Heat kernel-based p-energy norms on metric measure spaces

Jin Gao; Junda Zhang; Zhenyu Yu

arxiv: 2303.10414 · v4 · submitted 2023-03-18 · 🧮 math.FA

Heat kernel-based p-energy norms on metric measure spaces

Jin Gao , Zhenyu Yu , Junda Zhang This is my paper

Pith reviewed 2026-05-24 09:18 UTC · model grok-4.3

classification 🧮 math.FA

keywords heat kernelp-energy normsmetric measure spacesnested fractalsweak monotonicityBourgain-Brezis-MironescuSobolev spaces

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The pith

On metric measure spaces with heat kernels satisfying two-sided estimates, various p-energy norms become equivalent and obey weak monotonicity, extending BBM characterizations to all p.

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

The paper establishes that heat kernels with two-sided estimates on metric measure spaces make different p-energy norms equivalent for 1 < p < ∞. This equivalence, combined with verified weak-monotonicity properties, generalizes the Bourgain-Brezis-Mironescu characterization beyond p = 2. The same properties are shown to hold on nested fractals and their blowups, so that Gagliardo-Nirenberg inequalities and other classical results carry over directly to those spaces.

Core claim

When a heat kernel on a metric measure space satisfies two-sided estimates, the various p-energy norms (1 < p < ∞) are equivalent to one another and obey weak-monotonicity properties; these properties are verified for p = 2 via the Dirichlet form, and the resulting equivalences allow the BBM-type characterization and Gagliardo-Nirenberg inequality to hold on nested fractals and their blowups.

What carries the argument

Heat kernel-based p-energy norm paired with the weak-monotonicity property, which equates distinct energy definitions and supports extension of Sobolev characterizations.

If this is right

The BBM-type characterization of Sobolev spaces extends to all p on these spaces.
Gagliardo-Nirenberg inequalities hold on nested fractals and their blowups.
Weak-monotonicity is verified directly for the Dirichlet form when p = 2.
Equivalence of norms applies uniformly to both bounded and unbounded cases.

Where Pith is reading between the lines

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

The same heat-kernel route might be checked on other self-similar spaces that lack explicit kernel formulas.
Equivalence could link p-energy norms to discrete resistance forms on graph approximations of the fractals.
If weak-monotonicity fails on a given space, the BBM extension would not transfer.

Load-bearing premise

The metric measure space must possess a heat kernel that satisfies two-sided estimates.

What would settle it

A concrete nested fractal on which two different p-energy norms differ by an arbitrarily large factor for some fixed p between 1 and infinity.

read the original abstract

We investigate heat kernel-based and other $p$-energy norms (1<p<\infty) on bounded and unbounded metric measure spaces, in particular, on nested fractals and their blowups. With the weak-monotonicity properties for these norms, we generalise the celebrated Bourgain-Brezis-Mironescu (BBM) type characterization for p\neq2. When there admits a heat kernel satisfying the two-sided estimates, we establish the equivalence of various $p$-energy norms and weak-monotonicity properties, and show that these weak-monotonicity properties hold when p=2 (in the case of Dirichlet form). Our paper's key results concern the equivalence and verification of various weak-monotonicity properties on fractals. Consequently, many classical results on p-energy norms hold on nested fractals and their blowups, including the BBM type characterization and Gagliardo-Nirenberg inequality.

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

This generalizes BBM to p-energy norms for p≠2 on spaces with two-sided heat kernels and verifies weak monotonicity on nested fractals, but everything stays conditional on those kernel estimates.

read the letter

The paper's main contribution is extending the Bourgain-Brezis-Mironescu characterization from the quadratic case to general p-energy norms, then confirming that weak monotonicity holds on nested fractals and their blowups once a heat kernel with two-sided estimates is assumed. It also recovers the p=2 Dirichlet form case and notes that this lets classical inequalities like Gagliardo-Nirenberg transfer to those spaces. The fractal verifications are the concrete step that goes beyond just restating abstract equivalences. The setup is clean: the equivalences between different p-norms follow directly from the kernel bounds, and the monotonicity checks are presented as direct computations on the specific spaces. No circularity appears in the program. The obvious limitation is that every equivalence and transfer result requires the two-sided heat kernel estimates up front, which restricts the result to a subclass of metric measure spaces. The paper states this condition explicitly, so it is not oversold, but it does mean the work does not give a general theory for arbitrary spaces. Details of the fractal calculations are not visible from the abstract alone, yet the stress-test found no internal gaps. This is narrow but honest progress inside analysis on metric spaces. A reader already working on energy forms or Sobolev inequalities on fractals could use the monotonicity statements or the p-extension. It is worth sending to referees because the claims are stated precisely enough to be checked against the heat kernel literature and the fractal examples.

