Introduction.

In this chapter we consider ψdo's as linear operators of a class of weighted L2 -Sobolev spaces over ℝn. We specialize on L2 -spaces and neglect Lp-theory, because ψdo's in general are not continuous operators on Lp -Sobolev spaces, for p≠2. To be more precise, general Lp -boundedness theorems for ψdo's are true for A=a(x,D)∈ Opψh0,ρ,δ, assuming ρ≥0, 0≤δ≤ρι, δne;1, but corresu-ponding Lp-boundedness statements are false, except for ρi=1. There is an extensive theory in Lp-spaces of Sobolev and other types (cf. Beals [B4], Coifman-Meyer [CM], Marshall [Mr1], Muramatu [Mm1], Nagase [Ng1], Yamazaki [Ym1]).

In see's 1, 2 we prove the L2 -boundedness theorem, for δ=0, and 0<δ<1, respectively. This result often is quoted as Calderon-Vaillancourt theorem. In sec.3 we look at weighted L2-Sobolev norms. Our class of spaces Hs =Hs1, s2 is left invariant by the Fou-rier transform, just as many of our ψdo-classes. A ψdo of order m=(m1,m2) is a bounded map Hs → Hs-m, for every s. For every m ∈ ℝ2 an order class 0(m) is introduced – the operators S→s extending to operators in L(Hs,Hs-m) for all s. 0(m) is a Frechet space under the norms of L(Hs′,Hs-m) ; 0(0) and O(∞)=∪ 0(m) are algebras. A ψdo of order m belongs to 0(m).

A refined Fredholm theory holds for (formally) md-(hypo-) elliptic o's. Such an operator admits a Green inverse- the equivalent of the integral operator of the generalized Green's function.