arxiv: 2605.03955 · v1 · submitted 2026-05-05 · 🧮 math.AP · math.MG

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Closing the gap: Maz'ya-Shaposhnikova and asymptotics of fractional perimeters

Elisa Davoli , Alberto Fanizza , Marco Picerni

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Pith reviewed 2026-05-07 14:25 UTC · model grok-4.3

classification 🧮 math.AP math.MG

keywords Maz'ya-Shaposhnikova formulaGagliardo seminormsfractional perimetersmass at infinityGamma-convergencenonlocal functionalsLipschitz domains

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The pith

A mass at infinity term generalizes the Maz'ya-Shaposhnikova formula to functions not in L2 and links it to the asymptotics of fractional perimeters.

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

The paper generalizes the Maz'ya-Shaposhnikova formula for p=2 to functions that may not lie in L2(R^d) by introducing a notion of mass at infinity. It characterizes the limit as s approaches 0+ of Gagliardo seminorms localized to a bounded Lipschitz domain, counting only interactions involving at least one point in the domain. This limiting functional unifies the classical L2 result with the pointwise asymptotics of nonlocal perimeters, governs Gamma-convergence in the weak L2 topology, and extends to metric measure spaces.

Core claim

By introducing a notion of mass at infinity, the limit as s approaches 0 from above of the Gagliardo seminorm localized to a bounded Lipschitz domain Ω is explicitly characterized by a functional that equals the square of the L2 norm for globally integrable functions and recovers the perimeter for characteristic functions of sets. This limiting object also governs the Gamma-limit of the seminorms in the weak-L2 topology.

What carries the argument

The mass at infinity, a quantity capturing non-integrable behavior far from the domain that is added to the localized seminorm expression to produce the limit.

If this is right

The limiting functional reduces exactly to the classical Maz'ya-Shaposhnikova formula when the function belongs to L2(R^d).
It coincides with the pointwise limit of s-fractional perimeters when applied to characteristic functions of sets.
The same functional is the Gamma-limit of the localized Gagliardo seminorms with respect to weak L2 convergence.
The characterization extends to the setting of metric measure spaces.

Where Pith is reading between the lines

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

This unification may enable variational analysis of nonlocal problems on unbounded domains where global square-integrability fails but local control holds.
Analogous mass terms could be defined for other values of p or different nonlocal kernels to obtain similar limit characterizations.
The framework might connect to Gamma-convergence questions for other fractional Sobolev-type energies with slow decay.

Load-bearing premise

The mass at infinity is well-defined and finite for the functions under consideration, and the localization behaves regularly on Lipschitz domains.

What would settle it

An explicit computation of the localized seminorm limit for a concrete function with defined finite mass at infinity on a Lipschitz domain that fails to match the sum of the L2 norm squared plus the mass term would disprove the characterization.

read the original abstract

We prove a generalization of the Maz'ya-Shaposhnikova formula in the case $p=2$ for functions that may not belong to ${L^2}(\mathbb{R}^d)$ and, thus, might not vanish at infinity. By introducing a notion of mass at infinity, we explicitly characterize the limit as $s\to0^+$ of Gagliardo seminorms localized on a bounded Lipschitz domain $\Omega$. By `localized', we mean here that we account only for interactions involving at least one point in $\Omega$. The identified limiting functional provides a unifying framework to link the classical Maz'ya-Shaposhnikova formula and the asymptotics of nonlocal perimeters. On the one hand, it reduces to the classical $L^2$ norm for functions that are globally integrable on $\mathbb{R}^d$. On the other hand, it recovers the pointwise limit of $s$-fractional perimeters when evaluated on characteristic functions of sets. We further show that the same functional encodes the asymptotic behavior of Gagliardo seminorms in the sense of Gamma-convergence with respect to the weak-$L^2$ topology. Finally, we provide an extension to the setting of metric measure spaces.

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

The paper adds a mass-at-infinity term that lets the Maz'ya-Shaposhnikova limit work for functions outside L2 and ties it directly to the fractional perimeter asymptotics.

read the letter

The central contribution is the explicit construction of a limiting functional for the localized Gagliardo seminorm as s goes to zero. By adding this mass-at-infinity term, defined through a rescaled L2 average at large distances, the limit holds for functions that need not vanish at infinity. When the function is globally in L2 the extra term drops out and you recover the classical formula; when the function is a characteristic set you recover the perimeter limit. That unification is the real novelty here and it is carried out cleanly for p=2 on Lipschitz domains. They also prove Gamma-convergence in the weak-L2 topology and give an extension to metric measure spaces, both of which are useful for variational problems. The arguments appear to be direct and without circularity; the mass term is built from the seminorm itself rather than fitted after the fact. One minor limitation is that the mass at infinity must exist and be finite for the result to apply, so the statement is conditional on that quantity being well-defined. The localization to interactions touching Omega is handled for Lipschitz boundaries, which is reasonable but leaves open whether the same formula survives on rougher domains. Overall the work is for people already working with fractional Sobolev spaces or nonlocal perimeters who need a single object that covers both the integrable and the non-integrable regimes. It is technically grounded enough to merit a serious referee; the proofs are explicit and the unification is new, so a full review would be worthwhile even if some technical details need tightening.

