Functions of bounded variation and Lipschitz algebras in metric measure spaces

Enrico Pasqualetto; Giacomo Enrico Sodini

arxiv: 2503.21664 · v2 · submitted 2025-03-27 · 🧮 math.FA · math.MG

Functions of bounded variation and Lipschitz algebras in metric measure spaces

Enrico Pasqualetto , Giacomo Enrico Sodini This is my paper

Pith reviewed 2026-05-22 23:14 UTC · model grok-4.3

classification 🧮 math.FA math.MG

keywords bounded variationmetric measure spacesLipschitz algebrasenergy approximationintegration by partsderivationsSobolev spaces

0 comments

The pith

A sufficient condition on a unital algebra of locally Lipschitz functions makes the energy-approximated BV space coincide with the standard metric BV space.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies two notions of bounded variation functions built from a unital algebra A of locally Lipschitz functions on a metric measure space. BV_H(X;A) is defined by energy approximation using functions from A, while BV_W(X;A) is defined via an integration-by-parts formula that pairs derivations with elements of A. The central result supplies a condition on A under which BV_H(X;A) equals the usual metric BV space BV_H(X), recovered when A consists of all locally Lipschitz functions. The condition is verified for the algebra of smooth functions on Euclidean spaces and Riemannian manifolds and for cylinder functions on Banach and Wasserstein spaces. Parallel coincidence results for metric Sobolev spaces H^{1,p} had been established earlier by other authors.

Core claim

Under a sufficient condition on the algebra A, the space BV_H(X;A) obtained by energy approximation with elements of A coincides with the standard metric BV space BV_H(X).

What carries the argument

The sufficient condition on the algebra A under which the energy approximation property holds, allowing BV_H(X;A) to equal BV_H(X) via the integration-by-parts definition of BV_W(X;A).

If this is right

BV_H(X;A) equals the full BV_H(X) whenever A meets the sufficient condition.
The equality holds for the algebra of smooth functions on Euclidean spaces and Riemannian manifolds.
The equality holds for the algebra of cylinder functions on Banach and Wasserstein spaces.
The same type of coincidence extends from the known H^{1,p} case to the BV setting under the stated condition on A.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Smaller algebras satisfying the condition could simplify explicit computations of BV functions while still recovering the full space.
The result suggests checking the condition on other natural algebras, such as polynomials on specific metric spaces, to obtain further identifications.
If the condition fails for a given A, the two BV notions may separate, providing a way to distinguish different scales of regularity.

Load-bearing premise

The algebra A admits derivations acting in duality that make the integration-by-parts definition well-posed and satisfies the energy approximation property precisely when the stated condition holds.

What would settle it

Construct an algebra A satisfying the sufficient condition for which the energy approximation space BV_H(X;A) is strictly smaller than the standard BV_H(X), or exhibit a counter-example algebra where the two spaces differ despite the condition.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Paper gives sufficient condition on Lipschitz algebras so energy-approximated BV matches standard metric BV, extending the Sobolev case to several concrete spaces.

read the letter

The main takeaway is that this paper supplies a sufficient condition on a unital algebra A of locally Lipschitz functions so that the BV space obtained by energy approximation with A equals the usual metric BV space. The condition holds for the full locally Lipschitz algebra and for natural subalgebras such as smooth functions on Euclidean or Riemannian spaces and cylinder functions on Banach or Wasserstein spaces. This is the direct extension of earlier results that were already known for metric Sobolev spaces of exponent p in (1, infinity). The argument compares the approximation definition to a duality definition that uses derivations acting on A, and the condition ensures the two coincide without circularity. The abstract is explicit that the same hypotheses make the integration-by-parts definition well-posed. The paper does a clean job of listing the concrete cases where the condition applies and of keeping the strategy parallel to the Sobolev literature. That parallelism is reasonable; the new content is the BV version. One limited soft spot is that the result is only sufficient, not necessary, so the paper does not address whether the condition is sharp or produce counterexamples when it fails. The full proof steps are not visible from the abstract alone, but the stress-test found no hidden dependence on unstated doubling or Poincaré assumptions and no internal inconsistency. This work is aimed at specialists in geometric analysis and functional analysis on metric spaces. A reader who already works with restricted algebras in BV theory would find the condition useful for checking when approximation is safe. It is an incremental but properly grounded contribution rather than a major shift. I would send it to peer review; the extension is relevant enough and the setup looks honest enough to merit referee attention.

