One-sided Muckenhoupt weights and one-sided weakly porous sets in $\mathbb{R}$

Francisco Javier Mart\'in-Reyes; Hugo Aimar; Ignacio G\'omez Vargas; Ivana G\'omez

One-sided Muckenhoupt weights and one-sided weakly porous sets in $\mathbb{R}$

T0 review · 0 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves that a set $E$ is right-sided weakly porous exactly when some negative power of the distance to $E$ is a locally integrable one-sided $A_1^+$ weight.

desk verdict Solid, correct paper: the new one-sided weak porosity notion earns its keep and Theorem 4.5 is the right one-sided analogue; the only flagged gap is a minor, easily repairable estimate. read the letter →

arxiv 2411.19856 v1 pith:MQBHJDPI submitted 2024-11-29 math.CA math.MG

Hugo Aimar , Ivana Gómez , Ignacio Gómez Vargas , Francisco Javier Martín-Reyes This is my paper · ORCID

classification math.CAmath.MG MSC 28A8028A7542B37

keywords one-sidedMuckenhouptweightsA1right-sidedweaklyporoussetsleft-sideddistancefunctionsmaximalholefunctionHausdorffdimensionweakporosity

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 establishes a one-to-one correspondence between one-sided Muckenhoupt weights and one-sided porosity on the real line: a nonempty set $E\subset\mathbb{R}$ admits some $\alpha>0$ for which $d(\cdot,E)^{-\alpha}$ belongs to the right-sided Muckenhoupt class $A_1^+(\mathbb{R})$ and is locally integrable if and only if $E$ is right-sided weakly porous. The new geometric condition says that every interval's left half contains disjoint intervals avoiding $E$ whose total length is a fixed proportion of that half, and each such hole is comparable in size to the largest $E$-free interval in the right half. The proof turns this porosity into exponential decay of the measure of the thickened set $\{x: d(x,E)<\varepsilon\}$ near the left side, which yields the $A_1^+$ inequality and, as a corollary, a Hausdorff-dimension estimate for $E$. The paper also shows that being both left- and right-sided weakly porous is equivalent to the two-sided weak porosity already known to characterize ordinary $A_1$ distance weights, so the two theories are consistent. This matters because it gives a purely geometric description of exactly which distance functions make one-sided maximal operators weighted-bounded, and it exposes the one-sided anisotropy hidden in the classical condition.

What carries the argument

The carrying object is the maximal-hole function $\rho_E(I)$, the radius of the largest open interval centered somewhere in $I$ and contained in $I\setminus E$, and the new class of $(\sigma,\gamma)$-right-sided weakly porous sets built from it. The mechanism that does the heavy lifting is Lemma 4.3, an exponential-neighborhood contraction: for $\tilde I$ obtained from the largest hole in $I^+$, the measure of $F(\varepsilon)=E(\varepsilon)\cap\tilde I$ decays by a fixed factor $\beta_2$ when $\varepsilon$ is scaled by $\beta_1=\gamma/4$. Iterating that contraction supplies the integrability estimate that closes Case 3 of Theorem 4.5 and, through Corollary 4.4, the Hausdorff-dimension bound.

What would settle it

Take $E=\mathbb{N}_0$ from Example 3.9, pick an interval $I$ whose right half contains a pore of length $2\rho(I^+)$, and compute $|F(\gamma\varepsilon/4)|$ versus $|F(\varepsilon)|$ directly for small $\varepsilon$; a single instance where $|F(\gamma\varepsilon/4)|$ exceeds $\beta_2|F(\varepsilon)|$ would disprove the contraction lemma as stated, while uniform validity would confirm the geometric engine used to prove Theorem 4.5.

