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

Abstract

We construct a smooth Banach manifold BV$([a,b],M)$ whose elements are suitably-defined functions $f:[a,b]\to M$ 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$([0,1],G)$.

arXiv comment55 pages, LaTeX, v2: Corrected some typos and shortened the title, results remain unchanged

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