Real interpolation of Sobolev spaces

Abstract

We prove that W 1 p is a real interpolation space between W 1for p > q 0 and 1 ≤ p 1 < p < p 2 ≤ ∞ on some classes of manifolds and general metric spaces, where q 0 depends on our hypotheses.

Real Interpolation of Sobolev Spaces Associated to a Weight

“…The proof of the case when V ∈ RH ∞loc is the same as the one in section 4 in [6]. Consider now V ∈ RH qloc for some 1 < q < ∞.…”

“…We refer to [6] for the proof of Theorem 6.2. The proof of item 1. of Theorem 6.1 is the same as in the non-homogeneous case.…”

“…Examples: For examples of spaces on which our interpolation result applies see section 11 of [6]. Examples of RH q weights in R n for q < ∞ are the power weights |x| −α with −∞ < α < n q and positive polynomials for q = ∞.…”

“…The particular case V = 1 is treated in [6] (also V = 0). The method is actually valid on some Lie groups and even some Riemannian manifolds in which we place ourselves.…”

Real interpolation for grand Besov and Triebel-Lizorkin spaces on RD-spaces

“…Let H be a linear complex Hausdorff space, and let A 0 , A 1 be two complex quasi-Banach spaces such that A 0 ⊂ H and A 1 ⊂ H . Let A 0 + A 1 be the set of all elements a ∈ H which can be represented as a = a 0 + a 1 with a 0 ∈ A 0 and a 1 …”