Watch

Extended reading notes

Core claim

The central claim is Theorem 4.5: $d(\cdot,E)^{-\alpha}\in A_1^+(\mathbb{R})\cap L^1_{\mathrm{loc}}(\mathbb{R})$ for some $\alpha>0$ if and only if the nonempty set $E$ is right-sided weakly porous. In the forward direction, the right-sided porosity of $E$ is shown to control the distribution of points close to $E$: Lemma 4.3 gives a uniform contraction $|F(\beta_1\varepsilon)|\le \beta_2|F(\varepsilon)|$, with $F(\varepsilon)=E(\varepsilon)\cap\tilde I$, and summing the decayed layers produces the $A_1^+$ estimate on $I^-$. In the reverse direction, the one-sided $A_1^+$ inequality is read directly as the required abundance of large holes: points of $I^-\setminus E$ whose component is shorter than $2\gamma\rho(I^+)$ occupy a fraction of $I^-$ that can be made arbitrarily small by choosing $\gamma$ small, so the complementary long holes cover a fixed proportion of $I^-$. A consistency theorem identifies the two-sided weakly porous sets with the intersection of the left- and right-sided classes, and Corollary 4.4 bounds the Hausdorff dimension of any right-sided weakly porous set by $1-\log\beta_2/\log\beta_1$.

Load-bearing premise

The exponential-contraction estimate for neighborhoods of $E$ rests on the unproved assertion that each pore interval loses no more than half its length when the endpoint strips of width $\gamma\varepsilon/4$ are removed; if that assertion fails, the contraction inequality and the main theorem's Case 3 do not follow from the proof given.

Editorial extensions

If this is right

A right-sided weakly porous set is exactly the kind of set for which some negative power of the distance is a locally integrable $A_1^+$ weight; every set that produces such a weight must be right-sided weakly porous.
Every right-sided weakly porous set has zero Lebesgue measure and Hausdorff dimension at most $1-\log\beta_2/\log\beta_1$, with $\beta_1,\beta_2$ the contraction constants of Lemma 4.3.
A set is weakly porous in the two-sided sense if and only if it is simultaneously left- and right-sided weakly porous, so the classical $A_1$ distance-weight theorem is the symmetric special case of the one-sided theorem.
By reflection symmetry, the mirror statement holds: $E$ is left-sided weakly porous if and only if $d(\cdot,E)^{-\alpha}\in A_1^-(\mathbb{R})\cap L^1_{\mathrm{loc}}(\mathbb{R})$ for some $\alpha>0$.
Right-sided weak porosity is not inherited by arbitrary subsets, as the set $\mathbb{N}_0$ shows, even though every tail of a two-sided weakly porous set is right-sided weakly porous; the one-sided condition is genuinely anisotropic.

Reading between the lines

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

The contraction ratio in Lemma 4.3 likely ties the smallest admissible weight exponent $\alpha$ in Theorem 4.5 to the dimension gap $1-\dim_H E$; the paper states the decay exponent $\log\beta_2/\log\beta_1$ but does not identify which $\alpha$ is optimal.
The one-sided definitions are coordinate-dependent, but they suggest a directional version in higher dimensions: fixing a direction and requiring pores to accumulate on one side of every ball should characterize directional $A_1^+$ distance weights, a generalization not attempted here.
A direct check on the accumulated set of Example 3.10 would show whether the unproved half-length estimate in Lemma 4.3 is essential or merely an artifact of the proof, since that set is right-sided weakly porous and its neighborhoods can be computed explicitly.

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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

Quick take: this is a solid paper. The one-sided weak porosity notion is genuinely new, and Theorem 4.5—d(·,E)^-α ∈ A1+ ∩ L1_loc for some α > 0 iff E is right-sided weakly porous—is the natural one-sided counterpart to the Anderson–Lehrbäck–Mudarra–Vähäkangas result. The proofs are detailed, the constants are explicit, and the consistency diagram in Theorem 3.7 is a nice addition. The one soft spot the reader flagged is real but small. In Lemma 4.3, the bound |J_i \ E(ε')| ≥ 1/2 |J_i| is asserted without derivation. The stress-test note supplies the missing argument: since |J_i| ≥ γε = 4ε', the ε'-neighborhood of E can meet J_i only in two endpoint strips of total length at most 2ε' ≤ |J_i|/2. That closes the gap cleanly, so the contraction estimate and Corollary 4.4 stand. I also noticed a couple of typos—for instance, the Case 2 estimate in Theorem 4.5 writes the factor (1−α) in a way that could confuse a reader—but nothing that changes the conclusion. I disagree with any suggestion that the self-citation to [1] is a problem; it is used only for analogy, and the borrowed Lemma 3.6 from [2] is explicitly cited and load-bearing only in Theorem 3.7, where it is applied correctly. The examples (N0 right-sided but not left-sided; the cut-off of Z) are helpful and illustrate the anisotropy well. Bottom line: the central theorem is correct, the new definitions are coherent, and the paper deserves a serious referee. I would send it out and expect a minor-revision outcome. The intended audience is harmonic analysts working on Muckenhoupt weights and geometric measure theorists interested in porosity; they will both find this worth reading.

