In the present chapter we attend to two different, but related problems. First, we propose to also study comparison algebras in L(Hs), where Hs is an L2-Sobolev space on Ω of order s. Here the spaces Hs always are understood as spaces of the HS-chain induced by the (self-adjoint) Friedrichs extension H of H0, for a a given triple {Ω, dμ, H} (cf.I,6). In that respect we must assume cdn.(s) in order to be in agreement with the customary definition of Sobolev spaces.

Second, we will focus on more general N-th order expressions within reach of a given comparison algebra C (cf. V, def.6.2). Every compactly supported expression already is within reach of the minimal comparison algebra, as we know from V,6. However, for a general algebra C on a noncompact Ω the general expression L no longer is within reach. Thus we will ask for criteria to decide whether a given L is within reach.

The relation between these two tasks is discussed in sec.1 below. There we also discuss the organization of the present chapter. In particular we point out that some of the theorems have to be ‘recycled’, in the following sense. The first application only applies to the minimal comparison algebra J0 in L(H) (or in L(Hs)). That brings certain higher order differential expressions ‘within reach’ of J0, hence of larger algebras, so that, in turn, the same theorem now may be applied again, to get larger classes within reach, etc.