数理解析研究所講究録 1261 巻 2002 年 78-86

78

Gevrey Hypoellipticity for Extended
Grushin Class and
FBI-Transformation
Tadato MATSUZAWA\S 1. Introduction
In this monograph we shffi mention only the main results obtained recently on Gevrey hypoelipticity for extended class of Grushin operators.
Precise proof of them will be given in aforthcoming paper. We shall determine the non-isotropic Gevrey exponents for for Grushin operators by using
also the method of FBI-transformation given in [1] somewhat modifying it
asas by using the method of pseudodifferential operators.
get
an amelioration of the results obtained in the previous papers [3]and [4].
$\mathrm{w}\mathrm{e}\mathbb{I}$$\mathrm{T}\mathrm{h}\mathrm{u}\mathrm{s},\mathrm{w}\mathrm{e}$52. Gevrey functions and FBI-transformation
We denoteandWe remember the definition of Gevrey functions.
$x$

$=$

$(x_{1}, \ldots,x_{n})\in \mathrm{R}^{n}$$D_{\mathrm{j}}=-i\partial_{x_{\mathrm{j}}},j=1$,$\ldots$, , as usual.
$n$Definition 2.1. Let 0be an open set inand. Then wesay that,, $s_{j}>0$ , if for any compact subset $K$ of
0there are positive constants
andsuch that$\mathrm{R}^{n}$$\phi$$\in G^{\{\cdot\}}(\Omega)$$s$$\in C^{\infty}(\Omega)$$=(s_{1}, \ldots, s_{n})$$C_{0}$(2.1)$\phi$$C_{1}$$\sup_{x\in K}|D^{\alpha}\phi(x)|\leq C_{0}C_{1}^{|\alpha|}|\alpha|^{(\cdot,\alpha)}$,$\alpha\in \mathrm{Z}_{+}^{n}$,1991 Mathematics Subject Classifications: 34, 35, 46
Keywords and phrases: Gevrey hypoellipticity, degerenete elliptic operators, FBI-transformation

79
where

$\langle s, \alpha\rangle=s_{1}\alpha_{1}+\cdots+s_{n}\alpha_{n}$Proposition 2.1. Let
there are positive constants.

$\varphi\in C_{0}^{\infty}(\Omega)$$C_{0}$

and$C_{1}$$\sup_{x\in K}|D_{x_{\mathrm{j}}}^{k}\varphi(x)|\leq C_{0}C_{1}^{k}k!.j$

Then we have$\varphi\in G^{\{\cdot\cdot\cdots,.\}}1,2,n(\Omega)$,. If for any compact subset$K$

such thatof$\Omega$.

$j=1$ , 2, $\ldots,n$ , $k\in Z_{+}$.

The proof can be obtained by using FBI-transformation whose definition
will be given in (2.2).

Proposition 2.2. Let be a positive parameter. For any
such that
there exists a positive constant$\epsilon,0<\epsilon$

$a$

$<1$

,

$C_{e}$$|\partial_{x}^{k}e^{-ax^{2}}|\leq C_{e}^{k+1}a^{\frac{h}{2}}k!^{\frac{1}{2}}e^{-eax^{2}}$Let$u(x)\in C_{0}^{\infty}(\mathrm{R}^{n})$,$-\infty<x$ $<\infty$,$k=1$ , 2,.

$\ldots$. Then we have the Fourier inversion formula$u(x)=(2 \pi)^{-n}\int\int_{\mathrm{R}^{n}\mathrm{x}\mathrm{R}^{n}}u(y)e^{i(x-y,\xi\rangle}dyd\xi$

Now shift the contour of integration from.

$\mathrm{R}^{n}\cross \mathrm{R}^{n}\subset \mathrm{R}^{n}\cross \mathrm{C}^{n}$$\Gamma(y,\xi)=(y,\xi+i\langle\xi\rangle(x-y))$,$(y,\xi)\in \mathrm{R}^{n}\cross \mathrm{R}^{n}$to the contour.

