Independence on $p$ of weak upper gradients on RCD spaces

TL;DR

This paper proves that on RCD(K,∞) spaces, all p-weak gradients are identical for p>1, extending the result to BV functions on proper spaces, thus unifying gradient concepts in this setting.

Contribution

It establishes the equivalence of p-weak gradients for all p>1 on RCD spaces, including BV functions on proper spaces, providing a unified framework.

Findings

All p-weak gradients coincide for p>1 on RCD(K,∞) spaces.

The result extends to BV functions on proper spaces.

Provides a unified understanding of gradient concepts in metric measure spaces.

Abstract

We study $p$-weak gradients on RCD(K,$\infty$) metric measure spaces and prove that they all coincide for $p>1$. On proper spaces, our arguments also cover the extremal situation of BV functions.