Jul 18, 1980 - $\nu u(y;\eta)=\sim\int_{N}e^{-i(\eta|Y)}\nu(y, z)u(z)dz$ , ..... $\left\{\begin{array}{ll}\phi_{\alpha}(x;\xi|Y_{1})=\langle\varphi_{2}(x;\xi)|X\rangle+ ...
TOKYO J. MATH. VOL. 4, No. 2, 1981
On Regular Fr\’echet-Lie Groups II Composition Rules of Fourier-Integral Operators on a Riemannian Manifold
Hideki OMORI, Yoshiaki MAEDA and Akira YOSHIOKA Okayama University, Keio University and Tokyo Metropolitan University
Introduction
In the previous paper [8], we gave a differential geometrical expression of Fourier-integral operators on a closed riemannian manifold $N$ without using local coordinate pathches, which is expressed in the following relatively concrete form: (Cf. (19) for the precise meaning of the notations.) (1)
$(Fu)(x)$ $=\sum_{\alpha}\int\int\lambda_{\alpha}a(x;\xi, X)e^{-i\langle\varphi_{2}tx:\epsilon)|X\rangle-i|\xi|A_{\alpha}(X)}(\nu u)(\varphi_{1}(x;\xi);X)dXd\xi$
$+(K\circ u)(x)$
,
is a symplectic transformation of order 1 on $T^{*}N-\{0\}$ . Although our operators such as (1) form much narrower class than what was defined by H\"ormander [3] or Guillemin -Sternberg [2], our expression contains less ambiguities, and hence one can give a sort of coordinate system on a “vicinity” of the identity operator of the Fourier-integral operators of order (cf. Theorem 5.8 [8]). Moreover, the above expression seems to be convenient for concrete computation of the fundamental solution of the equation where
$\varphi=(\varphi_{1};\varphi_{2})$
$0$
(2)
$\frac{d}{dt}u=\sqrt{-1}$
Pu
for a pseudo-differential operator $P$ of order 1 with a real principal symbol. We shall state the reason in what follows. Let be the group generated by the invertible Fourier-integral operators of order , written in the form (1). We regard as if it $G_{L}\mathscr{F}^{0}$
$0$
Received July 18, 1980
$G\mathscr{F}^{0}$
222
H. OMORI, Y. MAEDA AND A. YOSHIOKA
were a locally connected topological group. Then, the identity component is generated by any neighborhood of the identity. shall prove the following:
In this paper,
$GF_{0}^{0}$
we
There is a vicinity of the identity in the space of Fourier-integral operators written in the form (1) such that every element in is invertible and the inverse is written in the form (1). Theorem A.
$\mathfrak{R}$
$\mathfrak{R}$
be the group generated by Theorem B. Let can be written in the form (1). element in
$\mathfrak{R}$
$G\mathscr{F}_{0^{0}}^{\prime}$
.
Then every
$G\mathscr{J}_{0}^{0}$
of (2) is contained in Remark that the fundamental solution , and hence it is written in the form (1). However, the concrete computation must contain the same difficulties as in the case $N=R^{n}$ . The advantage of our expression seems to be in the reduction of the difficulties in computing the fundamental solution to the same level as in the case $N=R$ “, even if $N$ is a closed manifold. For the proof of above theorems, we must establish the formulae of compositions, and inversions of Fourier-integral operators of order . is a regular Fr\’echetIt is in fact the first step of proving that differentiability of the group operations will Lie group. Although the be proved in forthcoming papers, it is not hard by the above formulae is a locally connected topological group. to see that Now, we would like to recall our situation, and several notations used in the previous paper. Let $S^{*}N$ be the unit cosphere bundle over $N$ imbedded naturally in the cotangent bundle $T^{*}N$, and the group of all contact transformations on $S^{*}N$. By Lemma 1.6 in [8], is naturally isomorphic to the group of all symplectic $T^{*}N-\{0\}$ , where a symplectic transformatransformations of order 1 on $\varphi:T^{*}N-\{0\}\rightarrow T^{*}N-\{0\}$ is ; to called be order 1, if tion satisfies $e^{\sqrt{-1}}tP$
$GF_{0}^{0}$
$0$
$G_{\backslash }\Psi_{0}^{0}$
$G\mathscr{G}_{0^{0}}^{\prime}$
$\mathcal{D}_{\omega}(S^{*}N)$
$\mathcal{D}_{\omega}(S^{*}N)$
$\mathcal{D}_{\Omega}^{(1)}$
$\varphi(x;\xi)=(\varphi_{1}(x;\xi)$
$\varphi_{2}(x;\xi))$
(3)
$\varphi_{1}(x;r\xi)=\varphi_{1}(x;\xi)$
,
$\varphi_{2}(x;r\xi)=r\varphi_{2}(x;\xi)$
,
for every $r>0$ , where $(y;\eta)$ means a point in $T^{*}N$ such that $y\in N,$ (the fibre of $T^{*}N$ at ). Since is a topological group under the through the identification mentioned above. -topology, so is By functions on $T^{*}N$ we denote the totality of all C-valued with the following asymptotic expansions: $\eta eT_{l}^{*}$
$\mathcal{D}_{\omega}(S^{*}N)$
$y$
$C^{\infty}$
$\mathcal{D}_{\Omega}^{(1)}$
$C^{\infty}$
$\sum_{C}^{0}$
(4)
where
$ a(x;\xi)\sim a_{0}(x;\hat{\xi})+a_{-1}(x;\hat{\xi})r^{-1}+\cdots+a_{-j}(x;\hat{\xi})r^{-j}+\cdots$
$r=|\xi|,$
$\xi=r^{-1}\xi$
and
$a_{-j}’ s$
are
$C^{\infty}$
functions on
.