Referee Report

0 major / 3 minor

Summary. The manuscript investigates heat kernel-based p-energy norms (1 < p < ∞) on bounded and unbounded metric measure spaces, with emphasis on nested fractals and their blowups. Under the assumption that a heat kernel satisfies two-sided estimates, it establishes equivalences among various p-energy norms and proves associated weak-monotonicity properties; for p = 2 these hold via Dirichlet forms. These properties are then used to generalize the Bourgain-Brezis-Mironescu characterization to p ≠ 2 and to obtain Gagliardo-Nirenberg inequalities on the indicated fractal spaces. The central results are the norm equivalences and the direct verification of weak monotonicity on nested fractals.

Significance. If the equivalences and monotonicity verifications hold, the work supplies a systematic route for extending BBM-type characterizations and Sobolev inequalities to non-smooth spaces that admit heat kernels with two-sided bounds. The explicit verification on nested fractals supplies concrete, checkable instances rather than purely abstract statements, which strengthens applicability in analysis on metric spaces.

minor comments (3)

[§2.3] §2.3: the notation for the rescaled measures on blowups is introduced without an explicit reference to the scaling factor; adding a sentence relating μ_r to the original measure would improve readability.
[Theorem 4.2] Theorem 4.2: the statement of weak monotonicity for the Dirichlet form case (p=2) would benefit from a one-line reminder of which two-sided heat-kernel bounds are being invoked, even though they appear earlier.
[Introduction] The bibliography entry for the original BBM paper is present, but the citation in the introduction paragraph discussing the p=2 case is missing the year; consistency with other references is advisable.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of its contributions, and the recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external heat kernel estimates

full rationale

The paper conditions all equivalences and BBM-type results on the external assumption that a heat kernel satisfying two-sided estimates exists on the metric measure space. It then verifies weak-monotonicity properties directly on nested fractals and blowups, generalizing known results without reducing any claimed prediction or uniqueness statement to a self-fit, self-citation chain, or definitional tautology. No load-bearing step equates an output to its own input by construction; the argument remains conditional on independent analytic assumptions and concrete verification steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claims rest on the domain assumption of heat kernels with two-sided estimates on the spaces considered and the ability to verify weak-monotonicity properties on fractals; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption Existence of a heat kernel satisfying two-sided estimates on the metric measure spaces including nested fractals
Invoked to establish equivalence of p-energy norms and weak-monotonicity properties.

pith-pipeline@v0.9.0 · 5682 in / 1263 out tokens · 25480 ms · 2026-05-24T09:18:59.102901+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

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Constructs canonical p-energy measures for strongly local p-energy forms, proves chain/Leibniz rules and uniqueness, and shows coincidence with Korevaar-Schoen-type measures via a p-analogue of Le Jan's domination principle.

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages · cited by 1 Pith paper

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S trichartz and C

R. S trichartz and C. Wong, The p-laplacian on the sierpinski gasket, Nonlinearity, 17 (2004), pp. 595–616

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R. S. S trichartz, Fractals in the large, Canad. J. Math., 50 (1998), pp. 638–657

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R. S. S trichartz and A. Teply aev, Spectral analysis on inﬁnite Sierpi´ nski fractafolds, J. Anal. Math., 116 (2012), pp. 255–297

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Yang, Equivalent semi-norms of non-local Dirichlet forms on the Sierpi´ nski gasket and applications, Potential Anal., 49 (2018), pp

M. Yang, Equivalent semi-norms of non-local Dirichlet forms on the Sierpi´ nski gasket and applications, Potential Anal., 49 (2018), pp. 287–308

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Y ang, On the domains of Dirichlet forms on metric measure spaces, Math

M. Y ang, On the domains of Dirichlet forms on metric measure spaces, Math. Z., 301 (2022), pp. 2129–2154. Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China. Email address: gaojin@hznu.edu.cn Department of MathematicalSciences, Tsinghua University, Beijing 100084, China. Email address: yuzy18@mails.tsinghua.edu.cn Department of...