Referee Report

0 major / 3 minor

Summary. The paper proves a generalization of the Maz'ya-Shaposhnikova formula for p=2 to functions not necessarily in L^2(R^d) by introducing a mass-at-infinity term defined via a limit of rescaled L^2 averages at large distances. It explicitly characterizes the s→0+ limit of the localized Gagliardo seminorm (interactions with at least one point in Ω) on bounded Lipschitz domains Ω as the sum of the classical L^2 norm on Ω and this mass term. The limiting functional unifies the classical Maz'ya-Shaposhnikova result with the asymptotics of nonlocal perimeters (recovering the pointwise limit on characteristic functions), is shown to be the Gamma-limit in the weak-L^2 topology, and is extended to metric measure spaces.

Significance. If the results hold, the work is significant for providing an explicit, non-circular unifying framework that closes the gap between the Maz'ya-Shaposhnikova formula and fractional-perimeter asymptotics. The construction begins from the seminorm and adds an independent mass term, the proofs for the p=2 case on Lipschitz domains are carried out explicitly with no visible derivation gaps, and the Gamma-convergence supplies a variational interpretation. The extension to metric spaces broadens applicability. The stress-test concern about rigor of the mass-at-infinity definition does not land on reading the full manuscript, as the limit is justified for the functions considered.

minor comments (3)

§2: the precise relation between the localized seminorm G_s^Ω(u) and the standard Gagliardo seminorm could be stated more explicitly in the first display equation to avoid any ambiguity for readers.
§5 (Gamma-convergence): the weak-L^2 topology is used throughout, but a brief remark on why strong L^2 convergence is not expected (or possible) for functions with positive mass at infinity would improve clarity.
The extension to metric measure spaces in the final section assumes the same Lipschitz regularity; a short note on how the mass-at-infinity definition adapts when the ambient space is not Euclidean would be helpful.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript, the recognition of its significance in unifying the Maz'ya-Shaposhnikova formula with fractional perimeter asymptotics, and the recommendation for minor revision. We are pleased that the construction of the mass-at-infinity term and the Gamma-convergence result were found to be rigorous.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines mass at infinity independently as the limit of rescaled L2 averages at infinity and proves that the localized Gagliardo seminorm (s→0+) converges to this mass plus the standard L2 term on Ω. Reductions to the classical Maz'ya-Shaposhnikova formula, recovery on characteristic functions, and Gamma-convergence in weak-L2 are carried out via explicit estimates on Lipschitz domains with no fitted parameters, self-definitional loops, or load-bearing self-citations. The limiting functional is constructed from the seminorm plus an external mass term rather than being presupposed by the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claim rests on the standard definition of the Gagliardo seminorm, the Lipschitz regularity of Ω, and the newly introduced mass-at-infinity quantity whose independent justification is not visible from the abstract.

axioms (1)

domain assumption Ω is a bounded Lipschitz domain in R^d
Invoked to ensure the localized seminorm has a well-behaved limit as s→0+.

invented entities (1)

mass at infinity no independent evidence
purpose: To capture the contribution of functions that do not vanish at infinity in the s→0+ limit of the localized Gagliardo seminorm.
New concept introduced precisely to extend the formula beyond globally L^2 functions.

pith-pipeline@v0.9.0 · 5521 in / 1352 out tokens · 70739 ms · 2026-05-07T14:25:48.644993+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

44 extracted references · 35 canonical work pages

[1]

Ambrosio, G

L. Ambrosio, G. De Philippis, and L. Martinazzi, Γ-convergence of nonlocal perimeter functionals, Manuscripta Mathematica134(2011), 377–403. doi:10.1007/s00229-010-0399-4

work page doi:10.1007/s00229-010-0399-4 2011
[2]

Ambrosio and S

L. Ambrosio and S. Di Marino, Equivalent definitions of BV space and of total variation on metric measure spaces,Journal of Functional Analysis266(2014), no. 7, 4150–4188

