Preduals of metric BV spaces

2025/11/20 by Pasqualetto, Enrico
Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #Geometric Analysis and Curvature Flows

paper · doi:10.48550/arxiv.2511.16255

Abstract

We study the predual of the space of functions of bounded variation defined over a metric measure space (\rm X,\sf d,\mathfrak m) with \mathfrak m finite. More specifically, for any exponent p∈(1,∞) we construct an isometric predual of the space \rm BV_p(\rm X) of p-integrable functions of bounded variation, which we equip with the norm ‖f‖_\rm BV_p(\rm X):=‖f‖_L^p(\rm X)+|Df|(\rm X). Moreover, we prove that the standard BV space \rm BV(\rm X):=\rm BV₁(\rm X), which fails to have a predual for some choices of the metric measure space, does have a predual in the case where (\rm X,\sf d,\mathfrak m) is a PI space (i.e. a doubling metric measure space supporting a weak (1,1)-Poincaré inequality) of finite diameter. Along the way, we also develop a basic theory of BV functions in the setting of extended metric-topological measure spaces, which is of independent interest.

Citations

• Functions of bounded variation and Lipschitz algebras in metric measure spaces
• Derivations and Sobolev functions on extended metric-measure spaces
• BV Functions and Sets of Finite Perimeter on Configuration Spaces
• Sobolev spaces in extended metric-measure spaces
• Rectifiability of the reduced boundary for sets of finite perimeter over RCD(K,N) spaces
• Rigidity of the 1-Bakry-Émery inequality and sets of finite perimeter in RCD spaces
• Optimal transport, Cheeger energies and contractivity of dynamic transport distances in extended spaces
• Sobolev and BV spaces on metric measure spaces via derivations and integration by parts
• Nonsmooth differential geometry - An approach tailored for spaces with Ricci curvature bounded from below
• An introduction to BV functions in Wiener spaces
• On sets of finite perimeter in Wiener spaces: reduced boundary and\n convergence to halfspaces
• Quasiconformal maps in metric spaces with controlled geometry

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• Functions of bounded variation on complete and connected one-dimensional metric spaces
• Approximation of BV by SBV functions in metric spaces
• The variational 1-capacity and BV functions with zero boundary values on metric spaces
• BV functions and nonlocal functionals in metric measure spaces
• A compactness result for BV functions in metric spaces