Let the manifold Ω satisfy the general assumptions of III,1. In particular we assume the existence of a countable locally finite atlas {Ωj : j=1,2,…}, where each is compactly contained in some Uj, where {uj} is another locally finite atlas of Ω. Let Ω∼ be an open subdomain of Ω (where Ω = Ω∼ is permitted.) Suppose f(x) and g(x) > 0 are functions over Ω∼ and Ω, respectively. We will use the Landau symbols in the following sense:

Write f=0(g) (in Ω∼) if f(x)/g(x) is bounded over Ω∼; write f=0Ω(g) (in Ω∼) if f=0(g) (in Ω∼) and (in Ω). (That is, for ε > 0 there exists a compact set K ⊂ Ω such that |f(x)/g(x)| < ε for all.) We shall write f=0(g), and f=0(g), (without “(in Ω∼)”, etc.) if no confusion can arise.

Lemma A.1. Let f, g be as above, and let g be continuous over Ω. If f = 0Ω(g), then there exists a positive C∞(Ω)-function ψ such that f = 0(ψ) (in Ω∼), and ψ = 0Ω(γ).

Proof. Let φ(x) = f(x)/g(x), so that we have φ(x) bounded over Ω∼ and limx→∞φ(x) = 0. Consider φ extended to Ω by setting φ(x) = 0 outside Ω, then the limit still is zero. With our partition ωj define ηj. = sup{|φ(x)| : x ∈ supp ωj}. Observe that ηj>0, and. For there exists a compact set Kε ⊂ Ω such that |φ|<ε outside Kε, for every ε>0.