Then we have the formula,$u(x)=(2 \pi)^{-n}\int\int_{\mathrm{R}^{f*}\mathrm{x}\mathrm{R}^{l*}}u(y)e-y,\xi)-\langle\xi\rangle(x-y)^{2}\alpha\langle x(:x -y,\xi)dyd\xi$

where$\langle\xi\rangle=$$(1 +\xi^{2})^{\frac{1}{2}}=(1 +\xi_{1}^{2}+\cdots+\xi_{n}^{2})^{\frac{1}{2}}$and$\alpha(x-y,\xi)=\prod_{j=1}^{n}(1+i(x_{j}-y_{j})\xi_{j}(1+\xi^{2})^{-\frac{1}{2}})$.

From this formula we define the FBI-transformation of $u(x)$ by
(2.2)

.

$Fu(x, \xi)=\int_{\mathrm{R}^{n}}u(y)e^{:(\approx-y,\xi)-(\xi)(x-y)^{2}}\alpha(x-y,\xi)dy$

80
The result of M. Christ, [1]is modified slightly in the following theorem
relating to the characterization of the class.
$G^{\{\ovalbox{\tt\small REJECT}\}}’$

Theorem 2.3. (cf. [1 ], Theorem 2.3.) Let$1,j$ $=1$ , 2,
, and
. Then the following four assertios are$equ\dot{l}vdent$
mutually:(a)in a neighborhood of.
(b) There exist
and a neighborhood of such that$s$$=(s_{1},s_{2}, \ldots, s_{n}),s_{\mathrm{j}}\geq$$u(x)\in C_{0}^{\infty}(\mathrm{R}^{n})$$\ldots,n$$u(x)\in G^{\{\cdot\}}$$x_{0}\in \mathrm{R}^{n}$$\mathrm{C},8$$V$$\in \mathrm{R}_{+}$,$|Fu(x,\xi)|\leq Ce^{-\delta\Sigma_{\mathrm{j}=1}^{n}|\xi_{\mathrm{j}}|^{\frac{1}{\mathrm{j}}}}.$$x_{0}$$(x,\xi)\in V\cross \mathrm{R}^{n}$.

(c) There exist an open neighborhoodof andsuch that, for each,, there exists a decomposition$U=U(x_{0})\subset \mathrm{C}^{n}$$\mathrm{R}_{+}$$\lambda\in \mathrm{R}_{+}^{n}$$x_{0}$$C,\mathit{5}$$\in$$|\lambda|\geq 1$in$u=g_{\lambda}+h_{\lambda}$$U\cap \mathrm{R}^{n}$such that $gx$ is holomorphic in $U$ ,$|g_{\lambda}(z)|\leq Ce^{C|\lambda||Im(z)|}$,$z\in U$and,$|h_{\lambda}(x)|\leq Ce^{-\delta\Sigma_{\mathrm{j}=1}^{n}x_{\mathrm{j}^{\mathrm{j}}}^{[perp]}}.$$x$$\in U\cap \mathrm{R}^{n}$.

(d) There exist an open neighborhoodof andsuch that for each,, there exists a decomposition$U=U(x_{0})\subset \mathrm{C}^{n}$$\lambda\in \mathrm{R}_{+}^{n}$$g_{\lambda}$$C,\delta$$\in \mathrm{R}_{+}$$|\lambda|\geq 1$in$u=g_{\lambda}+h_{\lambda}$such that$x_{0}$$U\cap \mathrm{R}^{n}$

is holomorphic in $\{z\in U;|Im(z_{\mathrm{j}})|\leq\langle\lambda\rangle.|\lambda|^{-1}\}\equiv U_{\lambda}$,$|g_{\lambda}(z)|\leq C$,$z\in U_{\lambda}$,

and$|h_{\lambda}(x)|\leq Ce^{-\delta\Sigma_{\mathrm{j}=1}^{n}x^{\frac{1}{\mathrm{j}^{\mathrm{j}}}}}.$,$x$$\in U\cap \mathrm{R}^{n}$.

