Sum-intersection property of Sobolev spaces

Abstract

If 1 < p < ∞ and 0 < λ < 1, we may always decompose any function f∈ W1,p(ℝN) as f = g + h with g ∈ Wλ, p∕λ and h ∈ Wp, 1. A stronger property, natural in the context of functional calculus in Sobolev spaces, is that we may choose g ∈ Wλ, p∕λ ∩ W1, p and h ∈ Wp, 1 ∩ W1, p. We address here the question of the validity of a similar result for three Sobolev spaces Ws1,p1, Ws, p, Ws2,p2 satisfying the proportionality relations (1)s=θs1+(1−θ)s2,(Formula presented)for some θ∈(0,1). For most of s1, …, p2 satisfying (1), we prove that (2)Ws,p(ℝN)=(Ws1,p1(ℝN)∩Ws,p(ℝN))+(Ws2,p2(ℝN)∩Ws,p(ℝN)). In some exceptional situations, this equality does not hold, and we derive an alternative decomposition. We also establish the validity of (2) when the first equality in (1) is replaced by the suboptimal condition s > θs1 + (1 − θ)s2.