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

TL;DR

The paper develops a BV-function framework for maps into Banach manifolds with a local addition, establishing BV([a,b],M) as a smooth Banach manifold and BV([a,b],G) as a Banach-Lie group when G is a Banach-Lie group. It introduces bilinear and right-hand side vector-measure constructions, proves differentiability of BV-function mappings, and builds the modeling spaces for bundle sections to treat BV-paths in bundles. A BV-differential-equation theory is developed for BV-functions on Banach spaces and manifolds, with local existence and uniqueness results. The central result is vector-measure regularity for Banach-Lie groups: Evol provides a smooth correspondence from RN-na vector measures to BV-paths in G, and δ^r acts as the inverse, yielding a diffeomorphism between measure spaces and BV-path spaces; this yields a semidirect-product structure and, ultimately, a homotopy equivalence BV([0,1],G) ≃ G. These results connect BV-paths, vector measures, and infinite-dimensional Lie group regularity, with implications for rough-path theory and infinite-dimensional geometric analysis.

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$([0,1],G)$.

Theorems & Definitions (128)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 2.8
• Remark 2.10
• Lemma 2.11
• proof
• Remark 2.13
• Lemma 2.14
• proof