Remark 2.1. By using appropriate cut-0ff functions for , the standard
method of calculation goes well to provewith the aid of PropOsition 2.2. Proof that (b)can be obtained by the$u$$(\mathrm{a})\Leftrightarrow(\mathrm{b})$$\Rightarrow(\mathrm{c})\Rightarrow(\mathrm{d})\Rightarrow(\mathrm{b})$81same method as in [1 ]. However, it might be needed to add asketch of the. it will be sufficient to consider one dimensional case.
proof of (a)(R) and let be real analytic in a
1. Let uE(i) The case ehere s
for same $45>0$ . Then forsay in ,(jneighborhood of$”\ovalbox{\tt\small REJECT}=(\mathrm{b})$$\ovalbox{\tt\small REJECT} \mathrm{u}$$C\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$$\{’\ovalbox{\tt\small REJECT}|" \mathrm{z}_{0}|<’\}$$\mathrm{z}_{\mathrm{C}\mathrm{h}}$$\mathcal{F}u(x,\xi)=\int u(y)e^{:(x-y,\xi)-(\xi\rangle(x-y)^{2}}\alpha(x-y,\xi)dy$

we make adeformation of the integral contour in$\omega_{\delta}$

and we have$(x,\xi)\in V\cross \mathrm{R}$$|\mathcal{F}u(x,\xi)|\leq Ce$ $-c(\xi\rangle,$,

and and are positive constants
where is asmall neighborhood of
independent of .
, so that(ii) The case where $s>1$ . We may suppose that
. By Proposition 2.2, taking , , C’
we havesufficiently large and sufficiently small we have
$V$

$C$

$x_{0}$

$c$

$\xi$

$u\in C_{0}^{\infty}(\omega_{\delta})\cap G^{\{\cdot\}}$$C_{1}$$u(y)\alpha(x-y,\xi)\in C_{0}^{\infty}(\omega_{\delta})\cap G^{\{\cdot\}}$

$C$

$c’$

$| \xi^{-N}\int e^{:(x-y)\xi}D_{y}^{N}(u(y)\alpha e^{-(\xi)(ax-y)^{2}})dy|$$.j!^{\frac{1}{2}}\langle\xi\rangle^{\mathrm{i}}2(N-j)!$$\leq|\xi|^{-N}C_{1}^{N+1}\sum_{j=0}^{N}$$(\begin{array}{l}Nj\end{array})$.

.

$\leq|\xi|^{-N}C^{N+1}N!.\frac{\langle\xi\rangle^{\mathrm{i}}2}{j!-\frac{1}{2}}$$\leq(\frac{CN}{|\xi|}.)^{N}e^{(\cdot-\frac{1}{2})|\xi|\neq_{-\mathrm{I}}}.$

Now take $N$ such thatabove quantity is estimated by$|N-(\epsilon|\xi|)^{\underline{1}}.|<1$with$C(C\epsilon)^{N}e^{(\cdot-\frac{1}{2})|\zeta|\cdot-\mathrm{T}}\tau^{\llcorner}\leq C’e^{-\mathrm{c}’|\xi|!}$,$\epsilon$.
sufficiently small. Then the.

$\square$$x\in\omega_{\frac{\delta}{2}},\xi\in \mathrm{R}$Remark 2.2. In Theorem 2.3, we can replace$\mathcal{F}u(x,\xi)$byfollows:(2.3)$\mathcal{F}.u(x,\xi)=\int u(y)e^{:(x-y,\xi\rangle-(\xi\rangle.(x-y)^{2}}\alpha.(x-y,\xi)dy$$\mathcal{F}.u(x,\xi)$

as

82
n
$\langle \mathrm{C}\rangle_{\mathrm{s}}\ovalbox{\tt\small REJECT}$$\mathrm{n}$$\mathrm{I}(1+4\mathrm{j})\ovalbox{\tt\small REJECT}_{\mathrm{t}}$$j\ovalbox{\tt\small REJECT} S$$\alpha.(x-y,\xi)=\coprod_{\mathrm{j}=1}^{n}(1+\frac{i}{s_{\mathrm{j}}}(x_{j}-y_{\mathrm{j}})\xi_{j}(1+\xi_{j}^{2})^{\frac{1}{2\cdot \mathrm{j}}-1})$.

The transform $Fu(z,\xi)$ extends, for each , to an entire holomorphic
function of. We can see that the same reasoning as in (c) and (d)
gives
$\xi$

$z\in \mathrm{C}^{n}$

(2.4)

for

$z$

$|Fu(z,\xi)|\leq Ce^{-\delta\Sigma_{=1}^{n}|\xi|^{[perp]_{\mathrm{j}}}}.e^{C|\xi||Im(z)|}$in asufficiently small neighborhood$V\subset \mathrm{C}^{n}$of$x_{0}$.

