Weak porosity in spaces of homogeneous type

Diego Maldonado

Weak porosity in spaces of homogeneous type

T0 review · 0 major / 5 minor · reviewed 2026-07-30 · grok-4.5

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Pith's one-line read Weak porosity plus a doubling hole function forces the distance function into the Muckenhoupt A1 class on spaces of homogeneous type.

desk verdict Clean short proofs of known porosity–A1 implications; useful for specialists, modest novelty, extra LDT hypothesis disclosed. read the letter →

arxiv 2607.23686 v1 pith:Q2JMI4D3 submitted 2026-07-26 math.CA

Diego Maldonado This is my paper · ORCID

classification math.CA MSC 28A7528A8042B3730L99

keywords MuckenhouptweightsweakporosityspacesofhomogeneoustypedistancefunctionreverseHölderinequalitydoublingmeasuresmaximalfree-hole

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper gives short proofs that link weak porosity of a set E to the Muckenhoupt A1 condition on negative powers of the distance to E. In a space of homogeneous type whose balls are open and where Lebesgue differentiation holds, if E is weakly porous and its maximal free-hole function is doubling, then some negative power of dist(·,E) belongs to A1. Conversely, once that power is already in A1, the free-hole function must itself be doubling, and this direction needs only open balls. The arguments work by showing that weak porosity plus doubling of the hole function makes dist(·,E) a doubling reverse-Hölder weight, which then sits in some Ap class and yields the A1 conclusion after taking a suitable power. The shorter route clarifies when the geometric porosity condition is equivalent to a classical weighted-norm inequality.

What carries the argument

The maximal E-free hole function ρd,E together with the reverse-Hölder inequality for dist(·,E). Weak porosity produces a uniform lower bound of the average of dist by a multiple of ρd,E; doubling of ρd,E upgrades this to the RH∞ condition on dist, which feeds the classical RH-to-Ap lemma and yields A1 after taking a power.

What would settle it

Exhibit a space of homogeneous type with open balls in which Lebesgue differentiation fails, together with a weakly porous set whose free-hole function is doubling, yet no negative power of the distance lies in A1.

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Extended reading notes

Core claim

If (X,d,μ) has open balls and satisfies Lebesgue differentiation, and if E is weakly porous with doubling maximal free-hole function ρd,E, then there exists α>0 (controlled only by the structural constants) such that dist(·,E)−α lies in A1(X,d,μ). Conversely, the mere membership of that power in A1 forces ρd,E to be doubling, without needing Lebesgue differentiation.

Load-bearing premise

For the direction from weak porosity to the A1 weight, the space must satisfy the Lebesgue differentiation theorem; open balls alone are not enough for the short argument given here.

Editorial extensions

If this is right

Weak porosity plus doubling of ρd,E becomes a practical geometric test for the A1 condition on distance weights.
Once dist(·,E)−α is known to be A1, the free-hole function is automatically doubling, simplifying later geometric arguments.
The equivalence yields a clean characterization of weak porosity under the standing doubling assumption on ρd,E.
The same circle of ideas produces a weak reverse-Hölder inequality for the distance even without doubling of ρd,E.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The conjecture that weak porosity is equivalent to the RH∞ condition on the distance alone, once balls are open, would remove the extra doubling hypothesis on the hole function in many settings.
Spaces that already possess annular decay automatically convert RH∞ weights into doubling weights, so the conjecture holds there by the paper’s own corollaries.
The short RH-to-Ap route may adapt to other geometric conditions (porosity, Ahlfors regularity) that produce reverse-Hölder control on a distance or gauge function.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Diego’s note re-proves the two directions linking weak porosity + ρ-doubling to dist^{-α} ∈ A1 on spaces of homogeneous type. The hard direction (Theorem 1.1) routes through a transparent RH_∞ + doubling argument for the distance function (Lemma 3.1 + Theorem 3.2), then quotes the standard upgrade to A_p. The converse (Theorem 1.3) is elementary casework and needs only open balls. Both proofs check line-by-line; constants are tracked; the three minor slips the stress-test flags (sign in (2.3), radius in Lemma 4.1, middle step in (4.2)) are typographical and do not touch the conclusions.

What is actually new is the shorter conceptual path and a few intermediate statements (weak RH_∞ without ρ-doubling, the characterization in Corollary 3.4, the equivalence of doubling for dist and for ρ). The equivalence itself is already in Mudarra and in Aimar–Gómez–Gómez-Vargas; the price of brevity is the extra Lebesgue-differentiation hypothesis that [1] avoided. Maldonado states this honestly in Remark 2.6. The conjecture in §5 is sensible and correctly scoped.

