Auxiliary results concerning functions on manifolds

Let the manifold Ω satisfy the general assumptions of III,1. In particular we assume the existence of a countable locally finite atlas {Ω j : j=1,2,…}, where each is compactly contained in some U j , where {u j } 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=0 Ω (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 = 0 Ω (g), then there exists a positive C ∞ (Ω)-function ψ such that f = 0(ψ) (in Ω ∼ ), and ψ = 0 Ω (γ). Proof. Let φ(x) = f(x)/g(x), so that we have φ(x) bounded over Ω ∼ and lim x→∞ φ(x) = 0. Consider φ extended to Ω by setting φ(x) = 0 outside Ω, then the limit still is zero. With our partition ω j define η j . = sup{|φ(x)| : x ∈ supp ω j }. Observe that η j >0, and. For there exists a compact set K ε ⊂ Ω such that |φ|<ε outside K ε , for every ε>0.