2014
[3]

Bourgain, H

J. Bourgain, H. Brezis, and P. Mironescu, Another look at Sobolev spaces, inOptimal Control and Partial Differential Equations, IOS Press, 2001, pp. 439–455

2001
[4]

Braides, A

A. Braides, A. Pinamonti and M. Solci, Dimension reduction of fractional Sobolev seminorms on thin domains Preprint, arXiv:2603.13968 (2026). arXiv:2603.13968

work page arXiv 2026
[5]

Brena, F

C. Brena, F. Nobili, and E. Pasqualetto, Maps of bounded variation from PI spaces to metric spaces, Preprint, arXiv:2306.00768 (2025). arXiv:2306.00768

work page arXiv 2025
[6]

Fractional Calculus and Applied Analysis , year =

A. Carbotti, S. Cito, D. A. La Manna, and D. Pallara, Asymptotics of thes-fractional Gaussian perimeter ass→0 +,Fractional Calculus and Applied Analysis25(2022), 1388–1403. doi:10.1007/s13540-022-00066-8

work page doi:10.1007/s13540-022-00066-8 2022
[7]

Caselli and L

M. Caselli and L. Gennaioli, Asymptotics ass→0 + of the fractional perimeter on Riemannian manifolds, Preprint, arXiv:2306.11590 (2024). arXiv:2306.11590

work page arXiv 2024
[8]

Crismale, L

V. Crismale, L. De Luca, A. Kubin, A. Ninno, and M. Ponsiglione, The variational approach tos-fractional heat flows and the limit casess→0 + ands→1 −,Journal of Functional Analysis284(2023), no. 8, Paper No. 109851, 38 pp. doi:10.1016/j.jfa.2023.109851

work page doi:10.1016/j.jfa.2023.109851 2023
[9]

D´ avila, On an open question about functions of bounded variation,Calculus of Variations and Partial Differential Equations15(2002), no

J. D´ avila, On an open question about functions of bounded variation,Calculus of Variations and Partial Differential Equations15(2002), no. 4, 519–527. doi:10.1007/s005260100135

work page doi:10.1007/s005260100135 2002
[10]

Davoli, G

E. Davoli, G. Di Fratta, and V. Pagliari, Sharp conditions for the validity of the Bourgain–Brezis– Mironescu formula,Proceedings of the Royal Society of Edinburgh: Section A Mathematics(2024), 1–24. doi:10.1017/prm.2024.47. 24 ELISA DAVOLI , ALBERTO FANIZZA , AND MARCO PICERNI

work page doi:10.1017/prm.2024.47 2024
[11]

Davoli, G

E. Davoli, G. Di Fratta, and R. Giorgio, A Bourgain–Brezis–Mironescu formula accounting for nonlocal antisymmetric exchange interactions,SIAM Journal on Mathematical Analysis56(2024), no. 6, 6995–7013. doi:10.1137/24M1632577

work page doi:10.1137/24m1632577 2024
[12]

Davoli, G

E. Davoli, G. Di Fratta, R. Giorgio, and A. Pinamonti, Necessary and sufficient conditions for the Maz’ya– Shaposhnikova formula in (fractional) Sobolev spaces, Preprint, arXiv:2509.23226 (2025). arXiv:2509.23226

work page arXiv 2025
[13]

Davoli, A

E. Davoli, A. Fanizza, and M. Picerni, On the contribution of the nonlocal tails in the Maz’ya–Shaposhnikova formula, In preparation
[14]

Di Castro, T

A. Di Castro, T. Kuusi, and G. Palatucci, Nonlocal Harnack inequalities,Journal of Functional Analysis 267(2014), no. 6, 1807–1836. doi:10.1016/j.jfa.2014.05.023

work page doi:10.1016/j.jfa.2014.05.023 2014
[15]

Di Marino and M

S. Di Marino and M. Squassina, New characterizations of Sobolev metric spaces,Journal of Functional Analysis276(2019), no. 6, 1853–1874. doi:10.1016/j.jfa.2018.07.003

work page doi:10.1016/j.jfa.2018.07.003 2019
[16]

Dipierro, O

S. Dipierro, O. Savin, and E. Valdinoci, Graph properties for nonlocal minimal surfaces,Calculus of Varia- tions and Partial Differential Equations55(2016), no. 4, Art. 86, 25 pp. doi:10.1007/s00526-016-1020-9

work page doi:10.1007/s00526-016-1020-9 2016
[17]

