Real interpoaltion of Sobolev spaces associated to a weight

TL;DR

This paper investigates the interpolation properties of Sobolev spaces associated with Schrödinger operators with positive potentials, establishing conditions under which these spaces form real interpolation spaces on certain manifolds and Lie groups.

Contribution

It demonstrates that Sobolev spaces $W^{1}_{p,V}$ are real interpolation spaces between $W_{p_1,V}^{1}$ and $W_{p_2,V}^{1}$ under specific conditions on the parameters and geometric settings.

Findings

$W^{1}_{p,V}$ is a real interpolation space between $W_{p_1,V}^{1}$ and $W_{p_2,V}^{1}$.

Interpolation holds on classes of manifolds and Lie groups.

Constants $s_0, q_0$ depend on the hypotheses.

Abstract

We study the interpolation property of Sobolev spaces of order 1 denoted by $W^{1}_{p,V}$, arising from Schr\"{o}dinger operators with positive potential. We show that for $1\leq p_1<p<p_2<q_{0}$ with $p>s_0$, $W^{1}_{p,V}$ is a real interpolation space between $W_{p_1,V}^{1}$ and $W_{p_2,V}^{1}$ on some classes of manifolds and Lie groups. The constants $s_{0}, q_{0}$ depend on our hypotheses.