Symbolic Calculus Near the Boundary

In this chapter we study symbolic calculus of the Laplace–Beltrami operator \(\Delta _{\varOmega }\) near the boundary \(\partial \varOmega \) . It should be emphasized that the symbols of asymptotic are explicitly given by functions with geometric terms in light of the symbolic calculus of pseudo-differential operators developed by Iwasaki (Osaka J. Math. 31:663–728, 1994).