On the asymptotic behaviour of the Green operators for elliptic boundary problems and the pure imaginary powers of some second order operators
Daisuke Fujiwara · Journal of the Mathematical Society of Japan · 1969
p_{0}(f, g, x, \rho, \sigma)=\sigma( 9))(x, \rho dg, \sigma)f(x)$ .We call $\sigma(\mathscr{L})$ the principal symbol of $\mathscr{L}P$ .Since $\sigma( 9))$ is independent of $s$ , we can consider $\sigma(9)$ as a section of the bundle $\Pi-1Hom(X\otimes 1_{R^{m}}, Y\otimes 1_{R^{m}})$ over$U_{2}\subset U_{1}$ , and for any section $v$ of $X_{1U}$ which is constant with respect to the trivialization of $X_{1U}$ and for the coordinate functions $(x_{1}, x_{2}, \cdots , x_{n})$ valid in $U_{1}$ and $\forall_{\xi}\in R^{n}$ , $\forall_{S},$ $\forall_{\sigma}\in R^{m},$ $e^{-i(x\cdot\xi\vdash s\cdot\sigma)}P(\Psi ve^{i(x\cdot\xi s\cdot\sigma)}\llcorner)$ depends linearly on $v$ .So we write this as $e^{-i(x\cdot\hat{\sigma}\dashv-s\cdot\sigma)}P(\Psi ve^{(x\cdot\xi\cdot\succ s\cdot a)})=P(\Psi;x, \xi, \sigma)v$ .We call $P(\Psi;x, \xi, \sigma)$ the Fourier integral kernel of $ g\Psi$ with respect to the local coordinates.Clearly $ P(\Psi$ ; $x,$ $\xi,$ $\sigma$ ) has an asymptotic expansion $P(\Psi;x, \xi, \sigma)\sim\sum_{j}p_{j}(\Psi;x, \xi, \sigma)$