_{n}+\overline{\sum_{k=1}^{n1}}b_{k}(x)Dk+c(x)$, $c(x)\in$, $a(x)$ ,is (hypoelliptic and locally solvable at the origin in if
andsuch that,is real-valued and there is( see, also, [13],Let A be an operator defined by Au(x) $=(x_{n}D_{n}u(x) \% D_{n}(x_{n}u(x)))/2$,, where$A=x_{n}D_{n}-\mathrm{i}/2$ .
Moreover, let
and
for
,
,
$\mathscr{D}’$$C^{\infty}(\mathbb{R}^{n})$$\mathrm{a}(\mathrm{x})$$b_{k}(x)$$\alpha(x)$$\gamma\in(\mathbb{Z}_{+})^{n}$$\gamma_{n}=0$$(D^{\gamma}\alpha)(0)$$\neq 0$$\mathrm{i}.e.$$Q(x,D)=D_{1}^{m}+ \sum_{|\alpha|\leq m,\alpha_{1}<m}a_{\alpha}D^{\alpha’}A^{a_{n}}$$m\in \mathrm{N}$$D_{1}^{\alpha_{1}}$...

$a_{\alpha}\in \mathbb{C}$$D_{n-1}^{\alpha_{n-1}}$$\alpha’=$$(\alpha_{1}, \cdots, \mathrm{o}\mathrm{e}_{\iota-1})$$\alpha=(\alpha_{1}, \cdots, \alpha_{n})\in(\mathbb{Z}_{+})^{n}$$D^{\alpha’}=$.

Theorem 1.5. $Q(x,D)$ is locally solvable at the origin in$\mathscr{D}’$.

Remark. By the above theorem the operator$P\equiv D_{1}^{2}+\cdots+D_{n-1}^{2}+x_{n}^{2}D_{n}^{2}+\overline{\sum_{k=1}^{n1}}a_{k}D_{k}+a_{n}x_{n}D_{n}+b$

C. (ii) In [13] and [14] Tahara
is locally solvable at the origin in ?’, where
studied more general operators and proved local solvability of those operators inin a germ sense, (iii) The argument used in the proof of Theorem 1.5 gives analternative proof of local solvability of differential operators with constant coefficients.
In \S 2 we shall give criteria ( abstruct necessary conditions and sufficient conditions) for local solvability. Using these results one can prove Theorems 1.3 and1.4. In \S 3 we shall prove Theorem 1.5.
$a_{k},b\in$$\mathscr{D}’$170

2. Outline of the proofs of Theorems 1.3 and 1.4
We begin with well-known results on local solvability in[19] and [6] .

$\mathscr{D}’$( see, e.g., [15],$)$, whereR. (i)
and $p(x,\xi)$ be a symbol in
Proposition 2.1. Let
If there is an open neighborhood of such that for any $s\geq 0$ there are$x^{0}\in \mathbb{R}^{n}$$m\in$$S_{\mathrm{I},0}^{m}$$U$$\ell\in \mathbb{R}$$x^{0}$

and $C>0$ satisfying$||\langle D\rangle^{s}u||\leq C\{||\langle D\rangle^{t\mathrm{r}}p(x,D)u||+||u||\}$

for any$u\in C_{0}^{\infty}(U)$, then $p(x,D)$ is locally solvable at$x^{0}$

in$\mathscr{D}’$denotes. Here. (ii) if $p(x,D)$ is$||f||$, $||f||=( \int|f(x)|^{2}dx)^{1/2}$ for
the -norm of $f$,
locally solvable at in ?’, then there is an open neighborhood $Uofx^{0}$ such thatand $C>0$ satisfying
for any $s\geq 0$ there are$L^{2}$$f\in L^{2}(\mathbb{R}^{n})$$\mathrm{i}.e.$$x^{0}$$\ell\in \mathbb{R}$$||\langle D\rangle^{s}u||\leq C||\langle D)^{t}{}^{t}p(x,D)u||$for any$u\in C_{0}^{\infty}(U)$.