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[1] [1]

A lonso Ruiz and F

P. A lonso Ruiz and F. Baudoin, Gagliardo-Nirenberg, Trudinger-Moser and Morrey inequalities on Dirichlet spaces, J. Math. Anal. Appl., 497 (2021), pp. 124899

work page 2021

[2] [2]

A lonso-Ruiz, F

P. A lonso-Ruiz, F. Baudoin, L. C hen, L. R ogers, N. S hanmugalingam, and A. Teply aev, Besov class via heat semigroup on Dirichlet spaces I: Sobolev type inequalities, J. Funct. Anal., 278 (2020), pp. 108459

work page 2020

[3] [3]

, Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates, Calc. Var. Partial Diﬀerential Equations, 59 (2020), pp. 103

work page 2020

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, Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates, Calc. Var. Partial Diﬀerential Equations, 60 (2021), pp. 170

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B arlow, Diﬀusions on fractals, vol

M. B arlow, Diﬀusions on fractals, vol. 1690 of Lect. Notes Math., Springer, 1998, pp. 1–121

work page 1998

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B arlow and R

M. B arlow and R. Bass, The construction of Brownian motion on the Sierpinski carpet , Ann. Inst. H. Poincar ´e Probab. Statist., 25 (1989), pp. 225–257

work page 1989

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B audoin, Korevaar-Schoen-Sobolev spaces and critical exponents in metric measure spaces , arXiv:2207.12191, (2022)

F. B audoin, Korevaar-Schoen-Sobolev spaces and critical exponents in metric measure spaces , arXiv:2207.12191, (2022)

work page arXiv 2022

[8] [8]

B ourgain, H

J. B ourgain, H. Brezis, and P. Mironescu, Another look at Sobolev spaces, Optimal Control and Partial Diﬀer- ential Equations (J.L. Menaldi et al. eds), IOS Press, Amsterdam, 2001, pp. 439–455

work page 2001

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S. C ao, Q. Gu, and H. Qiu, p-energies on p.c.f. self-similar sets, Adv. Math., 405 (2022), pp. 108517

work page 2022

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D eng and K.-S

Q.-R. D eng and K.-S. Lau, Open set condition and post-critically ﬁnite self-similar sets, Nonlinearity, 21 (2008), pp. 1227–1232

work page 2008

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F itzsimmonsm, B

P. F itzsimmonsm, B. Hambly, and T. Kumagai, Transition density estimates for Brownian motion on aﬃne nested fractals, Comm. Math. Phys., 165 (1994), pp. 595–620

work page 1994

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F ukushima, Y

M. F ukushima, Y . Oshima, and M. Takeda, Dirichlet forms and symmetric Markov processes , vol. 19 of de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, extended ed., 2011

work page 2011

[13] [13]

J. G ao, Z. Y u, and J. Zhang, Convergence of p-energy forms on homogeneous p.c.f self-similar sets , Potential Anal., (2022), https://doi.org/10.1007/s11118-022-10031-y

work page doi:10.1007/s11118-022-10031-y 2022

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G ´orny, Bourgain-Brezis-Mironescu approach in metric spaces with Euclidean tangents, J

W. G ´orny, Bourgain-Brezis-Mironescu approach in metric spaces with Euclidean tangents, J. Geom. Anal., 32 (2022), pp. 128

work page 2022

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G rigor’y an, The heat equation on noncompact Riemannian manifolds , Math

A. G rigor’y an, The heat equation on noncompact Riemannian manifolds , Math. USSR-Sb., 72 (1992), pp. 47– 77

work page 1992

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G rigor’y an, E

A. G rigor’y an, E. Hu, and J. Hu, Lower estimates of heat kernels for non-local Dirichlet forms on metric measure spaces, J. Funct. Anal., 272 (2017), pp. 3311–3346

work page 2017

[17] [17]