\S 3. Main results
We shall give the definition of the extended class of Grushin operators.
We write. Let $m$ be an even positive
integer and let
whose elements are$(x, y)=(x_{1}, \ldots, x_{k},y_{1}, \ldots,y_{n})\in \mathrm{R}^{k+n}$$\sigma=$$(\sigma_{1},\sigma_{2}, \ldots,\sigma_{k}),q=(q_{1},q_{2}, \ldots,q_{k})$rational numbers such that$\sigma_{1}$,$\ldots$,$\sigma_{p}>0$, $\sigma_{p+1}=\cdots=\sigma_{k}=0$ , $(0\leq p\leq k)$$q_{1}\geq q_{2}\geq\cdots\geq q_{p}\geq 0$,$q_{p+1}\geq\cdots\geq 0$,$q_{1}>0$.

Furthermore, we assume$mq_{\mathrm{j}}\in \mathrm{Z},j=1$,$\ldots,k$;$\frac{mq_{\dot{f}}}{\sigma_{j}}\in \mathrm{Z},j=1,2$,$\ldots,p$.

We pose the following major hypothesis:
Hypothesis (G) We suppose $1+q_{p}> \sigma_{0}=\max(\sigma_{1}, \ldots,\sigma_{p})$ .

Remark 3.1. Grushin’s original major hypothesis given in [2]was $1+$
. We shffi see that we can weaken this condition
as above. (See \S 5 and \S 6.) The assumption on , , given in [4]is alsoslightly weakened as above. When $p=0$ , we consider $q_{0}=0$ ,.
$q_{k}> \sigma_{0}=\max(\sigma_{1}, \ldots,\sigma)P$$q_{1}$$\ldots$$q_{k}$$\sigma_{0}=0$

83
into two parts such as $x=(x’, x^{\prime/})$ when $1\leq p<k$ , where$x’=$. We consider $x=x’$ when $p=k$
andand $x=x”$ when $\sigma=(0, \ldots,0)$ . Now we shall consider adifferential operator
with polynomial coefficients under the hypothesis (M):We divide$x$$x^{\prime/}=(x_{p+1}, \ldots, x_{k})$$(x_{1}, \ldots, x_{p})$

(3.1)

,

$a_{\alpha\beta\nu\gamma}\in \mathrm{C}$,$P(x’,y, D_{x}, D_{y})=( \sigma,\nu)+|\gamma|=(q,a)+|\alpha+\beta|-m\sum_{|\alpha+\beta|\leq m}a_{\alpha\beta\nu\gamma}x^{\prime\nu}y^{\gamma}D_{x}^{\alpha}D_{Y}^{\beta}$$\alpha,\nu\in \mathrm{Z}_{+}^{k}$,$\beta,\gamma\in \mathrm{Z}_{+}^{n}$,iscan be non zero only when
whereanon negative integer and we write such as $|\alpha+\beta|=|\alpha|+|\beta|$ . We may also.
consider $\nu=$$P(x’,y,\xi,\eta)$
satisfies the following condition.
We can see the symbol
Condition 1. (quasi-homogeneity) We have$|\gamma|=\langle q, \alpha\rangle+|\alpha+\beta|-m-\langle\sigma,\nu\rangle$$a_{\alpha\beta\nu\gamma}$$(\nu_{1}, \ldots, \nu_{p},0, \ldots,0)$$P(\lambda$ $-\sigma_{X’,\lambda^{-1}y,\lambda^{1+q}\xi,\lambda\eta)=\lambda^{m}P(x’,y,\xi,\eta)}$where$\lambda^{-\sigma}x’=$$(\lambda^{-\sigma_{1}}x_{1}, \ldots, \lambda^{-\sigma_{p}}x_{p})$and,$\lambda>0$,$x,\xi\in \mathrm{R}^{k},y,\eta\in \mathrm{R}^{n}$$\lambda^{1+q}\xi=(\lambda^{1+q1}\xi_{1}, \ldots, \lambda^{1+q\mathrm{t}}\xi_{k})$.

