Real interpolation of Sobolev spaces

TL;DR

This paper establishes that Sobolev spaces $W^{1}_{p}$ can be obtained via interpolation between other Sobolev spaces on certain manifolds and metric spaces, extending the understanding of their structure.

Contribution

It proves the interpolation property of Sobolev spaces $W^{1}_{p}$ on manifolds and metric spaces, generalizing previous results to broader settings.

Findings

$W^{1}_{p}$ is an interpolation space between $W^{1}_{p_{1}}$ and $W^{1}_{p_{2}}$

Results apply to classes of manifolds and metric spaces

Depends on a parameter $q_{0}$ related to hypotheses

Abstract

We prove that $W^{1}_{p}$ is an interpolation space between $W^{1}_{p_{1}}$ and $W^{1}_{p_{2}}$ for $p>q_{0}$ and $1\leq p_{1}<p<p_{2}\leq \infty$ on some classes of manifolds and general metric spaces, where $q_{0}$ depends on our hypotheses.