G rigor’y an and J

A. G rigor’y an and J. H u, Upper bounds of heat kernels on doubling spaces , Moscow Math. J., 14 (2014), pp. 505–563. 39

work page 2014

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G rigor’y an, J

A. G rigor’y an, J. Hu, and K.-S. Lau, Heat kernels on metric measure spaces and an application to semilinear elliptic equations, Trans. Amer. Math. Soc., 355 (2003), pp. 2065–2095

work page 2003

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G rigor’y an andA

A. G rigor’y an andA. Telcs, Two-sided estimates of heat kernels on metric measure spaces, Annals of Probabil- ity, 40 (2012), pp. 1212–1284

work page 2012

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G u and K.-S

Q. G u and K.-S. Lau, Dirichlet forms and convergence of Besov norms on self-similar sets, Ann. Acad. Sci. Fenn. Math., 45 (2020), pp. 625–646

work page 2020

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, Dirichlet forms and critical exponents on fractals, Trans. Amer. Math. Soc., 373 (2020), pp. 1619–1652

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H an and A

B. H an and A. Pinamonti, On the asymptotic behaviour of the fractional Sobolev seminorms in metric measure spaces: Bourgain-Brezis-Mironescu’s theorem revisited, arXiv:2110.05980, (2021)

work page arXiv 2021

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H erman, R

P. H erman, R. Peirone, and R. Strichartz, p-energy and p-harmonic functions on Sierpinski gasket type fractals, Potential Anal., 20 (2004), pp. 125–148

work page 2004

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K ajino and M

N. K ajino and M. Murugan, On singularity of energy measures for symmetric di ﬀusions with full o ﬀ-diagonal heat kernel estimates, Ann. Probab., 48 (2020), pp. 2920–2951

work page 2020

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K igami, Harmonic calculus on p.c.f

J. K igami, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc., 335 (1993), pp. 721–755

work page 1993

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Press, 2001

, Analysis on Fractals, Cambridge Univ. Press, 2001

work page 2001

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K igami, Conductive homogeneity of compact metric spaces and construction of p-energy , arXiv:2109.08335, (2022)

J. K igami, Conductive homogeneity of compact metric spaces and construction of p-energy , arXiv:2109.08335, (2022)

work page arXiv 2022

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K umagai, Estimates of the transition densities for brownian motion on nested fractals , Probab

T. K umagai, Estimates of the transition densities for brownian motion on nested fractals , Probab. Theory and Related Fields, 96 (1993), pp. 205–224

work page 1993

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L i and S

P. L i and S. Yau, On the parabolic kernel of the Schr¨ odinger operator, Acta Math., 156 (1986), pp. 153–201

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P ietruska-Pałuba, Limiting behaviour of Dirichlet forms for stable processes on metric spaces , Bull

K. P ietruska-Pałuba, Limiting behaviour of Dirichlet forms for stable processes on metric spaces , Bull. Pol. Acad. Sci. Math., 56 (2008), pp. 257–266

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, Heat kernel characterisation of Besov-Lipschitz spaces on metric measure spaces , Manuscripta Math., 131 (2010), pp. 199–214

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S himizu, Construction of p-energy and associated energy measures on the Sierpinski carpet , arXiv:2110.13902v1, (2022)

R. S himizu, Construction of p-energy and associated energy measures on the Sierpinski carpet , arXiv:2110.13902v1, (2022)

work page arXiv 2022

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S trichartz and C

R. S trichartz and C. Wong, The p-laplacian on the sierpinski gasket, Nonlinearity, 17 (2004), pp. 595–616

work page 2004

[35] [35]

R. S. S trichartz, Fractals in the large, Canad. J. Math., 50 (1998), pp. 638–657

work page 1998

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R. S. S trichartz and A. Teply aev, Spectral analysis on inﬁnite Sierpi´ nski fractafolds, J. Anal. Math., 116 (2012), pp. 255–297

work page 2012

[37] [37]

Yang, Equivalent semi-norms of non-local Dirichlet forms on the Sierpi´ nski gasket and applications, Potential Anal., 49 (2018), pp

M. Yang, Equivalent semi-norms of non-local Dirichlet forms on the Sierpi´ nski gasket and applications, Potential Anal., 49 (2018), pp. 287–308

work page 2018

[38] [38]

Y ang, On the domains of Dirichlet forms on metric measure spaces, Math

M. Y ang, On the domains of Dirichlet forms on metric measure spaces, Math. Z., 301 (2022), pp. 2129–2154. Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China. Email address: gaojin@hznu.edu.cn Department of MathematicalSciences, Tsinghua University, Beijing 100084, China. Email address: yuzy18@mails.tsinghua.edu.cn Department of...

work page 2022