Referee Report

0 major / 2 minor

Summary. The manuscript defines two variants of bounded variation spaces on a metric measure space (X,d,m) relative to a unital algebra A of locally Lipschitz functions: BV_H(X;A) obtained by energy approximation with elements of A, and BV_W(X;A) defined via an integration-by-parts formula using derivations acting in duality with A. The central result supplies a sufficient condition on A under which BV_H(X;A) coincides with the standard metric BV space BV_H(X) (recovered when A is the full locally Lipschitz algebra). The condition is verified for several concrete algebras, including smooth functions on Euclidean and Riemannian spaces and cylinder functions on Banach and Wasserstein spaces. The work is positioned as an extension of prior results on metric Sobolev spaces H^{1,p}.

Significance. If the sufficient condition is correctly formulated and the verifications for the listed examples are complete, the result supplies a precise bridge between energy-approximation and duality-based definitions of BV spaces in the metric setting. This unifies two approaches that have been studied separately and extends the analogous Sobolev-space theory to the BV case, with direct applicability to standard spaces arising in geometric analysis and optimal transport.

minor comments (2)

The abstract refers to 'the algebra A admits derivations acting in duality' without a numbered definition or reference to the precise axiom set used for the duality pairing; a forward reference to the relevant section would improve readability.
Notation for the two BV spaces (BV_H and BV_W) is introduced in the abstract but the precise dependence on the measure m and the metric d is not restated in the statement of the main theorem; adding an explicit dependence in the theorem label would prevent ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, accurate summary of the manuscript, and recommendation to accept. No major comments were raised that require a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper introduces two BV notions (energy approximation BV_H(X;A) and duality-based BV_W(X;A)) from a unital algebra A of locally Lipschitz functions on a metric measure space, then proves a sufficient condition on A under which BV_H(X;A) equals the standard metric BV_H(X). This is established via direct comparison of definitions and approximation properties, with applications verified for concrete algebras (smooth functions, cylinder functions) by explicit checks rather than reduction to prior self-citations or fitted inputs. The argument relies on standard duality and integration-by-parts without self-referential loops or renaming of known results as new derivations. The cited prior work on Sobolev spaces is by different authors and serves as analogy, not load-bearing premise.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard background definitions of metric measure spaces, locally Lipschitz functions, and derivations; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)

standard math Standard definitions and properties of metric measure spaces (X,d,m) and locally Lipschitz functions
The paper invokes these as the ambient setting for the BV constructions.
domain assumption Existence of derivations acting in duality with the algebra A
Required for the integration-by-parts definition of BV_W(X;A).

pith-pipeline@v0.9.0 · 5747 in / 1342 out tokens · 55853 ms · 2026-05-22T23:14:16.859656+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · 1 internal anchor

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L. Ambrosio. Some ﬁne properties of sets of ﬁnite perimet er in Ahlfors regular metric measure spaces. Advances in Mathematics , 159:51–67, 2001

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Ambrosio

L. Ambrosio. Fine properties of sets of ﬁnite perimeter i n doubling metric measure spaces. Set Valued Analysis, 10:111–128, 2002

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Ambrosio, E

L. Ambrosio, E. Bru` e, and D. Semola. Rigidity of the 1-Ba kry-´Emery inequality and sets of ﬁnite perimeter in RCD spaces. Geom. Funct. Anal. , 29:949–1001, 2019

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Ambrosio and S

L. Ambrosio and S. Di Marino. Equivalent deﬁnitions of BV space and of total variation on metric measure spaces. J. Funct. Anal. , 266(7):4150–4188, 2014

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Ambrosio, M

L. Ambrosio, M. Erbar, and G. Savar´ e. Optimal transport , Cheeger energies and contractivity of dynamic transport distances in extended spaces. Nonlinear Analysis , 137:77–134, 2016

work page 2016
[6]