Repeating the same argument as in the proof of Proposition 2.1 we shall prove. In
Theorems 2.4 and 2.5 below which give criteria for local solvability indoing so, we need the following$\mathscr{D}^{*/}$). Let $f(t)$ be a continuous functions on
Lemma 2,2 (Lemma 5,1.8 insuch that $f(t)\geq 0$ $(t \in[0,\infty))$ and $\lim_{t\prec\infty}f(t)/t=0$. Then there is ansatisfying the following: (i) $F(t)\geq$
analytic function $F(t)$ defined in$[1\mathrm{f}]$$[0, \infty)$$\mathbb{C}\backslash (-\infty,0]$$\max_{0<s\leq t}f(s)$for$t\geq 0$. (ii)$\lim_{tarrow+\infty}F(t)/t=0$. (iii)$\lim_{tarrow+\infty}t/(F(t)(1+$. (iv) $0<F’(t)\leq F(t)/t$ for $t>0$. (v) There is $C>0$ such that
$F(t)/t\leq CF’(t)$ for. (v)for $t>0$. (vii). (viii) There is $C>0$ such that$\log t)\overline{)}=0$$t\geq C$$\lim_{\mathrm{f}arrow+\infty}t^{2}F^{\prime/}(t)/F(t)=$$F^{\prime/}(t)<0$$0$$|(d/dt)^{k}F(t)|\leq C(2/t)^{k}k!F(t)$for $t>0$ and$k\in \mathbb{Z}+\cdot$

ifbelongs to
Definition 2,3. (i) We say that a symbol o). (ii) We say that a symbol
such that o)there issatisif there is a real analytic function $F(t)$ defined near
belongs to. (i) $F(t)\geq t/(1+\log t)$
fying the following conditions: (0)for $t\geq 1$ . (ii) $\lim_{tarrow+\infty}F(t)/t=0$ . (iii) $0<F’(t)\leq F(t)/t$ for $t\geq 1$ . (\‘iv)
for $t\geq 1$ .
There is $C>1$ such that $F(t)/t\leq CF’(t)$ for $t\geq C$ . (v)$|(d/dt)^{k}F(t)|\leq$(vi). (vii) There is $C>0$ such that$C(2/t)^{k}k!F(t)$ for.
and$(\xi)\in C^{\infty}(\mathbb{R}^{n})$$(\xi)=\epsilon\langle\xi\rangle^{1/\kappa}$$\epsilon\geq 1$$\mathscr{K}_{(\kappa)}’$$\omega(\xi)$$\in C^{\infty}(\mathbb{R}^{n})$$[1, \infty)$$\Psi_{\{\kappa\}}^{/}$$\mathit{0}\}(\xi)=F(\langle\xi\rangle^{1/\kappa})$$F^{\prime/}(t)<0$$\lim_{\mathrm{f}arrow+\infty}t^{2}F’(t)/F(t)=0$$t\geq 1$$k\in \mathbb{Z}_{+}$

Using the Hahn-Banach theorem and Poincar\’e’s inequality we can prove thefollowing

171

Theorem 2.4. Letthat for any$x^{0}\in \mathbb{R}^{n}$$\varpi(\xi)\in \mathscr{K}_{*}’$, and letthere are$\Omega$be an open neighborhood of$C>0$ such that$x^{0}$. Assume$\mu(\xi)\in \mathscr{K}_{*}’and$$||e^{q\}(D)}v||\leq C\{||e^{\mu(D)}{}^{t}p(x,D)v||+||v||\}$

for any$v\in \mathscr{D}^{(\kappa)}(\Omega)$. Then $p(x,D)$ is locally solvable at$x^{0}$

in$\mathscr{D}^{*\prime}$.

. (i) $Let*=(\kappa)$ , and assume that $p(x,D)$ is locally
Theorem 2.5. Let. Then there is an open neighborhood $Uofx^{0}$ such that forsolvable at inand $C>0$ satisfying
any
there are$x^{0}\in \mathbb{R}^{n}$$\mathscr{D}^{(\kappa)/}$$x^{0}$$\delta>0$$\epsilon>0$(2.1)

for any$||e^{\epsilon\langle D\}^{1/\kappa}}v||\leq C||e^{\delta\langle D\rangle^{1/\kappa}}{}^{t}p(x,D)v||$$v\in \mathscr{D}^{(\kappa)}(U)$(ii) $Let*=\{\kappa\}$ , and assume that $p(x,D)$ is locally solvable atthere is an open neighborhoodand $C>0$ satisfying
are

$U$

of

in
$\delta>0$$such$ thatwithfor any$x^{0}$$x^{0}$.

