Oct 6, 1980 - and $\theta eT_{x}^{*}$ ...... $\delta_{0}(f)=\xi_{\dot{f}}\partial^{\dot{f}}f+x^{i}\partial_{i}f-2f$ . .... $\Omega_{3}((\xi_{1})^{8}, (x^{1})^{\theta})(0)$.
TOKYO J. MATH. VOL. 4, No. 2, 1981
On Regular Fr\’echet-Lie Groups III A Second Cohomology Class Related to the Lie Algebra of Pseudo-Differential Operators of Order One
Hideki OMORI, Yoshiaki MAEDA, Akira YOSHIOKA and Osamu KOBAYASHI Okayama University, Keio University and Tokyo Metropolitan University
Introduction
Fourier integral operators have been defined by Hormander [5], and developed extensively by himself and many other authors as a tool of studying fundamental solutions of Cauchy problems of pseudo-differential equations of hyperbolic type. However, if we deal with a Fourier integral operator $F$ defined on a manifold, we see immediately that the expression of $F$ contains usually a huge ambiguity. Phase functions and amplitude functions do not have invariant meanings under tha change of local coordinate systems, and the rule of coordinate transformations is usually a very complicated one. Therefore, there arise several difficulties to define a topology, for instance, on the space of all Fourier integral operators of order . In [11], we gave a sort of global expression of Fourier integral operators and in [12] we defined a ”vicinity” of the identity operator $F^{0}$ in the space such that satisfies the properties of a topological local group. Moreover we have shown in [11] that can be expressed in an “almost“ unique fashion, if we fix a riemannian metric $\mathscr{G}^{-0}$
$0$
$\mathfrak{R}$
$\mathfrak{R}$
$F\in \mathfrak{R}$
$C^{\infty}$
on
.
$N$
Let us explain this situation at first. Let be the group of all $T^{*}N-\{0\}$ , where $T^{*}N$ is the symplectic transformations of order one on cotangent bundle a closed riemannian manifold $N$. It is known that of all contact transformations is isomorphic to the group $S^{*}N$ is a topological group . Since on the unit cosphere bundle through topology, we give the same topology on the $\mathcal{D}_{\rho}^{(1)}$
$C^{\infty}$
$\mathcal{D}_{\omega}(S^{*}N)$
$\mathcal{D}_{D}^{(1)}$
$\mathcal{D}_{\omega}(S^{*}N)$
under.
$C^{\infty}$
Received October 6, 1980
$\mathcal{D}_{\Omega}^{(1)}$
256
H. OMORI, Y. MAEDA, A. YOSHIOKA AND O. KOBAYASHI
the above isomorphism. $\mathcal{D}_{\rho}^{(1)}$
Let
.
$U$
be a neighborhood of the identity in
Let be the Fr\’echet space of all C-valued functions on $T^{*}N$ the closed unit disk bundle in . Define a diffeomorphism $\tau:D^{*}N\rightarrow T^{*}N$ by $\tau(x;\theta)=(x;(\tan(\pi/2)|\theta|)(\theta/|\theta|))$ , where indicates the point in $T^{*}N$ such that the base point is (the fibre and of $T^{*}N$ at ), and $D^{*}N$ is the open disk bundle in $T^{*}N$. We set . is a Fr\’echet space through the identification $C^{\infty}(\overline{D}^{*}N)$
$C^{\infty}$
$\overline{D}^{*}N$
$(x;\theta)$
$\theta eT_{x}^{*}$
$x$
$x$
$\sum_{c}^{0}$
$\sum_{C}_{\tau}^{0}=\tau^{-1*}C^{\infty}(\overline{D}^{*}N)$