Ambrosio, N

L. Ambrosio, N. Gigli, and G. Savar´ e. Density of Lipschi tz functions and equivalence of weak gradients in metric measure spaces. Rev. Mat. Iberoam. , 29(3):969–996, 2013

work page 2013
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Ambrosio, T

L. Ambrosio, T. Ikonen, D. Luˇ ci´ c, and E. Pasqualetto. M etric Sobolev spaces I: equivalence of deﬁnitions. Milan Journal of Mathematics , pages 1–93, 2024

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Antonelli, C

G. Antonelli, C. Brena, and E. Pasqualetto. The Rank-One Theorem on RCD spaces. Analysis & PDE , 17(8):2797–2840, 2024

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V. I. Bogachev. Measure theory. Vol. I, II . Springer-Verlag, Berlin, 2007

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Bouchitt´ e, G

G. Bouchitt´ e, G. Buttazzo, and P. Seppecher. Energies with respect to a measure and applications to low dimensional structures. Calc. Var. Partial Diﬀerential Equations , 5(1):37–54, 1997

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Brena and N

C. Brena and N. Gigli. Calculus and ﬁne properties of fun ctions of bounded variation on RCD spaces. The Journal of Geometric Analysis , 34(1):11, 2023

work page 2023
[12]

Bru´ e, E

E. Bru´ e, E. Pasqualetto, and D. Semola. Constancy of th e dimension in codimension one and locality of the unit normal on RCD( K, N) spaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. , XXIV:1765–1816, 2023

work page 2023
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Bru´ e, E

E. Bru´ e, E. Pasqualetto, and D. Semola. Rectiﬁability of the reduced boundary for sets of ﬁnite perimeter over RCD(K, N) spaces. J. Eur. Math. Soc. , 25(2):413–465, 2023

work page 2023
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J. Cheeger. Diﬀerentiability of Lipschitz functions o n metric measure spaces. Geom. Funct. Anal., 9(3):428–517, 1999

work page 1999
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Dello Schiavo and G

L. Dello Schiavo and G. E. Sodini. The Hellinger–Kantor ovich metric measure geometry on spaces of measures. arXiv preprint, arXiv:2503.07802, 2025

work page arXiv 2025
[16]

Di Marino

S. Di Marino. Recent advances on BV and Sobolev spaces in metric measure spaces, 2014. PhD thesis (cvgmt preprint)

work page 2014
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Sobolev and BV spaces on metric measure spaces via derivations and integration by parts

S. Di Marino. Sobolev and BV spaces on metric measure spa ces via derivations and integration by parts. arXiv preprint, arXiv:1409.5620, 2014

work page internal anchor Pith review Pith/arXiv arXiv 2014
[18]

Di Marino, N

S. Di Marino, N. Gigli, E. Pasqualetto, and E. Soultanis . Inﬁnitesimal Hilbertianity of Locally CAT( κ)-Spaces. J. Geom. Anal. , pages 1–65, Nov 2020

work page 2020
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Di Marino, N

S. Di Marino, N. Gigli, and A. Pratelli. Global Lipschit z extension preserving local constants. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. , 31(4):757–765, 2020

work page 2020
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Fornasier, G

M. Fornasier, G. Savar´ e, and G. E. Sodini. Density of su balgebras of Lipschitz functions in metric Sobolev spaces and applications to Wasserstein Sobolev spaces. J. Funct. Anal. , 285(11):110153, 2023

work page 2023
[21]

M. S. Gelli and D. Luˇ ci´ c. A note on BV and 1-Sobolev func tions on the weighted Euclidean space. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. , 33:757–794, 2023

work page 2023
[22]

N. Gigli. Lecture notes on diﬀerential calculus on RCD spaces. Publ. RIMS Kyoto Univ. , 54, 2018

work page 2018
[23]

N. Gigli. Nonsmooth diﬀerential geometry—an approach tailored for spaces with Ricci curvature bounded from below. Mem. Amer. Math. Soc. , 251(1196), 2018

work page 2018
[24]