$\mathscr{D}^{\{\kappa\}/}$. Then$\delta<\mathrm{a}$

)

there

$\epsilon>0$,
for any
where is a positive constant determined by $p(x,\xi)$ . If $p(x,D)$ is properlyon the right-hand side of (2.2).
ported, then one can drop the $tem$. Let, whereIn the rest of this section we assume that $p(x, \xi)$(2.1)$v\in \mathscr{D}^{(\kappa)}(U)$$||e^{\epsilon\langle D\}^{1/\kappa}}v||\leq C\{||e^{\delta\langle D\rangle^{1/\kappa}}{}^{t}p(x,D)v||+||v||\}$$\sup-$$\epsilon_{0}$$||v||$$\in S_{*}^{m;0}$$\mathit{0})(\xi)\in \mathscr{K}_{*}’$$m\in \mathbb{R}$, and put
$p_{\Phi}(x, D):=e^{-\omega(D)}p(x,D)e^{\omega(D)}$.

Then we have$p_{a\}}(x, \xi)\sim\sum_{\alpha}\frac{1}{\alpha!}e^{\omega(\xi)}(\partial^{\alpha}e^{-\omega(\xi)})p_{(\alpha)}(x, \xi)$

Let $p>0$ , and let$p_{ca}^{p}(x, \xi)$be a symbol in$\tilde{p}_{cv}(x,\xi)\equiv p_{\omega}^{\rho}(x, \xi)$$S_{1,0}^{m}$.

satisfying$(\mathrm{m}\mathrm{o}\mathrm{d} S_{1,0}^{m-\rho})$.

Theorem 2.4 gives the following
be an open neighborhood of, and let$(\xi)\in \mathscr{K}_{*}’and$ $a>0$ there is $C>0$ such that

Theorem 2.6. Let
thatfor any o)(2.3)$x^{0}\in \mathbb{R}^{n}$$\Omega$$x^{0}$. Assume$||^{t}p_{\omega}^{\rho}(x,D)u||\geq a||\langle D\rangle^{m-p}u||-C||\langle D\rangle^{m-\rho-1}u||$.
If one can obtain the estimates of type (2.3), one can prove Theorem 1.3,
applying Theorem 2.6. For the detail we refer to [17]. Repeating the argumentsfor$u\in C_{0}^{\infty}(\Omega)$. Then $p(x,D)$ is locally solvable atin Cardoso-Treves [2], Ivrii-Petkov [9] and Iv
solutions we can prove Theorem 1.4 ( see [17])$x^{0}$$\ddot{\mathrm{m}}[8]$in$\mathscr{D}^{*/}$and constructing asymptotic

172

3. Proof of Theorem 1.5
and is a Hilbert space. So
Let$||(f,g)||_{X}=(||f||^{2}+||g||^{2})^{1/2}$. Let :
with norm $||(f,g)$ defined by
be a linear operator defined by$X=L^{2}(\mathbb{R}^{n})\mathrm{x}$$X=L^{2}(\mathbb{R}^{n})\oplus L^{2}(\mathbb{R}^{n})$$L^{2}(\mathbb{R}^{n})$$X$$\mathscr{F}$$||_{X}$$L^{2}(\mathbb{R}^{n})arrow$$X$$( \mathscr{F}u)(y)=(e^{y_{\hslash}/2}u(\oint,e^{y_{\hslash}}),e^{\mathrm{y}_{ll}/2}u(y^{t}, -e^{y_{n}}))$

Then$\mathscr{F}$is a unitary operator. We note that
. For
and
if

.

$\mathscr{F}(C_{0}^{\infty}(\mathbb{R}^{n}))\subset \mathscr{S}(\mathbb{R}^{n})\mathrm{x}$$|y_{n}|^{f}e^{y_{n}/2}\leq C_{l,k}$$y_{n}\leq k$$\ell_{7}k\in \mathbb{Z}_{+}$$(f,g)\in \mathscr{S}(\mathbb{R}^{n})\mathrm{x}$$a(y)D^{\alpha}(f,g)=(a(y)D^{\alpha}f(y),a(y)D^{a}g(y))$

RecaU that $A=x_{n}D_{n}-i/2$ and${}^{t}A=-A$ , we haveover, we have$\mathscr{S}(\mathbb{R}^{n})$$\mathscr{S}(\mathbb{R}^{n})$we define.

