Functions of bounded variation and Lipschitz algebras in metric measure spaces

¹ Department of Mathematics and Statistics, PO Box 35 (MaD), FI-40014 University of Jyvaskyla, Finland
² Institut fur Mathematik – Fakultät für Mathematik - Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 3 September 2025
Accepted: 8 February 2026

Abstract

Given a unital algebra 𝒜 of locally Lipschitz functions defined over a metric measure space (X, d, m), we study two associated notions of function of bounded variation and their relations: the space BV_H(X; 𝒜), obtained by approximating in energy with elements of 𝒜, and the space BV_W (X; 𝒜), defined through an integration-by-parts formula that involves derivations acting in duality with 𝒜. Our main result provides a sufficient condition on the algebra 𝒜 under which BV_H(X; 𝒜) coincides with the standard metric BV space BV_H(X), which corresponds to taking as 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 H^1,p of exponent p ∈ (1, ∞) were previously obtained by several different authors.