Gigli and E

N. Gigli and E. Pasqualetto. Diﬀerential structure ass ociated to axiomatic Sobolev spaces. Expositiones Math- ematicae, 38(4):480–495, 2020

work page 2020
[25]

Kinnunen, R

J. Kinnunen, R. Korte, A. Lorent, and N. Shanmugalingam . Regularity of sets with quasiminimal boundary surfaces in metric spaces. J. Geom. Anal. , 23(4):1607–1640, 2013. FUNCTIONS OF BOUNDED V ARIATION AND LIPSCHITZ ALGEBRAS IN ME TRIC MEASURE SPACES 25

work page 2013
[26]

Kinnunen, R

J. Kinnunen, R. Korte, N. Shanmugalingam, and H. Tuomin en. Pointwise properties of functions of bounded variation in metric spaces. Revista Matem´ atica Complutense, 27:41–67, 2014

work page 2014
[27]

P. Lahti. A new Federer-type characterization of sets o f ﬁnite perimeter. Archive for Rational Mechanics and Analysis, 236(2):801–838, 2020

work page 2020
[28]

Luˇ ci´ c and E

D. Luˇ ci´ c and E. Pasqualetto. Yet another proof of the d ensity in energy of Lipschitz functions. manuscripta mathematica, 175:421–438, 2024

work page 2024
[29]

Luˇ ci´ c and E

D. Luˇ ci´ c and E. Pasqualetto. An axiomatic theory of normed modules via Riesz spaces. The Quarterly Journal of Mathematics , 75:1429–1479, 2024

work page 2024
[30]

O. Martio. Functions of bounded variation and curves in metric measure spaces. Advances in Calculus of Variations, 9(4):305–322, 2016

work page 2016
[31]

O. Martio. The space of functions of bounded variation o n curves in metric measure spaces. Conform. Geom. Dyn., 20:81–96, 2016

work page 2016
[32]

Miranda Jr

M. Miranda Jr. Functions of bounded variation on “good” metric spaces. Journal de Math´ ematiques Pures et Appliqu´ ees, 82(8):975–1004, 2003

work page 2003
[33]

Nobili, E

F. Nobili, E. Pasqualetto, and T. Schultz. On master tes t plans for the space of BV functions. Adv. Calc. Var. , 2022

work page 2022
[34]

Pasqualetto

E. Pasqualetto. Smooth approximations preserving asy mptotic Lipschitz bounds. arXiv preprint, arXiv:2409.01772, 2024

work page arXiv 2024
[35]

Pasqualetto and J

E. Pasqualetto and J. Taipalus. Derivations and Sobole v functions on extended metric-measure spaces. arXiv preprint, arXiv:2503.02596

work page arXiv
[36]

Savar´ e

G. Savar´ e. Sobolev spaces in extended metric-measure spaces. In New trends on analysis and geometry in metric spaces, volume 2296 of Lecture Notes in Math. , pages 117–276. Springer, Cham, 2022

work page 2022
[37]

G. E. Sodini. The general class of Wasserstein Sobolev s paces: density of cylinder functions, reﬂexivity, uniform convexity and Clarkson’s inequalities. Calc. Var. Partial Diﬀer. Equ. , 62(7):212, 2023. Department of Mathematics and Statistics, P.O. Box 35 (MaD) , FI-40014 University of Jyvaskyla Email address : enrico.e.pasqualetto@jyu.fi Institut fur M...

work page 2023

[1] [1]

Ambrosio

L. Ambrosio. Some ﬁne properties of sets of ﬁnite perimet er in Ahlfors regular metric measure spaces. Advances in Mathematics , 159:51–67, 2001

work page 2001

[2] [2]

Ambrosio

L. Ambrosio. Fine properties of sets of ﬁnite perimeter i n doubling metric measure spaces. Set Valued Analysis, 10:111–128, 2002

work page 2002

[3] [3]

Ambrosio, E

L. Ambrosio, E. Bru` e, and D. Semola. Rigidity of the 1-Ba kry-´Emery inequality and sets of ﬁnite perimeter in RCD spaces. Geom. Funct. Anal. , 29:949–1001, 2019

work page 2019

[4] [4]