$S^{*}N$
, There is a
FR\’ECHET-LIE GROUPS II
natural linear mapping
223
of into the space of all series of functions on indexed by non-positive integers. It is not hard to see that is naturally isomorphic to the space of all functions on the unit closed disk bundle (cf. (11) of [8]), and the above asymptotic expansion (4) corresponds to Taylor’s expansion in the radial direction at $r=1$ . There exists a mapping such that $\alpha\beta=identity$ by a slight modification of the proof in [4] p. 35. However, it seems for us impossible to choose to be linear, (and this causes some troubles in making a Fr\’echet-Lie group). Remark that for every , $\beta\alpha(a)-a$ is rapidly decreasing in . Let be the space of all C-valued functions on $N\times N$. For each $K(x, y)\in C^{\infty}(N\times N)$ , we defined a smoothing operator by $C^{\infty}(S^{*}N)^{\infty}$
$\sum_{c}^{0}$
$\alpha$
$S^{*}N$
$C^{\infty}$
$\sum_{c}^{0}$
$C^{\infty}$
$\overline{D}^{*}N$
$\beta:C^{\infty}(S^{*}N)^{\infty}\rightarrow\sum_{c}^{0}$
$\beta$
$G_{L}\mathscr{F}_{0}^{0}$
$a(x;\xi)\in\sum_{c}^{0}$
$\xi$
$C^{\infty}(N\times N)$
$C^{\infty}$
$ K\circ$
(5)
where A
$C^{\infty}$
is the volume element of $N$ defined by the riemannian metric. function $\nu(x, y)$ on $N\times N$ will be called a cut off function if $dy$
(a)
$0\leqq\nu(x, y)\leqq 1,$
$\nu(x, y)=\nu(y, x)$
.
There is a sufficiently small number such that $\nu(x, y)=1$ if the distance (b)
$v$
,
$(K\circ u)(x)=\int_{N}K(x, y)u(y)dy$
$\epsilon>0$
$\rho(x, y)\leqq\epsilon/3$
, called the
$b\gamma eadth$
.
of
if $\rho(x, y)>2\epsilon/3$ . Now, if our Fourier-integral operator $F$ written in the form (1) is in a vicinity of the identity, then $F$ can be rewritten in the form: (c)
$\nu(x, y)=0$
$\mathfrak{R}$
(6)
$(Fu)(x)=\int_{\tau_{x}^{*}}a(x;\xi)^{\sim}\nu u(\varphi(x;\xi))d\xi+(K\circ u)(x)$
where form of
$\varphi\in \mathcal{D}_{o^{t1)}},$
(7)
$a\in\sum_{c}^{0},$
$u\in C^{\infty}(N)$
$K\in C^{\infty}(N\times N)$
and
$\nu u\sim$
is a sort of Fourier trans-
defined by
$\nu u(y;\eta)=\sim\int_{N}e^{-i(\eta|Y)}\nu(y, z)u(z)dz$
,
$yY=z$ (i.e., $Exp,$ $Y=z$ ).
