Calculus of Variations and Geometric Measure Theory

E. Caputo - M. Lučić - E. Pasqualetto - I. Vojnović

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

created by pasqualetto on 24 Aug 2023
modified by caputo on 13 May 2024


Published Paper

Inserted: 24 aug 2023
Last Updated: 13 may 2024

Journal: Revista Matemática Complutense
Year: 2023

ArXiv: 2308.12042 PDF


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.