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

TL;DR

This paper introduces the Bounded Interpolation Property, a first-order condition ensuring the independence of weak gradients from the parameter p on metric measure spaces, with stability under convergence and applicability to curvature-dimension spaces.

Contribution

The paper identifies the Bounded Interpolation Property as a key condition that guarantees p-independence of weak gradients and demonstrates its stability and prevalence in curvature-dimension spaces.

Findings

Bounded Interpolation Property ensures p-independence of weak gradients.

The property is stable under pointed measure Gromov-Hausdorff convergence.

It holds on a broad class of curvature-dimension spaces.

Abstract

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, guarantees that in fact such dependence is not present. 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.