Metric Space Theory of BV Functions

• Metric space theory of BV functions is a framework that extends classical bounded variation analysis to spaces with doubling measures and 1-Poincaré inequalities.
• It employs techniques such as pointwise characterizations, Lusin-type approximations, and modulus methods to articulate fine properties and approximation results.
• This theory enhances the study of geometric measure theory and potential theory by providing rigorous compactness, trace, and duality results in non-Euclidean settings.

A metric space theory of functions of bounded variation (BV) generalizes classical BV analysis from Euclidean spaces to the setting of metric measure spaces equipped with a doubling measure and supporting a 1-Poincaré inequality. This framework extends definitions, approximation techniques, fine properties, and analytical tools to a broad class of non-smooth spaces, enabling a deep geometric-measure-theoretic and potential-theoretic study of BV functions, sets of finite perimeter, and associated capacities.

1. Foundational Structure: Doubling, Poincaré, and BV Functions

Let $(X, d, \mu)$ be a complete metric space with a Borel regular measure $\mu$.

• Doubling: There exists $C_d \geq 1$ such that $\mu(B(x,2r)) \leq C_d\,\mu(B(x,r))$ for all $x \in X$ and $r>0$.
• 1-Poincaré Inequality: For constants $C_P>0$, $\lambda\geq 1$,

$$\int_B |u - u_B|\, d\mu \leq C_P\, r\, \int_{\lambda B} \mathrm{Lip}\,u\, d\mu,$$

with $u_B = \frac{1}{\mu(B)} \int_B u\, d\mu$ and local Lipschitz constant $\mathrm{Lip}\,u$.

Given these, the space supports a rich potential theory, Newton–Sobolev spaces $N^{1,1}(X)$, and a robust BV theory.

• BV Functions: For an open $\Omega\subset X$, $f\in L^1_\mathrm{loc}(\Omega)$ is in $\BV(\Omega)$ if the total variation

$$|Df|(\Omega) = \inf\Big\{ \liminf_{i\to\infty}\int_\Omega \mathrm{Lip}\,f_i\,d\mu : f_i\in \mathrm{Lip}_c(\Omega),\ f_i\to f\ \text{in}\ L^1_\mathrm{loc}\Big\}$$

is finite. The $\BV$-norm is $\|f\|_{\BV(\Omega)} = \int_\Omega |f|\,d\mu + |Df|(\Omega)$ (Lahti et al., 13 Jan 2025).

• Approximate Limits: For $x\in\Omega$,

$$f^\vee(x) = \inf\left\{ t\in\mathbb{R} : \limsup_{r\to 0} \frac{\mu(\{y\in B(x,r): f(y) > t\})}{\mu(B(x,r))} = 0 \right\},$$

$$f^\wedge(x) = \sup\left\{ t\in\mathbb{R} : \limsup_{r\to 0} \frac{\mu(\{y\in B(x,r): f(y) < t\})}{\mu(B(x,r))} = 0 \right\}.$$

2. Characterizations and Equivalence of BV Notions

Relaxation and modulus approaches yield equivalent classes and measures in the presence of doubling and a 1-Poincaré inequality:

• Miranda Jr. Definition:

$$|Du|(X) = \inf\left\{ \liminf_{i\to\infty} \int_X g_{u_i}\,d\mu : u_i\in \mathrm{Lip}_\mathrm{loc}(X),\ u_i\to u\ \text{in}\ L^1_\mathrm{loc},\ g_{u_i}\ \text{a 1-weak upper gradient} \right\}$$

• AM-modulus Definition (Martio): An AM-modulus control on the variation along almost all curves.

Equivalence Theorem (Durand-Cartagena et al., 2018): $\BV(X) = \BV_\mathrm{AM}(X),\quad N^{1,1}(X) = N^{1,1}_\mathrm{AM}(X)$ with comparable norms. The equivalence is mediated by the existence of a Semmes family of curves, enabled by the doubling and Poincaré conditions.

Pointwise Characterization (Lahti et al., 2013): $u\in BV(X)$ if and only if there exists a finite positive measure $\nu$ and constants $\sigma \geq 1$, $C_0 > 0$ such that for $\mu$-almost every pair $x, y \in X$,

$$|u(x)-u(y)| \leq C_0\, d(x,y)\left[\mathcal{M}_{\sigma d(x,y), \nu}(x) + \mathcal{M}_{\sigma d(x,y), \nu}(y)\right]$$

where $\mathcal{M}_{R, \nu}$ is the maximal function of the measure $\nu$.

3. Lusin-Type and SBV Approximations

Lusin Approximation Theorem (Lahti et al., 13 Jan 2025): For every $\varepsilon>0$, $f\in\BV(\Omega)$, there is a function $f_\varepsilon\in\BV(\Omega)$ and an open set $U_\varepsilon\subset\Omega$ such that:

• $\mathrm{Cap}_1(U_\varepsilon)<\varepsilon$, where

$$\mathrm{Cap}_1(E) = \inf \left\{ \|u\|_{N^{1,1}(X)} : u\in N^{1,1}(X),\ u\ge1\ \text{near}\ E\right\}$$

• $\|f-f_\varepsilon\|_{\BV(\Omega)}<\varepsilon$,
• $f^\vee = f_\varepsilon^\vee$, $f^\wedge = f_\varepsilon^\wedge$ on $\Omega\setminus U_\varepsilon$,
• $f_\varepsilon^\vee$ is upper semicontinuous, $f_\varepsilon^\wedge$ is lower semicontinuous on $\Omega$.

