Inserted: 28 dec 2021
Last Updated: 28 dec 2021
It is well known that on arbitrary metric measure spaces, the notion of minimal $p$-weak upper gradient may depend on $p$. In this paper we investigate how a first-order condition of the metric-measure structure, that we call Bounded Interpolation Property, grants that in fact such dependence is not present. The kind of independence we obtain is stronger than previously available results in the setting of PI spaces. We also show that the Bounded Interpolation Property is stable for pointed measure Gromov Hausdorff convergence and holds on a large class of spaces satisfying curvature dimension conditions.