No circularity, no invented objects, citations are appropriate. This is pure analysis fully contained in the text. It will not change practice outside weighted harmonic analysis on homogeneous spaces, but anyone working on porosity or distance weights will want the short proofs on the shelf.

I would send it to a referee. Accept after light polishing of the three slips.

Referee Report

0 major / 5 minor

Summary. The manuscript gives short proofs of two implications relating weak porosity of a set E in a space of homogeneous type (X,d,μ) to the Muckenhoupt A1 property of powers of its distance function. Theorem 1.1: if d-balls are open, the Lebesgue differentiation theorem (LDT) holds, E is weakly porous, and the maximal E-free hole function ρ_{d,E} is doubling, then dist(·,E)^{-α} ∈ A1 for some α>0 depending only on the structural constants; the route is to show dist(·,E) is a doubling RH_∞ weight (Lemma 3.1, Theorem 3.2) and then invoke the doubling+RH_q ⇒ A_p implication (Lemma 2.1, quoted from [4],[8]). Theorem 1.3: conversely, under openness of balls alone, dist(·,E)^{-α} ∈ A1 implies ρ_{d,E} doubling. Corollaries 3.3–3.4 and 4.3 record byproducts (weak RH_∞ without ρ-doubling; a characterization; dist doubling ⇔ ρ doubling), and §5 states a conjecture with partial evidence.

Significance. The equivalence (A)⇔(B) itself is known ([7, Theorem 1.1] in complete metric spaces with doubling Borel measure; [1, Theorem 1.1] under openness of balls), so the contribution is methodological rather than a new characterization: the proof of the hard direction (A)⇒(B) is reduced to a transparent two-case argument plus a standard reverse-Hölder-to-A_p lemma, with all constants tracked explicitly (α and the A1 constant depend only on K_d, C_μ, C_{ρ,E}, γ, σ). This is a genuine simplification over [3, §5], [7, §§5–6], [1, §5]. The extra LDT hypothesis relative to [1] is disclosed honestly in Remark 2.6, and its exact points of use (Lemma 2.1; μ(Ē)=0 via Remark 2.5) are identifiable. The conjecture in §5 is falsifiable and comes with verified special cases (Remark 5.2). The proofs are self-contained apart from Lemma 2.1 and use only standard quasi-triangle, doubling, and mean-value comparisons; I verified the main derivations line by line.

minor comments (5)

[Remark 2.2, display (2.3)] The printed chain has the wrong sign: from the A_p condition one obtains (⨍_B dist^{-α})^{1/α} ≤ [dist]_{A_p} (⨍_B dist)^{-1}, and then (⨍_B dist)^{-1} ≤ [dist]_{RH_∞} (ess sup_B dist)^{-1}, not ≤ [dist]_{A_p} ⨍_B dist ≤ ... ess sup dist as written. The final bound [dist^{-α}]_{A1} ≤ ([dist]_{A_p}[dist]_{RH_∞})^α is exactly what the corrected chain yields, so only the intermediate display needs fixing.
[Lemma 4.1, proof of (4.1)] Applying (2.8) with C = 2K_d gives the lower bound dist ≥ s/(2K_d) on the shrunken ball B(y, s/(2K_d)), but the text writes B(y, s/K_d). Since only the smaller ball is used (as F in (2.7)), the conclusion (4.1) is unaffected; please correct the displayed radius.
[Theorem 1.3, Case I, display (4.2)] The middle inequality ess sup_{B'} dist ≤ ρ_{d,E}(B') is not what (2.9) gives. The endpoint chain σ_{d,E}(B') ≤ ρ_{d,E}(B(x, r/2)) does follow, because (2.9) applied to B' = B(x, r/K') yields σ(B') ≤ ρ(B(x, K_d(2K_d+1)r/K')) and K_d(2K_d+1)/K' = 1/2; only the endpoints are used afterward. In Case II, the passage from ρ(B) ≤ C4 ρ(B(x, 2r/K')) to (1.5) also implicitly uses monotonicity of ρ in the ball together with 2/K' ≤ 1/3 < 1/2; a half-line of justification would help the reader.
[Lemma 2.1 / Remark 2.3] Lemma 2.1 is the only ingredient not proved in the paper and it carries the LDT hypothesis. The citations ([4, p.3391] and [8, Ch. 1]) are plausible, but a more precise pointer (theorem number, and confirmation that the result is stated for quasi-metric doubling spaces rather than only metric ones) would make the dependence checkable at a glance.
[General] Minor typos: 'homogenous' throughout (Theorems 1.1, 3.2, Corollaries 3.3, 4.3, §5); 'Lebegue' in Remark 2.6; 'ρ_{d,E}( \tfrac12 B)' versus the doubling form (1.3) — a remark that (1.5) and (1.3) are equivalent formulations would prevent confusion. In Lemma 2.7(ii) the conclusion (2.7) is stated with averages while the proof derives it via u(F) ≤ [u]_{RH_∞} u(B) μ(F)/μ(B); stating both forms would clarify.