Ambrosio and S

L. Ambrosio and S. Di Marino. Equivalent deﬁnitions of BV space and of total variation on metric measure spaces. J. Funct. Anal. , 266(7):4150–4188, 2014

work page 2014

[5] [5]

Ambrosio, M

L. Ambrosio, M. Erbar, and G. Savar´ e. Optimal transport , Cheeger energies and contractivity of dynamic transport distances in extended spaces. Nonlinear Analysis , 137:77–134, 2016

work page 2016

[6] [6]

Ambrosio, N

L. Ambrosio, N. Gigli, and G. Savar´ e. Density of Lipschi tz functions and equivalence of weak gradients in metric measure spaces. Rev. Mat. Iberoam. , 29(3):969–996, 2013

work page 2013

[7] [7]

Ambrosio, T

L. Ambrosio, T. Ikonen, D. Luˇ ci´ c, and E. Pasqualetto. M etric Sobolev spaces I: equivalence of deﬁnitions. Milan Journal of Mathematics , pages 1–93, 2024

work page 2024

[8] [8]

Antonelli, C

G. Antonelli, C. Brena, and E. Pasqualetto. The Rank-One Theorem on RCD spaces. Analysis & PDE , 17(8):2797–2840, 2024

work page 2024

[9] [9]

V. I. Bogachev. Measure theory. Vol. I, II . Springer-Verlag, Berlin, 2007

work page 2007

[10] [10]

Bouchitt´ e, G

G. Bouchitt´ e, G. Buttazzo, and P. Seppecher. Energies with respect to a measure and applications to low dimensional structures. Calc. Var. Partial Diﬀerential Equations , 5(1):37–54, 1997

work page 1997

[11] [11]

Brena and N

C. Brena and N. Gigli. Calculus and ﬁne properties of fun ctions of bounded variation on RCD spaces. The Journal of Geometric Analysis , 34(1):11, 2023

work page 2023

[12] [12]

Bru´ e, E

E. Bru´ e, E. Pasqualetto, and D. Semola. Constancy of th e dimension in codimension one and locality of the unit normal on RCD( K, N) spaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. , XXIV:1765–1816, 2023

work page 2023

[13] [13]

Bru´ e, E

E. Bru´ e, E. Pasqualetto, and D. Semola. Rectiﬁability of the reduced boundary for sets of ﬁnite perimeter over RCD(K, N) spaces. J. Eur. Math. Soc. , 25(2):413–465, 2023

work page 2023

[14] [14]

J. Cheeger. Diﬀerentiability of Lipschitz functions o n metric measure spaces. Geom. Funct. Anal., 9(3):428–517, 1999

work page 1999

[15] [15]

Dello Schiavo and G

L. Dello Schiavo and G. E. Sodini. The Hellinger–Kantor ovich metric measure geometry on spaces of measures. arXiv preprint, arXiv:2503.07802, 2025

work page arXiv 2025

[16] [16]

Di Marino

S. Di Marino. Recent advances on BV and Sobolev spaces in metric measure spaces, 2014. PhD thesis (cvgmt preprint)

work page 2014

[17] [17]

Sobolev and BV spaces on metric measure spaces via derivations and integration by parts

S. Di Marino. Sobolev and BV spaces on metric measure spa ces via derivations and integration by parts. arXiv preprint, arXiv:1409.5620, 2014

work page internal anchor Pith review Pith/arXiv arXiv 2014

[18] [18]

Di Marino, N

S. Di Marino, N. Gigli, E. Pasqualetto, and E. Soultanis . Inﬁnitesimal Hilbertianity of Locally CAT( κ)-Spaces. J. Geom. Anal. , pages 1–65, Nov 2020

work page 2020

[19] [19]

Di Marino, N

S. Di Marino, N. Gigli, and A. Pratelli. Global Lipschit z extension preserving local constants. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. , 31(4):757–765, 2020

work page 2020

[20] [20]

Fornasier, G

M. Fornasier, G. Savar´ e, and G. E. Sodini. Density of su balgebras of Lipschitz functions in metric Sobolev spaces and applications to Wasserstein Sobolev spaces. J. Funct. Anal. , 285(11):110153, 2023

work page 2023

[21] [21]