$Q(x,D)=D_{1}^{m}+ \sum_{|\alpha|\leq m,\alpha_{1}<m}a_{a}D^{\alpha’}A^{\alpha_{n}}$${}^{t}Q(x,D)=(-1)^{m}D_{1}^{m}+ \sum_{|\alpha|\leq m,\alpha_{1}<m}(-1)^{|\alpha|}a_{q}D^{\alpha’}A^{a_{n}}$$(\mathscr{F}(D_{k}u))(y)=D_{k}(\mathscr{F}u)(y)$

(
(3.1)

for

$\mathscr{F}$(An))(y)$(1 \leq k\leq n-1)$, where,,. Write$\overline{Q}(\eta)=\prod_{j=1}^{m}(\eta_{1}-\lambda_{j}(\eta’))$,is enumerated as$\{\lambda_{j}(\eta’)\}$${\rm Re}\lambda_{1}(\eta’)\leq{\rm Re}\ (\eta^{\prime/})\leq\cdots\leq{\rm Re}\lambda_{m}(\eta’’)$${\rm Im}\lambda_{j}(\eta^{\prime/})\leq{\rm Im}\lambda_{k}(\eta’)$It is obvious that${\rm Re}\lambda_{j}(\eta’)$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}v\subset\{y\in \mathbb{R}^{n};|y_{1}|\leq T\}$if,${\rm Re}\lambda_{j}(\eta^{\prime/})={\rm Re}\lambda_{k}(\eta’)$is continuous. Let $T>0$ , and let. Then we have

and $j<k$ .
$v\in \mathscr{S}(\mathbb{R}^{n})$$| \theta(\eta)|^{2}=|\int_{-T}^{T}e^{-iy_{1}\eta_{1}}\tilde{v}(y_{1}, \eta^{\prime/})dy_{1}|^{2}\leq 2T||\tilde{v}(y_{1_{7}}\eta’’)||_{L^{2}(\mathbb{R}_{y_{1}})}^{2}$

whereable set of;$\mathbb{R}^{n}$$(\eta_{1},\eta")$satisfy,, and let A be a Lebesgue measur. Let, wherefor $a.e$ .
such thatindenotesthemeasureR. Then (3.3) yields
Lebesgueand$\tilde{v}(y_{1}, \eta")$$\mathbb{R}$. More-$\overline{Q}(\eta)=\eta_{1}^{m}+\sum_{|\alpha|\leq m,\alpha_{1}<m}(-1)^{m-|\alpha|}a_{\alpha}\eta^{\alpha}$

(3.2)

(3.3)

. Since$(\mathscr{F}(^{t}Q(x,D)u))(y)=(-1)^{m}\overline{Q}(D)(\mathscr{F}u)(y)$$u\in \mathscr{S}(\mathbb{R}^{n})$where$=D_{n}(\mathscr{F}u)(y)$, since$=\mathscr{F}_{y’}[v(y_{1},y’)](\eta^{l\mathit{1}})$$\in\Lambda\}$$\mu(\Lambda(\eta’))\leq\epsilon$$\epsilon>0$$\eta’\in \mathbb{R}^{n-1}$$\mu$$||v||^{2}=(2 \pi)^{-n}\int_{\Lambda}|\hat{v}(\eta)|^{2}d\eta+(2\pi)^{-n}\int_{\mathbb{R}^{n}\backslash \Lambda}|\hat{v}(\eta)|^{2}d\eta$$\Lambda(\eta^{t/}):=\{\eta_{1}\in$173$\leq 2T(2\pi)^{-n}\int_{\mathbb{R}^{n-1}}(\int_{\Lambda(\eta’)}||\tilde{v}(\mathrm{y}_{1},\eta^{\prime/})||_{L^{2}(\mathbb{R}_{y_{1}})}^{2}d\eta_{1})d\eta’$$+(2 \pi)^{-n}\oint_{\mathbb{R}^{n}\backslash \Lambda}|\hat{v}(\eta)|^{2}d\eta$$\leq(T\epsilon/\pi)(2\pi)^{-n+1}\oint_{\mathbb{R}^{n-1}}||\tilde{v}(y_{1},\eta^{l/})||_{L^{2}(\mathbb{R}_{y_{1}})}^{2}d\eta’+(2\pi)^{-n}\int_{\mathbb{R}^{n}\backslash \Lambda}|\hat{v}(\eta)|^{2}d\eta$$=(T \epsilon/\pi)||v||^{2}+(2\pi)^{-n}\int_{\mathbb{R}^{n}\backslash \Lambda}|\hat{v}(\eta)|^{2}d\eta$.