be the Fr\’echet space of all C-valued functions on $N\times N$. For each $KeC^{\infty}(N\times N)$ , we define usually a smoothing operator with kernel $K$. A function $\nu(x, y)eC^{\infty}(N\times N)$ will be called a cut of function of breadth , if (i) $\nu(x, y)\geqq 0,$ $\nu(x, y)=\nu(y, x)$ , (ii) $\nu(x, y)\equiv 0$ if , where is the distance function. $\nu(x, y)\equiv 1$ (iii) if . Usually, we fix a cut off function with sufficiently small breadth , say where is the injectivity radius of $N$. Under these notations, one can define precisely a “vicinity” Let be a sufficiently small neighborhoods of the identity in and let be sufficiently small neighborhoods of 1 in and of in $F$ respectively. A Fourier integral operator is said to be $aeU_{1}$ and contained in if and only if there are $K\in V_{0}$ such that $F$ can be written in the form Let
$C^{\infty}(N\times N)$
$C^{\infty}$
$ K\circ$
$\epsilon$
$\rho(x, y)\geqq(2/3)\epsilon$
$\rho$
$\rho(x, y)\leqq(1/3)\epsilon$
$\nu$
$\epsilon0\\\Xi_{j}(x;r\xi)=r\Xi_{j}(x;\xi)\end{array}\right.$
Remark that (16)
$\frac{d}{dt}|_{t=0}\nu\sim u(\varphi_{t}(x;\xi))=\mathfrak{X}^{i}(x;\xi)\frac{\partial}{\partial X}|_{x=0}vu(.x(X, \xi))\sim$
$+\Xi_{j}(x;\xi)\frac{\partial\nu^{\sim_{u}}}{\partial\xi_{\dot{f}}}(x;\xi)$
,
where. $ae(X, \xi)=(ExpaeX;(dExp_{x}^{-1})_{X}^{*}\xi)$ . For more precise computations of the right hand members of (16), we need several notations as follows: We have used a brief notation and $x^{X}Y=.Zx$ then $Y$ can be written instead of $Exp$ $X$. If Ye by using $X$ and $Z$, which we shall denote by
.
$\# X$
$T_{x^{X}}$
’
(cf.
$Y=S(x;Z, X)$
(17)
\S 1. [11]).
We shall use also the following normal coordinate expressions around $(_{g}X;Y)=.\#(X,\tilde{Y})$
where
,
$(_{x}X;\eta)=_{x}(X, \xi)$
.
:
means $(Exp_{x}X;(dExp_{x})_{X}\tilde{Y})$ , and $\sim(X, \xi)=(Exp$ $X$; is given by (17), then as was already mentioned. If
$x(X,\tilde{Y})$
$(dExp_{x}^{-1})_{X}^{*}\xi)$
,
$x$
$Y$
261
FR\’ECHET-LIE GROUPS III
the normal coordinate expression of Using these notations, we see
(18)
$SwIl1$
be denoted by
$\tilde{S}(x;Z, X)$
.
$\left\{\begin{array}{l}\nu u(x;\xi)=\sim\int_{N}\nu(x, z)u(z)e^{-i\langle\epsilon|Z\rangle}dz,z=.Zx\\vu(.x(X, \xi))=\sim\int_{N}\nu(\cdot xX, z)u(z)e^{-i\langle\epsilon|Stx;z,x)\rangle}dz\sim\end{array}\right.$
Therefore,
(19)
we have
$\left\{\begin{array}{l}--f\frac{\partial\nu u}{\partial\xi_{\partial}}(x;\xi)=\Xi_{j}(x;\xi)\int_{N}\nu(x, z)u(z)\frac{\partial}{\partial\xi_{\dot{f}}}e^{-\sqrt{}\overline{-1}\langle\xi|Z\rangle}dz\\\Re^{l}\frac{\partial}{\partial X^{l}}|_{x=0}^{\sim}x\int_{N}\mathfrak{X}^{i}\frac{\partial v}{\partial X^{i}}|_{X=0}u(z)e^{-\sqrt{}\overline{-1}\langle\xi|Z\rangle}dz\\-\sqrt{-1}\int_{N}\mathfrak{X}^{l}\langle\xi|\frac{\partial\tilde{S}}{\partial X^{l}}(x;Z, 0)\rangle e^{-\sqrt{}\overline{-1}\langle|Z\rangle}\nu(x, z)u(z)dz\end{array}\right.$
Remark that
$\partial\tilde{S}^{\dot{f}}/\partial X^{l}(x;0,0)=-\delta_{l}^{j}$
(20)
.