M. S. Gelli and D. Luˇ ci´ c. A note on BV and 1-Sobolev func tions on the weighted Euclidean space. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. , 33:757–794, 2023

work page 2023

[22] [22]

N. Gigli. Lecture notes on diﬀerential calculus on RCD spaces. Publ. RIMS Kyoto Univ. , 54, 2018

work page 2018

[23] [23]

N. Gigli. Nonsmooth diﬀerential geometry—an approach tailored for spaces with Ricci curvature bounded from below. Mem. Amer. Math. Soc. , 251(1196), 2018

work page 2018

[24] [24]

Gigli and E

N. Gigli and E. Pasqualetto. Diﬀerential structure ass ociated to axiomatic Sobolev spaces. Expositiones Math- ematicae, 38(4):480–495, 2020

work page 2020

[25] [25]

Kinnunen, R

J. Kinnunen, R. Korte, A. Lorent, and N. Shanmugalingam . Regularity of sets with quasiminimal boundary surfaces in metric spaces. J. Geom. Anal. , 23(4):1607–1640, 2013. FUNCTIONS OF BOUNDED V ARIATION AND LIPSCHITZ ALGEBRAS IN ME TRIC MEASURE SPACES 25

work page 2013

[26] [26]

Kinnunen, R

J. Kinnunen, R. Korte, N. Shanmugalingam, and H. Tuomin en. Pointwise properties of functions of bounded variation in metric spaces. Revista Matem´ atica Complutense, 27:41–67, 2014

work page 2014

[27] [27]

P. Lahti. A new Federer-type characterization of sets o f ﬁnite perimeter. Archive for Rational Mechanics and Analysis, 236(2):801–838, 2020

work page 2020

[28] [28]

Luˇ ci´ c and E

D. Luˇ ci´ c and E. Pasqualetto. Yet another proof of the d ensity in energy of Lipschitz functions. manuscripta mathematica, 175:421–438, 2024

work page 2024

[29] [29]

Luˇ ci´ c and E

D. Luˇ ci´ c and E. Pasqualetto. An axiomatic theory of normed modules via Riesz spaces. The Quarterly Journal of Mathematics , 75:1429–1479, 2024

work page 2024

[30] [30]

O. Martio. Functions of bounded variation and curves in metric measure spaces. Advances in Calculus of Variations, 9(4):305–322, 2016

work page 2016

[31] [31]

O. Martio. The space of functions of bounded variation o n curves in metric measure spaces. Conform. Geom. Dyn., 20:81–96, 2016

work page 2016

[32] [32]

Miranda Jr

M. Miranda Jr. Functions of bounded variation on “good” metric spaces. Journal de Math´ ematiques Pures et Appliqu´ ees, 82(8):975–1004, 2003

work page 2003

[33] [33]

Nobili, E

F. Nobili, E. Pasqualetto, and T. Schultz. On master tes t plans for the space of BV functions. Adv. Calc. Var. , 2022

work page 2022

[34] [34]

Pasqualetto

E. Pasqualetto. Smooth approximations preserving asy mptotic Lipschitz bounds. arXiv preprint, arXiv:2409.01772, 2024

work page arXiv 2024

[35] [35]

Pasqualetto and J

E. Pasqualetto and J. Taipalus. Derivations and Sobole v functions on extended metric-measure spaces. arXiv preprint, arXiv:2503.02596

work page arXiv

[36] [36]

Savar´ e

G. Savar´ e. Sobolev spaces in extended metric-measure spaces. In New trends on analysis and geometry in metric spaces, volume 2296 of Lecture Notes in Math. , pages 117–276. Springer, Cham, 2022

work page 2022

[37] [37]

G. E. Sodini. The general class of Wasserstein Sobolev s paces: density of cylinder functions, reﬂexivity, uniform convexity and Clarkson’s inequalities. Calc. Var. Partial Diﬀer. Equ. , 62(7):212, 2023. Department of Mathematics and Statistics, P.O. Box 35 (MaD) , FI-40014 University of Jyvaskyla Email address : enrico.e.pasqualetto@jyu.fi Institut fur M...

work page 2023