# Quasi-continuous vector fields on RCD spaces

## TL;DR

This paper introduces a new notion of quasi-continuous tensor fields on RCD spaces, extending the existing framework where tensor fields are only defined almost everywhere, and explores the uniqueness of Sobolev vector field representatives.

## Contribution

It develops the concept of tensor fields defined '2-capacity-a.e.' on RCD spaces and analyzes the quasi-continuity of Sobolev vector fields, advancing tensor calculus in metric measure spaces.

## Key findings

- Defined tensor fields '2-capacity-a.e.' on RCD spaces
- Established conditions for the quasi-continuous representatives of Sobolev vector fields
- Enhanced the tensor calculus framework on RCD spaces

## Abstract

In the existing language for tensor calculus on RCD spaces, tensor fields are only defined m-a.e.. In this paper we introduce the concept of tensor field defined `2-capacity-a.e.' and discuss in which sense Sobolev vector fields have a 2-capacity-a.e. uniquely defined quasi-continuous representative.