We add the two more conditions on .
Condition 2. (elipticity) The operator is elliptic for $|x’|+|y|=1$ .
Condition 3. (non-zero eigenvalue) For all , $|\omega|=1$ , the equation$P$$\mathrm{P}$$\omega$$P(x’,y,\omega, D_{y})v(y)=0$

has no non-trivial solution in$S(\mathrm{R}_{y}^{n})$in$\mathrm{R}_{y}^{n}$.

We set the Gevrey indices as follows.
$\theta_{j}=\max(\frac{1+q_{j}}{1+q_{k}}, \frac{1+q_{p}}{1+q_{p}-\sigma_{0}})$$\theta_{j}=\frac{1+q_{\mathrm{j}}}{1+q_{k}}$for $j=p+1$ ,

, ,
$k$

$\ldots$

for $j=1$ ,$d= \max\{^{\underline{\theta_{j}+q_{j}}}\}\cdot I_{n}=(d, \ldots,d)$$1+qj$,$d= \max\{\frac{\theta_{\mathrm{j}}+q_{\mathrm{j}}}{1+qj}\}1\leq j\leq k$

We also denot$1\leq j\leq k$$\ldots,p$

.

’

,

84
Theorem 3.2. (cf. [4 ]) Let 0be an open neighborhood of (0,and consider the equation(3.2)

where $u(x,y)\in \mathcal{D}(\Omega)$ and$G_{x,y}^{\{\theta,d\}}(\Omega)$in 0,$P(x’,yD_{x},D_{y})u(x,y)=f(x,y)$$f(x, y)\in G_{x,y}^{\{\theta,d\}}.(\Omega)$.

. Then we have

Remark 3.3. In the above theorem we can see that
(i)
(ii)

$p=0,\theta_{1}=1\Leftrightarrow(\theta,d)=(1, \ldots, 1)$$p=0,\theta_{1}>1$$\Rightarrow 1$$<d= \frac{\theta_{1}+q_{1}}{1+q_{1}}<\theta_{1}$Examples (a) For the operator$p=0$ , $q_{1}=k$ ,, $d=1$ .
and$P_{1}=D_{y}^{2}+y^{2k}D_{x}^{2}$$\sigma_{1}=0$,.

, ( $k=1,2$ ,$\ldots$$\theta_{1}=1$(b) For the operator $P_{2}=D_{y}^{2}+(x^{2l}+y^{2k})D_{x}^{2}$ , $(k, l=1, 2, \ldots)$ ,$q_{1}=k,\sigma_{1}=k/l$

and$\theta_{1}=\frac{l(1+k)}{l(1+k)-k}$,$d= \frac{\theta_{1}+k}{1+k}$.

(c) For the operator $P_{3}=D_{y}^{2}+(x^{2l}+y^{2k})(D_{x}^{2}+D_{z}^{2})$ , $(k,$ $l=1$ ,

have

,

$q_{1}=q_{2}=k$ $\sigma_{1}=k/l,\sigma_{2}=0,x’=x,x"=z;\theta_{1}=\frac{l(1+k)}{l(1+k)-k},\theta_{2}=$(d) For the operator $P_{4}=D_{y}^{2}+(x^{2l}+y^{2k})D_{x}^{2}+D_{z}^{2}$ , $(k,$ $l=1$ ,have$q_{1}=k,q_{2}=0,\sigma_{1}=k/l,\sigma_{2}=0;\theta_{1}=1$ $+k,\theta_{2}=1$

We remark that this operatorof Grushin$P_{4}$,$d= \frac{\theta_{1}+k}{1+k}=$does not satisfy the original hypo

85(e) An example with $1<d_{1}<d_{2}$ is given by $P_{4}=D_{y}^{2}+(x^{4}+y^{4})D_{x}^{2}+$, where we have$(x^{2}+y^{2})D_{z}^{2}$,$q_{1}=2$ $q_{2}=1$,$\sigma_{1}=\sigma_{2}=1$;$\theta_{1}=\theta_{2}=2$,$d_{1}= \frac{4}{3}<d_{2}=\frac{3}{2}$,$d= \frac{3}{2}$.

