Local vector measures

TL;DR

This paper develops a framework for defining and analyzing vector measures in metric spaces, extending classical measure theory and nonsmooth analysis concepts to more general and less regular geometric contexts.

Contribution

It introduces a new notion of measures acting on sections of bundles in metric spaces, unifying various concepts like currents and BV functions in nonsmooth geometry.

Findings

Classical measure theory results have natural counterparts in this new setting.

The framework applies effectively to $ ext{RCD}$ spaces, linking differential calculus in different nonsmooth geometries.

Abstract

Consider a BV function on a Riemannian manifold. What is its differential? And what about the Hessian of a convex function? These questions have clear answers in terms of (co)vector/matrix valued measures if the manifold is the Euclidean space. In more general curved contexts, the same objects can be perfectly understood via charts. However, charts are often unavailable in the less regular setting of metric geometry, where still the questions make sense. In this paper we propose a way to deal with this sort of problems and, more generally, to give a meaning to a concept of `measure acting in duality with sections of a given bundle', loosely speaking. Despite the generality, several classical results in measure theory like Riesz's and Alexandrov's theorems have a natural counterpart in this setting. Moreover, as we are going to discuss, the notions introduced here provide a unified framework for several key concepts in nonsmooth analysis that have been introduced more than two decades ago, such as: Ambrosio-Kirchheim's metric currents, Cheeger's Sobolev functions and Miranda's BV functions. Not surprisingly, the understanding of the structure of these objects improves with the regularity of the underlying space. We are particularly interested in the case of $\RCD$ spaces where, as we will argue, the regularity of several key measures of the type we study nicely matches the known regularity theory for vector fields, resulting in a very effective theory. We expect that the notions developed here will help creating stronger links between differential calculus in Alexandrov spaces (based on Perelman's DC charts) and in $\RCD$ ones (based on intrinsic tensor calculus).