A first-order condition for the independence on p of weak gradients

It is well known that on arbitrary metric measure spaces, the notion of minimal p-weak upper gradient may depend p. In this paper we investigate how a first-order condition metric-measure structure, call Bounded Interpolation Property, guarantees in fact such dependence not present. We also show Property stable for pointed Gromov Hausdorff convergence and holds large class spaces satisfying curvature dimension conditions.