hypoellipticity of the operator
Remark 3.4. We omit the proof ofgiven in Theorem 3.2 since it is much simpler than that of Gevrey hypoellipticity. Then by using acut-0ff function for , we may suppose that
. By Thenin aneighborhood of
and,orem 2.3,(b), our main purpose becomes to prove that there exist asmalneighborhood of $(0,0)$ and positive constants and such that
$C^{\infty}$

$P$

$u$

$u$

$(0, 0)\in \mathrm{R}_{x_{1}y}^{k+n}$$f\in G_{x,y}^{\{\theta.d\}}$$f\in C_{0}^{\infty}(\Omega)$

$C$

$V$

$|Fu(\tilde{x},\tilde{y},\xi,\eta)|\leq e^{-\delta(\sum_{1}\epsilon_{j}^{+_{J+|\eta|\partial)}}}|- 1$

(3.3)

,

$\delta$,$(\tilde{x},\tilde{y},\xi,\eta)\in V\cross \mathrm{R}_{\xi,\eta}^{k+n}$where$F( \tilde{x},\tilde{y},\xi,\eta)=\int u(x,y)^{:}e^{((\tilde{x}-x,\xi)+\{\tilde{y}-y,\eta))-\langle\mu\rangle((\tilde{x}-x)^{2}+(\tilde{y}-y)^{2})}\alpha(\tilde{x}-x,\xi)$$.\alpha(\tilde{y}-y,\eta)dxdy$,$\mu=(\xi,\eta)$,$\alpha(\tilde{x}-x,\xi)=\prod_{j=1}^{k}(1+i)(\tilde{x}_{j}-x_{j})\xi_{j}(1+\xi^{2})^{-\frac{1}{2}}$$\alpha(\tilde{y}-y,\eta)=\prod_{\mathrm{j}=1}^{n}(1+i)(\tilde{y}_{j}-y_{j})\eta_{j}(1+\eta^{2})^{-\frac{1}{2}}$,.

We can prove the inequality (3.3) in three steps. We prove first the inequality(3.3) in the elliptic region:$R_{E}=\{(\xi,\eta);(\xi,\eta)\in \mathrm{R}_{\xi,\eta}^{k+n}, |\xi|\leq|\eta|\}$.

Next, we prove the inequalty (3.3) in the subellptic region:

,

$c>0$

$R_{S}= \{(\xi,\eta);(\xi,\eta)\in \mathrm{R}_{\xi,\eta}^{k+n}, (\frac{1}{c}\sum_{j=1}^{k}|\xi_{j}|^{\frac{1}{\theta_{\mathrm{j}}}})^{d}\leq|\eta|\leq|\xi|\}$

Finally, we obtain the inequality of the kind (3.3) in thedegenerate region:,$R_{D}= \{(\xi,\eta);(\xi,\eta)\in \mathrm{R}_{\xi,\eta}^{k+n},c|\eta|^{\frac{1}{d}}\leq\sum_{\mathrm{j}=1}^{k}|\xi_{j}|^{\frac{1}{\theta_{\mathrm{j}}}}\}$$L^{2}$.

since in

$c>0$

.

the

86
These steps will be completed by aprecision of the method given in [1]and[4].

REFERENCES[1] CHRIST, M.: INTERMEDIATE OPTIMAL GEVREY EXPONENTS occur,Comm. in Partial Differential Equations, 22(3-4),1997(359-379).
[2] GRUSHIN, V. V.: Hypoelliptic differential equations and pseudodifferential operators with operator-valued symbols, Math. USSR Sbornik,
17, N0.2, 1972(497-514).

[3] MATSUZAWA, T.: GEVREY HYPOELLIPTICITY FOR GRUSHINTORS, Publ. RIMS, Kyoto Univ., 33,1997(775-799).

$\mathrm{O}\mathrm{P}\mathrm{E}\mathrm{R}\mathrm{A}\sim$[4] HASHIMOTO, Y. HOSHINO, T. AND T. MATSUZAWA: NON-ISOTROPICGEVREY HYPOELLIPTICITY FOR GRUSHIN OPERATORS, to appear in
Publ. RIMS, Kyoto Univ.