As a matter of course, the expression (6) still contains some ambiguities, but and are uniquely determined by $F$ (cf. Proposition 5.3 in [8]). Hence if we replace by , then the smoothing term $K$ is uniquely $K$ ) as a sort of a determined also. Therefore one may regard local coordinate system of $F$. As it was mentioned above, one of the main purpose of this paper is to give the formulae of compositions and inversions of Fourier-integral operators. Let $F=F(\psi, a, K)$ be the operator given by (6). Although $\varphi$
$\alpha(a)$
$a$
$\beta\alpha(a)$
$(\varphi, \alpha(a),$
224
H. OMORI, Y. MAEDA AND A. YOSHIOKA
is not exactly a local coordinate of $F$, it is convenient to use this as if it were a local coordinate of $F$. The reason of such a is neither linear sophisticated manner is based on that by its nor differentiable. If is differentiable, then one can replace $G=F(\psi^{\prime}, b, L)$ be another to get a linear splitting. Let derivative operator contained in . Then, we shall obtain in this paper formulae such as $(\psi, a, K)$
$\beta:C^{\infty}(S^{*}N)^{\infty}\rightarrow\sum_{c}^{0}$
$\beta$
$\beta$
$(d\beta)_{0}$
$\mathfrak{R}$
(8)
$\left\{\begin{array}{l}F(\psi, a, K)F(\psi’, b, L)=F(\psi^{\prime}\psi, c, K’)\\F(\psi, a, K)^{-}=F(\psi^{-1}, a^{\prime}, L’)\end{array}\right.$
, which will be denoted by is given as a function of $K’=K^{\prime}(\psi, \psi^{\prime}, a, b, K, L),$ and $L’=$ . Similarly, . In near future, we shall prove that the above functions are smooth in some sense. However, the continuity in the -topology of these functions is not difficult to prove. Thus, one can get that is a locally connected topological group. For the proof of Theorem , the above composition rules (8) are not is not assumed to enough. We have to get a composition rule where En be close to the identity. Indeed, we need to compute $FG$ , where $F$ is a Fourier-integral operator written in the form (1). The and essence of the proof of Theorem is seen in the following:
where
$\psi,$
$c$
$\psi,$
$a,$
$b$
$a^{\prime}=a’(\psi, a)$
$c=(\psi, \psi^{\prime}, a, b)$ $L^{\prime}(\psi, a, K)$
$C^{\infty}$
$G_{L}\mathscr{F}_{0}^{0}$
$B$
$\psi$
$ G\in$
$B$
. Then, there are a be an element of Proposition A. Let of the identity in neighborhood and a neighborhood of $F$ is an operator given by uniform topology such that if under the (1) with , and if $G$ is an operator given by (6) with $\psi eU$ , then by $FG$ is an operator written in the same shape as in (1) replacing $\mathcal{D}_{sJ}^{(1)}$
$\varphi_{0}$
$\mathfrak{U}$
$\mathfrak{B}_{\varphi_{0}}$
$\mathcal{D}_{\rho}^{(1)}$
$\varphi_{0}$
$C^{1}$
$\varphi\in \mathfrak{B}_{\varphi_{0}}$
$\varphi$
$\psi\varphi$
.
Remark. It is an open question for us whether a composition of two Fourier-integral operators in the form (1) is again expressed by the
same form. \S 1. Notations, remarks and the summary of the previous paper. In general, we use the same notations used in [8], but since some of them are not familiar, we repeat these notations here. $N$ is a compact n-dimensional riemannian manifold without boundary. Let be the riemannian metric tensor with respect to a normal chart $(X^{1}, \cdots, X^{n})$ around $x\in N$ and let $g(x)=\det(g_{ij}(x))$ . We $C^{\infty}$
$g_{ij}$
use
225
FR\’ECHET-LIE GROUPS II $dX=\frac{V\overline{g(x)}}{\sqrt{2\pi}n}dX^{1}\wedge\cdots\wedge dX^{n}$
as volume forms on
,
$d\xi=\frac{1}{\sqrt{2\pi}n}\frac{1}{\sqrt{g(x)}}d\xi_{1}\wedge\cdots\wedge d\xi_{n}$
) is the tangent (resp. cotangent) space of $N$ at We use also the notation $dx=(1/\sqrt{2\pi}^{n})dx$ , where $dx$ is the volume element on $N$. Since we use normal charts very often, we have to use exponential mappings $Exp_{x}$ in the expressions of Fourier-integral operators. Thus, for simplicity of the notations, we use . $xX,$ $*.y$ instead of Exp $xX$, $Exp_{x}^{-1}y$ . Moreover, we denote $(y;Y)=x(X, Z),$ $(y;\eta)=x(X, \zeta)$ , if $(y;Y)=$ $(Exp_{x}X;(dExp_{x})_{X}Z)$ , $(y;\eta)=(Exp_{x}X:(dExp_{x}^{-1})_{X}^{*}\zeta)$ respectively. (X, $Z$ ) and (X, ) will be called normal coordinate expressions of $(y;Y)\in TN$, $T_{x},$
$T_{x}^{*}$
respectively, where $x$
$T_{x}$
.
(resp.
$T_{x}^{*}$
$\zeta$
$(y;\eta)\in T^{*}N$
respectively.
Coordinate transformations between two normal charts will be denoted by . Namely, if , then $Y=S(x;X,\overline{X})$
$y=x\overline{X}$
(9)
$yS(x;X,\overline{X})=\cdot Xx$
Since
$S(x;\overline{X},\overline{X})\equiv 0,$
(10)
$S$
.
can be written in the form
$S(x;X,\overline{X})=S_{1}(x;X,\overline{X})(X-\overline{X})$
(cf. (4), (5) in [8])
.
Obviously . Hence has an invariant meaning as a linear mapping of into . However, if we fix $X$ and vary , then we get a vector field. Note that $S(x;X,\overline{X})$ is defined for such that $|X|+|\overline{X}|