Circularity Check

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No circularity: ordinary implication proofs from independent definitions of weak porosity, ρ-doubling, and A1

full rationale

The paper proves two one-way implications between independently defined notions (weak porosity of E plus doubling of the maximal free-hole function ρ_{d,E}, versus membership of dist(·,E)^{-α} in the Muckenhoupt class A1). The arguments are direct measure-theoretic estimates (Lemmas 3.1, 2.9, 4.1; Theorems 3.2 and 1.3) that never redefine the target in terms of itself, never fit a parameter to data and re-label it a prediction, and never rest the central claim on a self-citation of the same porosity–A1 equivalence. The sole external analytic ingredient is the classical reverse-Hölder-plus-doubling ⇒ A_p fact (Lemma 2.1), quoted from the literature ([4],[8]) as a general weight-theory lemma whose hypotheses do not include the porosity conclusion. Prior characterizations in [1],[3],[7] are cited only as external benchmarks and historical context. Consequently the derivation chain is self-contained and non-circular.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper works entirely inside the standard axiomatic framework of spaces of homogeneous type and Muckenhoupt weight theory. No numerical parameters are fitted. The only non-standard inputs are the structural hypotheses (open balls, Lebesgue differentiation, doubling of ρ) that are stated explicitly as assumptions of the theorems rather than derived.

assumptions (5)

domain assumption d is a K-quasi-distance and μ is a doubling measure on the σ-algebra containing all d-balls (space of homogeneous type).
Standing hypothesis of the whole paper; standard definition recalled in §1.
domain assumption d-balls are open in the topology they generate.
Used for measurability of dist(·,E) and for the statements of both main theorems (see Remark 2.5 and Theorems 1.1, 1.3).
domain assumption The Lebesgue differentiation theorem holds in (X,d,μ).
Required only for Theorem 1.1 / the hard implication; equivalent to Borel-semiregularity by the cited [2, Thm 3.14].
standard math A nonnegative doubling weight that satisfies a reverse-Hölder inequality of order q>1 belongs to some A_p class (Lemma 2.1).
Classical fact for spaces of homogeneous type, cited to [4, p.3391] and [8, Ch.1]; used as the black-box upgrade from RH_∞ to A_p.
standard math If w ∈ A1 and w^{-1} ∈ L^1_loc then w^{-1} is doubling and satisfies RH_∞ (Lemma 2.7).
Elementary consequence of the A1 condition and the harmonic-arithmetic mean inequality; proved in the text.

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Cite this review

Pith. "Pith review of Weak porosity in spaces of homogeneous type." pith.science (2026). https://pith.science/paper/Q2JMI4D3

@misc{pith2026260723686,
  author       = {Pith},
  title        = {Pith review of: Weak porosity in spaces of homogeneous type},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q2JMI4D3}},
  note         = {Machine review of arXiv:2607.23686}
}

abstract

Based on the theory of Muckenhoupt weights, short and conceptually simpler proofs are provided for the following two implications: (1) if $(X, d, \mu)$ is a space of homogeneous type where $d$-balls are open sets and the Lebesgue differentiation theorem holds true and if $E \subset X$ is a weakly porous set whose maximal $E$-free hole function $\rho_{d, E}$ is doubling, then $\dist{\cdot, E}^{-\alpha} \in A_1(X, d, \mu)$ for some $\alpha > 0$; and (2) now without the assumption on the validity of Lebesgue's differentiation theorem, if $\dist{\cdot, E}^{-\alpha} \in A_1(X, d, \mu)$ for some $\alpha > 0$, then $\rho_{d, E}$ is doubling.

