Functions of bounded variation and Lipschitz algebras in metric measure spaces
Enrico Pasqualetto, Giacomo Enrico Sodini
TL;DR
The article develops two coherent notions of functions of bounded variation on general metric measure spaces: ${\rm BV}_{\rm H}({\rm X};\mathscr A)$ defined by energy relaxation with an approximating Lipschitz algebra $\mathscr A$, and ${\rm BV}_{\rm W}({\rm X};\mathscr A)$ defined via an integration-by-parts framework with Lipschitz derivations. It proves a general duality-based inclusion ${\rm BV}_{\rm W}({\rm X};\mathscr A)\subseteq {\rm BV}_{\rm H}({\rm X};\mathscr A)$ and a bound $\|{\bf D}f\|_{*,\mathscr A}\le |{\bf D}f|_{\mathscr A}({\rm X})$ for all $f$, using derivation-based arguments. Under completeness or Radon hypotheses, the classical BV spaces coincide: ${\rm BV}_{\rm H}({\rm X})={\rm BV}_{\rm W}({\rm X})$ with $|{\bf D}f|=|{\bf D}f|_*$, and when the Lipschitz algebra $\mathscr A$ is a good algebra on a complete space, ${\rm BV}_{\rm H}({\rm X};\mathscr A)={\rm BV}_{\rm H}({\rm X})$ and ${\|}{\bf D}f{\|}_{*,\mathscr A}={\|}{\bf D}f{\|}_{*}$; similarly for ${\rm BV}_{\rm W}$. The framework applies to Euclidean spaces, Riemannian manifolds, Banach and Wasserstein spaces with cylinder or smooth algebras, thereby unifying BV theory beyond complete/separable settings and linking to ${\rm H}^{1,1}$-type results. The introduction of good algebras and derivation-based BV provides a robust, flexible approach to BV function theory in broad metric measure spaces, with potential extensions to extended metric-topological spaces.
Abstract
Given a unital algebra $\mathscr A$ of locally Lipschitz functions defined over a metric measure space $({\mathrm X},{\mathsf d},\mathfrak m)$, we study two associated notions of function of bounded variation and their relations: the space ${\mathrm BV}_{\mathrm H}({\mathrm X};\mathscr A)$, obtained by approximating in energy with elements of $\mathscr A$, and the space ${\mathrm BV}_{\mathrm W}({\mathrm X};\mathscr A)$, defined through an integration-by-parts formula that involves derivations acting in duality with $\mathscr A$. Our main result provides a sufficient condition on the algebra $\mathscr A$ under which ${\mathrm BV}_{\mathrm H}({\mathrm X};\mathscr A)$ coincides with the standard metric BV space ${\mathrm BV}_{\mathrm H}({\mathrm X})$, which corresponds to taking as $\mathscr A$ the collection of all locally Lipschitz functions. Our result applies to several cases of interest, for example to Euclidean spaces and Riemannian manifolds equipped with the algebra of smooth functions, or to Banach and Wasserstein spaces equipped with the algebra of cylinder functions. Analogous results for metric Sobolev spaces ${\mathrm H}^{1,p}$ of exponent $p\in(1,\infty)$ were previously obtained by several different authors.