Real Interpolation of Sobolev Spaces Associated to a Weight

Abstract

Abstract. We study the interpolation property of Sobolev spaces of order 1 denoted by W 1 p,V , arising from Schrödinger operators with positive potential. We show that for 1 ≤ p 1 < p < p 2 < q 0 with p > s 0 , W 1 p,V is a real interpolation space between W 1 p1,V and W 1 p2,V on some classes of manifolds and Lie groups. The constants s 0 , q 0 depend on our hypotheses.

Abstract Hardy–Sobolev spaces and interpolation

“…This proposition is very similar to the ones of [7,8]. So we do not detail the proof and just explain the modifications.…”

“…Then the end of the proof is classical and is exactly the same as that of the decompositions proved in [7,8]. We do not repeat it.…”

“…One of the motivations is our Sobolev interpolation result [7,8] in different geometric frames, under the doubling property and Poincaré inequalities. After this result, it is interesting to consider a "nice" subspace of W 1,1 -as is the Hardy space for L 1 -and study the interpolation of this space with Sobolev spaces.…”

“…First as done in [7] and [8], we want to prove interpolation results using an adapted "Calderón-Zygmund" decomposition for Sobolev functions.…”

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

“…It was also pointed out in [10] that KS Some other recent developments on the real interpolation theory of Sobolev spaces on metric spaces were made by Badr [1,2] and Badr-Bernicot [3]. Badr in [1,2] obtained the interpolation properties between two Sobolev spaces both with order 1 on some classes of manifolds, Lie groups and metric spaces satisfying certain doubling properties, while Badr and Bernicot [3] studied the real interpolation between Hardy-Sobolev spaces and Sobolev spaces both with order 1 on doubling Riemannian manifolds via an atomic decomposition. Notice that the Triebel-Lizorkin spaces coincide with Sobolev spaces for parameters s = 1 and certain p, q.…”