Therefore, we have
(3.4)

if

$||v||^{2}/2 \leq(2\pi)^{-n}\int_{\mathbb{R}^{n}\backslash \Lambda}|\hat{v}(\eta)|^{2}d\eta$$T\epsilon/\pi\leq 1/2$.

Now we choose
A

(3.5)

{

$=$

$\eta\in \mathbb{R}^{n};|\eta_{1}-{\rm Re}\lambda_{j}(\eta’)|\leq\epsilon/(2m)$for some$j$}.

for each
Then A is a Lebesgue measurable set of and
(3.5)
have
(3.4)
(3.2),
weandFrom
is continuous.
since$\mathbb{R}^{n}$$\mu(\Lambda(\eta"))$$\leq\epsilon$$\eta^{ll}\in \mathbb{R}^{n-1}$,${\rm Re}\lambda_{j}(\eta’)$$|| \tilde{Q}(D)v||^{2}\geq(2\pi)^{-n}\int_{\mathbb{R}^{n}\backslash \mathrm{A}}|\tilde{Q}(\eta)v(\mathrm{A}\eta)|^{2}d\eta$$\geq(\epsilon/(2m))^{2m}(2\pi)^{-n}\oint_{\mathbb{R}^{n}\backslash \Lambda}|\hat{v}(\eta)|^{2}d\eta\geq 2^{-2m-1}(\epsilon/m)^{2m}||v||^{2}$if$v\in \mathscr{S}(\mathbb{R}^{n})$,$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}v\subset\{y\in \mathbb{R}^{n};|y_{1}|\leq T\}$and

$2T\epsilon\leq\pi$. This, together with

(3.1), gives(3.6)$||^{t}Q(x,D)u||^{2}=||\tilde{Q}(D)\mathscr{F}u||_{X}^{2}\geq 2^{-2m-1}(\epsilon/m)^{2m}||\mathscr{F}u||_{X}^{2}$$=2^{-2m-1}(\epsilon/m)^{2m}||u||^{2}$if$u\in C_{0}^{\infty}(\mathbb{R}^{n})$,$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}u\subset\{x\in \mathbb{R}^{n};|x_{1}|\leq T\}$

$AD^{\gamma}=D^{\gamma}(A+\mathrm{i}\gamma_{n})$and$2T\epsilon\leq\pi$. Let$\gamma\in(\mathbb{Z}_{+})^{n}$. Since, we have $D^{\gamma}{}^{t}Q(x,D)u={}^{t}Q^{\gamma}(x,D)D^{\gamma}u$, where $Q^{\gamma}(x,D)=$. (3.6) with $Q(x,D)$ replaced by$Q^{\gamma}(x,D)$$D_{1}^{m}+ \sum_{|\alpha|\leq m,\alpha_{1}<m}a_{a}D^{a’}(A+\mathrm{i}\gamma_{n})^{\alpha_{n}}$yields$||D^{\gamma}u||\leq 2^{2m+1/2}(mT/\pi)^{m}||D^{\gamma t}Q(x,D)u||$

with
for
is $C_{s}>0$ such taht$u\in C_{0}^{\infty}(\mathbb{R}^{n})$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}u\subset\{x\in \mathbb{R}^{n};|x_{1}|\leq T\}$

. Therefore, for any$s\in \mathbb{Z}_{+}$

there$||\langle D\rangle^{s}u||\leq C_{s}T^{m}||\langle D\rangle^{s}{}^{t}Q(x,D)u||$for$u\in C_{0}^{\infty}(\mathbb{R}^{n})$with$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}u\subset\{x\in \mathbb{R}^{n};|x_{1}|\leq T\}$

2.1, proves Theorem 1.5. This, together with Proposition

174

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