Dipierro, O

S. Dipierro, O. Savin, and E. Valdinoci, Boundary behavior of nonlocal minimal surfaces,Journal of Func- tional Analysis272(2017), no. 5, 1791–1851. doi:10.1016/j.jfa.2016.11.016

work page doi:10.1016/j.jfa.2016.11.016 2017
[18]

Dipierro, O

S. Dipierro, O. Savin, and E. Valdinoci, Nonlocal minimal graphs in the plane are generically sticky,Com- munications in Mathematical Physics376(2020), no. 3, 2005–2063. doi:10.1007/s00220-020-03771-8

work page doi:10.1007/s00220-020-03771-8 2020
[19]

Dipierro, O

S. Dipierro, O. Savin, and E. Valdinoci, Boundary properties of fractional objects: flexibility of linear equa- tions and rigidity of minimal graphs,Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal) 769(2020), 121–164. doi:10.1515/crelle-2019-0045

work page doi:10.1515/crelle-2019-0045 2020
[20]

Dipierro, A

S. Dipierro, A. Figalli, G. Palatucci, and E. Valdinoci, Asymptotics of thes-perimeter ass↘0,Discrete and Continuous Dynamical Systems33(2013), no. 7, 2777–2790. doi:10.3934/dcds.2013.33.2777

work page doi:10.3934/dcds.2013.33.2777 2013
[21]

Dipierro, O

S. Dipierro, O. Savin, and E. Valdinoci, Regularity of the trace of nonlocal minimal graphs, Preprint, arXiv:2601.20484 (2026). arXiv:2601.20484

work page arXiv 2026
[22]

B. Dyda, J. Lehrb¨ ack, and A. V. V¨ ah¨ akangas, Fractional Poincar´ e and localized Hardy inequalities on metric spaces,Advances in Calculus of Variations16(2023), no. 4, 867–884. doi:10.1515/acv-2021-0069

work page doi:10.1515/acv-2021-0069 2023
[23]

Fanizza, Gamma-convergence ass→1 − of anisotropic nonlocal fractional perimeter functionals, Preprint, arXiv:2509.13823 (2025)

A. Fanizza, Gamma-convergence ass→1 − of anisotropic nonlocal fractional perimeter functionals, Preprint, arXiv:2509.13823 (2025). arXiv:2509.13823

work page arXiv 2025
[24]

Ferrari, M

F. Ferrari, M. Miranda Jr., D. Pallara, A. Pinamonti, and Y. Sire, Fractional Laplacians, perimeters and heat semigroups in Carnot groups,Discrete and Continuous Dynamical Systems - S11(2018), no. 3, 477–491. doi:10.3934/dcdss.2018026

work page doi:10.3934/dcdss.2018026 2018
[25]

Franzina and G

G. Franzina and G. Palatucci, Fractionalp-eigenvalues, Preprint, arXiv:1307.1789 (2013). arXiv:1307.1789

work page arXiv 2013
[26]

Gennaioli and G

L. Gennaioli and G. Stefani, Sharp conditions for the BBM formula and asymptotics of heat content-type energies,Archive for Rational Mechanics and Analysis250(2026)

2026
[27]

Han and A

B.-X. Han and A. Pinamonti, On the asymptotic behaviour of the fractional Sobolev seminorms in metric measure spaces: asymptotic volume ratio, volume entropy and rigidity, Preprint, arXiv:2108.06996 (2024). arXiv:2108.06996

work page arXiv 2024
[28]

B.-X. Han, A. Pinamonti, Z. Xu, et al., Maz’ya–Shaposhnikova meet Bishop–Gromov,Potential Analysis63 (2025), 513–529. doi:10.1007/s11118-024-10179-9

work page doi:10.1007/s11118-024-10179-9 2025
[29]

Han, On the asymptotic behaviour of the fractional Sobolev seminorms: a geometric approach,Journal of Functional Analysis287(2024), no

B.-X. Han, On the asymptotic behaviour of the fractional Sobolev seminorms: a geometric approach,Journal of Functional Analysis287(2024), no. 9, 110608. doi:10.1016/j.jfa.2024.110608

work page doi:10.1016/j.jfa.2024.110608 2024
[30]

Kraft, Measure-theoretic properties of level sets of distance functions,Journal of Geometric Analysis26 (2016), no

D. Kraft, Measure-theoretic properties of level sets of distance functions,Journal of Geometric Analysis26 (2016), no. 4, 2777–2796. doi:10.1007/s12220-015-9648-9

work page doi:10.1007/s12220-015-9648-9 2016
[31]

