Manifolds of continuous BV-functions and vector measure regularity of Banach-Lie groups

TL;DR

This paper develops a smooth Banach manifold structure for BV-functions with values in Banach manifolds and groups, and introduces a smooth evolution map linking vector measures to BV-group functions, enhancing regularity theory.

Contribution

It constructs a new smooth Banach manifold of BV-functions into Banach manifolds and groups, and establishes a smooth evolution map from vector measures to BV-group functions.

Findings

Constructed a smooth Banach manifold BV([a,b], M) for BV-functions into Banach manifolds.

Established a Banach-Lie group BV([a,b], G) with a corresponding Lie algebra of BV-functions.

Developed a smooth evolution map from vector measures to BV(G) functions.

Abstract

We construct a smooth Banach manifold BV$([a,b], M)$ whose elements are suitably-defined functions $f:[a,b] \rightarrow M$ of bounded variation with values in a smooth Banach manifold $M$ which admits a local addition. If the target manifold is a Banach-Lie group $G$, with Lie algebra $\mathfrak{g}$, we obtain a Banach-Lie group BV$([a,b], G)$ with Lie algebra BV$([a, b], \mathfrak{g})$. Strengthening known regularity properties of Banach-Lie groups, we construct a smooth evolution map from a Banach space of $\mathfrak{g}$-valued vector measures on $[0,1]$ to BV$([0,1],G)$.