In the Euclidean setting, $f_\varepsilon$ can be chosen smooth on the complement of a small-capacity set, and the non-centered Hardy–Littlewood maximal function $M(f_\varepsilon)$ is continuous.

SBV Approximation (Lahti, 2018): Any $u\in\BV(\Omega)$ can be approximated strictly (i.e., in the sense $u_i\to u$ in $L^1$, $Du_i(\Omega)\to Du(\Omega)$) and uniformly by a sequence $u_i \in \mathrm{SBV}(\Omega)$, where the Cantor part of the variation vanishes. Moreover, the approximations do not introduce new significant jumps.

4. Fine Properties, Capacity, and Compactness

• Lower semicontinuity: $|Du|(U)$ is lower semicontinuous with respect to $L^1$-convergence in every 1-quasiopen set $U$ (Lahti, 2017).
• Uniform absolute continuity: Given strict convergence in $\BV(\Omega)$, the variation measures $|Du_i|$ are uniformly absolutely continuous with respect to 1-capacity.
• BV-compactness: On fixed $(X,d,\mu)$ with varying metrics $(d_j)$ converging locally uniformly and uniform local doubling/Poincaré, a sequence of bounded variation is precompact in $L^1_\mathrm{loc}$ (Don et al., 2018).
• Capacitary approximation: Small sets in capacity allow for local modifications of BV functions with small energy cost; enables approximation and regularization arguments (Lahti et al., 13 Jan 2025, Lahti, 2018).

5. Fine Structure: Jump Sets, Traces, and Extension

• Jump set structure: BV theory on metric spaces supports a countably Hausdorff-rectifiable jump set, rectifiability, and Federer–Vol'pert-type decompositions (Lahti et al., 13 Jan 2025).
• Approximate continuity and jump representation: Points outside the jump set admit precise representatives; the jump part of the total variation is computed via a density and the size of the jump.
• Traces on boundaries and extension: For domains of finite perimeter in doubling+Poincaré spaces, the trace operator $T:\BV(\Omega) \to L^1(\partial\Omega)$ is well-posed, as are bounded linear extensions (from certain domains) (Lahti, 2014, Malý et al., 2015).
• Maz'ya-type inequalities: If a BV function vanishes on a set of positive capacity, its BV energy controls its norm in certain functional spaces, generalizing classical Sobolev inequalities for Dirichlet boundary conditions (Lahti et al., 2015).

6. Vectorial and Metric-Valued BV, Duality, and Further Generalizations

• Banach and metric-space-valued BV: Several definitions coincide in PI-spaces (uniformly locally doubling + weak local Poincaré), including relaxation via post-composition with Lipschitz functions, approximation by simple maps, and weak dual formulations using test plans (Brena et al., 2023). For Banach targets, the different constructions yield the same class and comparable energies; for general metric spaces, divergence arises (Caamano et al., 2023).
• Preduals and duality: On PI-spaces of finite diameter, the classical BV space admits a canonical isometric predual, constructed via derivations and related projective tensor product spaces (Pasqualetto, 20 Nov 2025). For $p>1$ the predual is always available; for $p=1$, the existence may fail without the PI property.
• Algebraic test space reductions: The theory allows for the replacement of the full algebra of locally Lipschitz functions by smaller subalgebras (e.g., smooth, cylindrical, or cylinder functions) without loss of generality for the total variation and functional structure, under explicit density approximation properties (Pasqualetto et al., 27 Mar 2025).

7. Compactness and Metric-Measure Theoretic Embeddings

In one dimension and for certain classes (e.g., Jordan, Waterman, Young, integral variations), total boundedness (compactness) in the appropriate BV-type Banach norm is characterized by seminorm approximation on finitely many subintervals, and, for some classes, uniform integrability in $L^q$ (Gulgowski, 2022). This description highlights the intrinsic metric nature of total boundedness in Banach spaces of BV-type functions.

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References:

• "Lusin approximation for functions of bounded variation" (Lahti et al., 13 Jan 2025)
• "Approximation of BV by SBV functions in metric spaces" (Lahti, 2018)
• "Equivalence of two BV classes of functions in metric spaces, and existence of a Semmes family of curves under a $1$-Poincaré inequality" (Durand-Cartagena et al., 2018)
• "Preduals of metric BV spaces" (Pasqualetto, 20 Nov 2025)
• "Functions of bounded variation and Lipschitz algebras in metric measure spaces" (Pasqualetto et al., 27 Mar 2025)
• "A sharp Leibniz rule for BV functions in metric spaces" (Lahti, 2018)
• "A pointwise characterization of functions of bounded variation on metric spaces" (Lahti et al., 2013)
• "Compactness in the spaces of functions of bounded variation" (Gulgowski, 2022)
• "Quasiopen sets, bounded variation and lower semicontinuity in metric spaces" (Lahti, 2017)
• "Extensions and traces of functions of bounded variation on metric spaces" (Lahti, 2014)
• "Trace and extension theorems for functions of bounded variation" (Malý et al., 2015)
• "Trace theorems for functions of bounded variation in metric spaces" (Lahti et al., 2015)
• "Rough traces of $BV$ functions in metric measure spaces" (Buffa et al., 2019)
• "Functions of bounded variation on complete and connected one-dimensional metric spaces" (Lahti et al., 2019)
• "Maps of bounded variation from PI spaces to metric spaces" (Brena et al., 2023)
• "Fine properties of metric space-valued mappings of bounded variation in metric measure spaces" (Caamano et al., 2023)
• "On $BV$ functions and essentially bounded divergence-measure fields in metric spaces" (Buffa et al., 2019)