So,
we set
$\frac{\partial\tilde{S}^{\dot{f}}}{\partial X^{l}}(x;Z, 0)+\delta_{l}^{j}=T_{lk}^{j}(x;Z)Z^{k}$
,
and we obtain the following: (21)
$\int_{\tau_{x}^{s}}\frac{d}{dt}|_{\iota=0}\nu u(\varphi_{t}(x;\xi))d\xi\sim$
$=\int_{\tau_{\dot{x}}}\int_{N}\mathfrak{X}^{i}(x;\xi)\frac{\partial\nu}{\partial X^{i}}|_{X=0}u(z)e^{-Y^{--1\langle\xi|Z\rangle}}dz+\sqrt{-1}\int_{\tau_{x}^{\iota}}\mathfrak{X}^{l}(x;\xi)\xi_{l}\nu^{\sim}u(x;\xi)d\xi$
-I
$\int[\frac{\partial \mathfrak{X}^{l}}{\partial\xi_{k}}\xi_{j}T_{lk}^{j}+\mathfrak{X}^{l}\tau_{lk}^{k}]e^{-\sqrt{}\overline{-1}\langle\xi|Z\rangle}\nu(x, z)u(z)dzd\xi$
$-\int_{\tau_{x}^{s}}\frac{\partial\Xi_{\dot{f}}}{\partial\xi_{j}}(x;\xi)\nu u(x;\xi)d\xi\sim$
.
for sufficiently small $X$, we see that the first term of Since the right hand side is a smoothing operator. We denote this operator by . Set $\partial v/\partial X^{i}\equiv 0$
$K_{z^{\circ}}u$
(22)
$b(x;\xi)=-|\int[\frac{\partial \mathfrak{X}^{l}}{\partial\xi_{k}}(x;\xi+\eta)(\xi_{j}+\eta_{j})T_{lk}^{\dot{f}}(x;Z)$
$+X^{l}(x;\xi+\eta)T_{lk}^{k}(x;Z)]e^{-i\langle\eta|7\rangle}dZd\eta$
and we get
,
262
H. OMORI, Y. MAEDA, A. YOSHIOKA AND
(23)
$0$
.
KOBAYASHI
$\int_{\tau_{\dot{x}}}\frac{d}{dt}|_{t=0}vu(\varphi(x;\xi))d\xi\sim$
$=\sqrt{-1}\int_{\tau_{\dot{x}}}X^{l}(x;\xi)\xi_{l}v^{\sim}u(x;\xi)d\xi+\int_{\tau_{\dot{x}}}b(x;\xi)^{\sim}vu(x;\xi)d\xi$
$-\int_{\tau_{x}^{*}}\frac{\partial\Xi_{\dot{f}}}{\partial\xi_{\dot{f}}}(x;\xi)vu(x;\xi)d\xi+(K_{x^{\circ}}u)(x)\sim$
,
and hence (24)
$\frac{d}{dt}|_{t=0}(F_{t}u)(x)$
$=\sqrt{-1}\int_{\tau_{x}^{*}}\mathfrak{X}^{l}(x;\xi)\xi_{l}v^{\sim}u(x;\xi)d\xi$
$+\int_{r_{x}}.[\frac{d}{dt}|_{t=0}a,(x;\xi)+b(x;\xi)-\frac{\partial\Xi_{j}}{\partial\xi_{\dot{f}}}(x;\xi)]vu(x;\xi)d\xi\sim$
$+((\frac{d}{dt}|_{t=0}K_{t}+K_{f})\circ u)(x)$
.
The first term (resp. second term) is a pseudo-differential operator of order 1 (resp. ), and the last term is a smoothing operator. Hence, we get . Conversely, let , and let be the principal symbol of P. is real valued and positively homogeneous of degree 1. By the definition of pseudo-differential operators, there are and $K\in C^{\infty}(N\times N)$ such that $0$
$(d/dt)|_{t=0}F_{t}e\sqrt{-1}\ovalbox{\tt\small REJECT}^{1}$
$Pe\sqrt{-1}\ovalbox{\tt\small REJECT}^{1}$
$\sqrt{-1}a_{1}$
$a_{1}(x;\xi)$
$\tilde{a}(x;\xi)e\sum_{C}^{0}$
$(Pu)(x)=\int_{\tau_{\dot{x}}}(\sqrt{-1}a_{1}+\tilde{a})v^{\sim}u(x;\xi)d\xi+(K\circ u)(x)$
(Warning:
pansion of
$\tilde{a}$
$\tilde{a}$
and
$K$
.
are not necessarily unique, but the asymptotic ex-
is uniquely determined by
$P.$
)
Now, set
$\mathfrak{X}(x;\xi)=\frac{\partial a_{1}}{\partial\xi_{\dot{f}}}(x;\xi)\frac{\partial}{\partial X^{J}}|_{X\Leftarrow 0}-\frac{\partial}{\partial X}|_{X=0}a_{1}(.ae(X, \xi))\frac{\partial}{\partial\xi}$
.
