Pseudodifferential Operators. In this section, we use ϕ-symbols and phase functions introduced previously to propose an alternative method to define pseudodifferential operators on manifolds. In order to provide an intrinsic integral representation, we deal with operators acting in the space of densities, according to the approach already proposed by Xxxxxxx in [16]. Roughly speaking, a density is a spatially varying quantity on the manifold, that behaves nicely under change of coordinates.
Pseudodifferential Operators. Consider the oscillatory integral (3.1.9) to be the distribution kernel of an oper- ator A, which is then expressed explicitly as Au(x) = ∫ ei(x—y)αξαa(x, y, ξ)u(y)dyd¯ξ (3.1.10) where x, y are coordinates on Ω ⊂ M and u(x) = (u1(x) . . . um(x))T is a column of complex-valued half-densities, cf. Subsection 1.3. An operator A of the form (3.1.10) is called an m×m matrix pseudodifferential operator of order l on Ω acting in the space of half-densities. Throughout Chapter 3 we will always consider operators acting on m-columns of half-densities. We can extend this definition to the whole manifold M as follows. We say the operator A is an m × m matrix pseudodifferential operator of order l on M if its distribution kernel (3.1.9) is infinitely smooth outside the diagonal {(x, y) ∈ M × M : x = y} , (3.1.11) and if, for any point on the diagonal, there exists a neighbourhood Σ ⊂ M × M such that the distribution kernel of A can be written in the form (3.1.9) with a ∈ Sl and Σ ⊂ Ω × Ω. We can remove the dependence on the variable y (or x) in the amplitude of the pseudodifferential operator (3.1.10). Indeed, if A is an m × m pseudodifferential operator of order l with amplitude a ∈ Sl, then A differs by an integral operator with a C∞ kernel from the pseudodifferential operator with amplitude A(x, ξ), (x, ξ) ∈ TrM , given by A(x, ξ) ∼ ∂ξ ∂y a(x, y, ξ) . (3.1.12) y=x Σ i—|α| α α α α! This is achieved by expanding the amplitude a at the point y = x using Xxxxxx’x formula and then integrating by parts in the variable ξ. Equivalently, we can express (3.1.10) as follows:
