Introduction.

In the present chapter we will focus on pseudo-differential operators on differentiable manifolds. We assume either that ω is compact -then our theory will not differ from others – or that ω is a noncompact Riemannian space with conical ends.

While the Fourier transform and, correspondingly, the concept of Fourier multiplier a(D) = F−1 a(M)F is meaningfull only for functions or distributions defined on ℝn, the kind of ψdo's we introduced has a natural environment on a type of differentiable manifold, to be studied. The reason: Our ψdo's of ch.2 are invariant under a type of coordinate transform (discussed in sec.3) while Fourier multipliers do not have this property.

In seel we discuss distributions on manifolds. A special type of ‘S-manifolds’ is preferred, allowing the definition of a class S(Ω) of rapidly decreasing functions. The linear functionals on S(Ω) will be our temperate distributions. For simplicity we will consider only manifolds Ω allowing a compactification Ω° to which the C∞ -structure can be extended – making Ω° a compact manifold with boundary. In essence then S(Ω) will be the class of functions over Ω vanishing of all orders on ∂Ω.

In sec.2 we will introduce ‘admissible’ charts, cut-off's, partitions, as well as admissible coordinate transforms, all designed to give S(Ω) similar properties than S(ℝn). In particular a Riemannian metric is introduced, making ω a space with conical ends. In sec.3 we prove invariance of pseudo-differential operators under admissible coordinate transforms.

See's 4 and 5 generalize the calculus of ψdo's to spaces with conical ends.