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Reference graph

Works this paper leans on

8 extracted references · 1 linked inside Pith

[1]

Aimar, I

H. Aimar, I. G´ omez, and I. G´ omez-Vargas,Weakly porous sets andA 1 Muckenhoupt weights in spaces of homogeneous type, arXiv 2406.14369. 2024. arXiv: 2406.14369 [math.CA]. url: https://arxiv.org/abs/2406.14369

arXiv 2024
[4]

Indratno, D

S. Indratno, D. Maldonado, and S. Silwal,On the axiomatic approach to Harnack’s inequality in doubling quasi-metric spaces, J. Differential Equations, 254 (8), (2013), 3369–3394

2013
[8]

J. O. Str¨ omberg and A. Torchinsky, Weighted Hardy Spaces. Lecture Notes in Math., vol. 1381, Springer-Verlag, Berlin/New York (1989). Diego Maldonado, Kansas State University, Department of Mathematics. 138 Card- well Hall, Manhattan, KS-66506, USA. Email address:dmaldona@ksu.edu

1989
[2]

Alvarado and M

R. Alvarado and M. Mitrea, Hardy spaces on Ahlfors-regular quasimetric spaces. A sharp theory. Lecture Notes in Mathematics, Vol. 2142. Springer, 2015

2015
[3]

T. C. Anderson, J. Lehrb¨ ack, C. Mudarra, and A. V¨ ah¨ akangas,Weakly porous sets and Muckenhoupt Ap distance functions, J. Funct. Anal. 287 (2024), no. 8, Paper No. 110558, 34

2024
[5]

Dra. Eleonor Harboure

I. G´ omez-Vargas, Funciones distancia, pesos de Muckenhoupt y porosidad de conjuntos, Doctoral Thesis, Instituto de Matem´ atica Aplicada del Litoral “Dra. Eleonor Harboure”, CONICET, UNL, IMAL, Argentina, 2026

2026
[6]

Luukkainen,Assouad dimension: antifractal metrization, porous sets, and homogeneous measures, J

J. Luukkainen,Assouad dimension: antifractal metrization, porous sets, and homogeneous measures, J. Korean Math. Soc. 35 (1) (1998) 23–76

1998
[7]

Mudarra,Weak porosity on metric measure spaces, Proceedings of the Royal Society of Edinburgh: Section A Mathematics (2025), 1–48

C. Mudarra,Weak porosity on metric measure spaces, Proceedings of the Royal Society of Edinburgh: Section A Mathematics (2025), 1–48

2025

Pith tools

Reviewed July 30, 2026 · model on record in the stance chip above.

[1] [1]

Aimar, I

H. Aimar, I. G´ omez, and I. G´ omez-Vargas,Weakly porous sets andA 1 Muckenhoupt weights in spaces of homogeneous type, arXiv 2406.14369. 2024. arXiv: 2406.14369 [math.CA]. url: https://arxiv.org/abs/2406.14369

arXiv 2024

[2] [4]

Indratno, D

S. Indratno, D. Maldonado, and S. Silwal,On the axiomatic approach to Harnack’s inequality in doubling quasi-metric spaces, J. Differential Equations, 254 (8), (2013), 3369–3394

2013

[3] [8]

J. O. Str¨ omberg and A. Torchinsky, Weighted Hardy Spaces. Lecture Notes in Math., vol. 1381, Springer-Verlag, Berlin/New York (1989). Diego Maldonado, Kansas State University, Department of Mathematics. 138 Card- well Hall, Manhattan, KS-66506, USA. Email address:dmaldona@ksu.edu

1989

[4] [2]

Alvarado and M

R. Alvarado and M. Mitrea, Hardy spaces on Ahlfors-regular quasimetric spaces. A sharp theory. Lecture Notes in Mathematics, Vol. 2142. Springer, 2015

2015

[5] [3]

T. C. Anderson, J. Lehrb¨ ack, C. Mudarra, and A. V¨ ah¨ akangas,Weakly porous sets and Muckenhoupt Ap distance functions, J. Funct. Anal. 287 (2024), no. 8, Paper No. 110558, 34

2024

[6] [5]

Dra. Eleonor Harboure

I. G´ omez-Vargas, Funciones distancia, pesos de Muckenhoupt y porosidad de conjuntos, Doctoral Thesis, Instituto de Matem´ atica Aplicada del Litoral “Dra. Eleonor Harboure”, CONICET, UNL, IMAL, Argentina, 2026

2026

[7] [6]

Luukkainen,Assouad dimension: antifractal metrization, porous sets, and homogeneous measures, J

J. Luukkainen,Assouad dimension: antifractal metrization, porous sets, and homogeneous measures, J. Korean Math. Soc. 35 (1) (1998) 23–76

1998

[8] [7]

Mudarra,Weak porosity on metric measure spaces, Proceedings of the Royal Society of Edinburgh: Section A Mathematics (2025), 1–48

C. Mudarra,Weak porosity on metric measure spaces, Proceedings of the Royal Society of Edinburgh: Section A Mathematics (2025), 1–48

2025