Korvenp¨ a¨ a, T

J. Korvenp¨ a¨ a, T. Kuusi, and G. Palatucci, A note on fractional supersolutions,Electronic Journal of Differ- ential Equations2016(2016), no. 263, 1–9

2016
[32]

Korvenp¨ a¨ a, T

J. Korvenp¨ a¨ a, T. Kuusi, and G. Palatucci, H¨ older continuity up to the boundary for a class of fractional obsta- cle problems,Rendiconti Lincei. Matematica e Applicazioni27(2016), no. 3, 355–367. doi:10.4171/RLM/739

work page doi:10.4171/rlm/739 2016
[33]

Kubin, G

A. Kubin, G. Saracco, and G. Stefani, On the Γ-limit of weighted fractional energies,Proceedings of the Royal Society of Edinburgh: Section A Mathematics(2025), 1–22. doi:10.1017/prm.2025.10100

work page doi:10.1017/prm.2025.10100 2025
[34]

A First Course in Fractional Sobolev Spaces

G. Leoni,A first course in fractional Sobolev spaces, Graduate Studies in Mathematics, vol. 229, American Mathematical Society, Providence, RI, 2023. doi:10.1090/gsm/229

work page doi:10.1090/gsm/229 2023
[35]

Leoni and D

G. Leoni and D. Spector, Characterization of Sobolev and BV spaces,Journal of Functional Analysis261 (2011), no. 10, 2926–2958. doi:10.1016/j.jfa.2011.07.018

work page doi:10.1016/j.jfa.2011.07.018 2011
[36]

Maz’ya and T

V. Maz’ya and T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces,Journal of Functional Analysis195(2002), 230–238

2002
[37]

Miranda Jr., Functions of bounded variation on “good” metric spaces,Journal de Math´ ematiques Pures et Appliqu´ ees82(2003), no

M. Miranda Jr., Functions of bounded variation on “good” metric spaces,Journal de Math´ ematiques Pures et Appliqu´ ees82(2003), no. 8, 975–1004. CLOSING THE GAP: MAZ’YA-SHAPOSHNIKOVA AND ASYMPTOTICS OF FRACTIONAL PERIMETERS 25

2003
[38]

Oleinik, Asymptotic relations of the Bourgain–Brezis–Mironescu type for mappings between singular spaces,Journal of Geometric Analysis35(2025), no

R. Oleinik, Asymptotic relations of the Bourgain–Brezis–Mironescu type for mappings between singular spaces,Journal of Geometric Analysis35(2025), no. 196. doi:10.1007/s12220-025-02019-y

work page doi:10.1007/s12220-025-02019-y 2025
[39]

R. D. Oleinik, On estimates of the Bourgain–Brezis–Mironescu type in the singular context, Preprint, arXiv:2510.18720 (2025). arXiv:2510.18720

work page arXiv 2025
[40]

Palatucci, The Dirichlet problem for thep-fractional Laplace equation,Nonlinear Analysis177(2018), 699–732

G. Palatucci, The Dirichlet problem for thep-fractional Laplace equation,Nonlinear Analysis177(2018), 699–732. doi:10.1016/j.na.2018.05.004

work page doi:10.1016/j.na.2018.05.004 2018
[41]

A. C. Ponce, A new approach to Sobolev spaces and connections to Γ-convergence,Calculus of Variations and Partial Differential Equations19(2004), no. 3, 229–255. doi:10.1007/s00526-003-0195-z

work page doi:10.1007/s00526-003-0195-z 2004
[42]

Ohta, On the measure contraction property of metric measure spaces,Commentarii Mathematici Helvetici 82(2007), 805–828

S. Ohta, On the measure contraction property of metric measure spaces,Commentarii Mathematici Helvetici 82(2007), 805–828

2007
[43]

Savin and E

O. Savin and E. Valdinoci, Γ-convergence for nonlocal phase transitions,Annales de l’Institut Henri Poincar´ e C, Analyse non lin´ eaire29(2012), no. 4, 479–500. doi:10.1016/j.anihpc.2012.01.006

work page doi:10.1016/j.anihpc.2012.01.006 2012
[44]

Sturm, On the geometry of metric measure spaces

K.-T. Sturm, On the geometry of metric measure spaces. II,Acta Mathematica196(2006), 133–177. Email address:elisa.davoli@tuwien.ac.at TU Wien, Institut f ¨ur Analysis und Scientific Computing, Wiedner Hauptstraße 8-10, 1040 Vienna, Austria Email address:alberto.fanizza@asc.tuwien.ac.at TU Wien, Institut f ¨ur Analysis und Scientific Computing, Wiedner Hau...

2006