Real interpoaltion of Sobolev spaces associated to a weight

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.

💡 Research Summary

The paper investigates the real interpolation property of first‑order Sobolev spaces associated with a positive potential (V) that appears in Schrödinger operators of the form (L=-\Delta+V). The authors work on two broad classes of underlying spaces: complete Riemannian manifolds that satisfy a volume doubling condition, a Poincaré inequality, and a suitable heat kernel upper bound; and Lie groups equipped with a left‑invariant distance and Haar measure, for which analogous analytic tools are available.

The weighted Sobolev space is defined as
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