Extremal Sobolev Inequalities and Applications

Proof Let f : R → R be any smooth function that is identically 1 for t ≤ 0 and identically 0 for t ≥ 1. Since H k(R) is the completion of C∞(Rn), it is enough to show that every function φ ∈ C∞(Rn) ∩ H k(R) can be approximated in H k(R n) by functions in C∞ c (Rn). Consider the sequence φj(x) := φ(x)f(|x| − j). We have that φj ∈ C∞ c (Rn): in fact |x| is not differentiable at x = 0, but f(t) is identically 1 for t ≤ 0 so that φj is smooth for j > 0. As j → ∞, φj(x) → φ(x) for every x ∈ Rn, and |φj(x)| ≤ |φ(x)| which belongs to Lp(Rn), so by Lebesgue dominated convergence theorem we have ‖φj − φ‖p → 0. For every fixed k and every multiindex α of length k we have∇φj(x) → ∇αφ(x) as j →∞, and by induction