Then, ee is a Hamiltonian vector field on $T^{*}N-\{0\}$ satisfying (15). Moreover, . ee generates a one parameter symplectic transformation group . Also using , we define $b(x;\xi)$ by (22), and by the first term of (21). Now, set $\mathfrak{X}^{\dot{f}}\xi_{j}=(\partial a_{1}/\partial\xi_{j})\xi_{j}=a_{1}$
$\varphi_{t}e\mathcal{D}_{\rho}^{(1)}$
$\mathfrak{X}$
$K_{x}$
$(F_{t}u)(x)=\int_{\tau_{x}}.(1+t\tilde{a}-tb+t\frac{\partial\Xi_{;}}{\partial\xi_{\dot{f}}})vu(\varphi_{t}(x;\xi))d\xi+((tK-tK_{x})u)(x)\sim$
.
263
FR\’ECHET-LIE GROUPS III
Then the same computation as in (21) $\sim(24)$ leads us to the conclusion $(d/dt)|_{t=0}F_{t}=P$ . This completes the proof of Proposition A. is a Lie algebra under the usual It is well-known that commutator bracket. $\sqrt{-1}\ovalbox{\tt\small REJECT}^{1}$
REMARK. According to the statement of Theorem $A$ , the exact sequence (7) splits as Lie algebras. However this splitting does not necessarily imply the existence of a splitting of the following exact sequence: (25)
,
$1\rightarrow G\ovalbox{\tt\small REJECT}^{0}/G\ovalbox{\tt\small REJECT}^{-1}\rightarrow G_{L}\pi_{0}^{0}/G\mathscr{J}^{-1}\rightarrow \mathcal{D}_{\omega}(S^{*}N)\rightarrow 1$
be the group of all invertible operators written in where . If an infinite dimensional analogue of the the form $I+P,$ results of [4] or [13] would hold in this case, then we should see that is not simply connected. (25) splits as groups or that $G\ovalbox{\tt\small REJECT}^{-l}(l\geqq 0)$
$Pe\ovalbox{\tt\small REJECT}^{-l}$
$\mathcal{D}_{\omega}(S^{*}N)$
\S 2. Local cohomology group of the Lie algebra of contact
vector
fields.
It is easy to see that the cohomology groups related to the exact sequence (7) or (9) can be defined also locally or formally at an arbitrarily fixed point in $N$. So, at first, in this section we shall deal with the cohomology group of the Lie algebra of formal symplectic vector fields. be the ring of all formal power series Let $\Phi=C|[x^{1},$ , and let be of complex coefficients with variables . consisting of all polynomials of the subalgebra of are Lie algebras under the Poisson bracket , defined as and follows: $x$
$\cdots,$
$x^{n},$
$\xi_{1},$
$\xi_{n}\ovalbox{\tt\small REJECT}$
$\cdots,$
$\Phi^{\prime}$
$x^{1},$
$\cdots,$
$x^{n},$
$\xi_{1},$
$\cdots,$
$x^{1},$
$\Phi$
$\cdots,$ $\{$
$\Phi^{\prime}$
$\Phi$
$\{f, g\}=\partial^{t}f\partial_{i}g-\partial^{i}g\partial_{l}f$
$\partial^{i}=\partial/\partial\xi_{i},$
$C^{q}(\Phi)$
$\partial_{i}=\partial/\partial x^{i}$
$\Phi\times\cdots\times\Phi$
$C^{0}(\Phi)=\Phi,$
$\Phi’$
$\Phi$
) the (resp.
$\cdots,$
$\xi_{n}$
$C^{0}(\Phi^{\prime})=\Phi^{\prime}$
$\left\{\begin{array}{ll}dc(f_{1f}\cdots, f_{q+1})=\sum_{i=1}^{q+1}(-1 & )^{i+1}\{f_{i}, c(f_{1}, \cdots,\hat{f}_{i}, \cdots, f_{q+1})\}\\+\sum_{i
