On the integration of Banach modules and its applications to vector calculus on ${\sf RCD}$ spaces

TL;DR

This paper develops a theory for integrating Banach modules to connect vector calculus on leaves of a foliation in ${ m RCD}$ spaces with that on the base space, enhancing the understanding of geometric measure theory in these spaces.

Contribution

It introduces a general framework for integrating $L^0$-Banach modules, enabling the patching of vector fields across leaves in ${ m RCD}$ spaces, which was previously not well-understood.

Findings

Established a theory for integrating Banach modules in ${ m RCD}$ spaces.

Connected vector calculus on leaves with the base space.

Provided tools for measure-theoretic analysis of foliations.

Abstract

A finite-dimensional ${\sf RCD}$ space can be foliated into sufficiently regular leaves, where a differential calculus can be performed. Two important examples are given by the measure-theoretic boundary of the superlevel set of a function of bounded variation and the needle decomposition associated to a Lipschitz function. The aim of this paper is to connect the vector calculus on the lower dimensional leaves with the one on the base space. In order to achieve this goal, we develop a general theory of integration of $L^0$-Banach $L^0$-modules of independent interest. Roughly speaking, we study how to `patch together' vector fields defined on the leaves that are measurable with respect to the foliation parameter.