Referee Report

0 major / 5 minor

Summary. This paper introduces one-sided weakly porous sets in R and proves a geometric characterization of when distance powers d(·,E)^{-α} are one-sided A1 weights. The main result (Theorem 4.5) states that for a non-empty set E, d(·,E)^{-α} ∈ A_1^+(R) ∩ L^1_loc(R) for some α>0 if and only if E is right-sided weakly porous. The paper also proves (Theorem 3.7) that a set is weakly porous in the sense of Anderson et al. if and only if it is both left- and right-sided weakly porous. The proof of the main theorem is built on a one-sided maximal-hole function and a decay estimate for the level sets of the distance function (Lemma 4.3), and it yields a Hausdorff dimension bound for right-sided weakly porous sets (Corollary 4.4).

Significance. If correct, the main theorem gives a complete answer to the question of which subsets of R have distance powers in the one-sided A1 class, extending the recent two-sided results of Anderson, Lehrbäck, Mudarra, and Vähäkangas. The manuscript is careful and largely self-contained: it supplies detailed estimates for the weight condition, explicit parameter choices in the porosity direction, and a transparent reduction to closed sets. The examples (N0 and the asymmetric sequence E) are instructive and correctly illustrate the difference between one-sided and two-sided porosity. The paper also ships a nontrivial dimensional consequence (Corollary 4.4), which lends additional support to the naturalness of the definition. The main equivalence is derived directly from the definitions, without circular use of the conclusion, and the only external load-bearing input is Lemma 3.6 from [2].

minor comments (5)

[Lemma 4.3] In Lemma 4.3, after the intervals J_i^j are introduced, the estimate |J_i^j \ E(ε')| ≥ (1/2)|J_i^j| is asserted without proof; it is true because |J_i^j| ≥ γε = 4ε' and E(ε') can meet J_i^j only in the two endpoint strips of total length at most 2ε' ≤ |J_i^j|/2, but this justification should be included explicitly.
[Theorem 4.5, Case 2] In the computation of the average of d(x,∂I^-)^{-α}, the equality 2/(b-a) ∫_0^{(b-a)/2} u^{-α} du = (1-α)((b-a)/2)^{-α} is incorrect; the correct factor is 1/(1-α). The subsequent inequality remains valid with the corrected constant, so this is a local error in the displayed formula.
[Introduction] The text refers to a 'consistency diagram' that can be obtained from the main results, but no diagram appears in the manuscript; either include the diagram or delete the reference.
[Theorem 4.5 (II⇒I)] The proof begins 'Assume now that E is a closed (σ,γ,+)-w.p. set' without explicitly justifying the reduction from an arbitrary right-sided weakly porous set; the reduction follows from Proposition 3.5(i) and d(·,E)=d(·,\bar E), and it should be stated.
[Proposition 3.8(a)] In the case I ⊂ (x0,∞), the sentence 'E+ satisfies the w.p. condition on I ... and therefore also satisfies the right-sided w.p. condition' uses a local version of the argument in Theorem 3.7; a brief explanation of why Lemma 3.6 applies to the restricted set on the half-line would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the main equivalence is an independent derivation from the weight characterization and the geometric porosity definition.

full rationale

The derivation chain is self-contained and non-circular. Definition 3.4 introduces one-sided weak porosity purely geometrically, using E-free intervals and the maximal hole function ρ; it does not mention A1+ weights. Theorem 4.5 is then proved in two independent directions. In I⇒II, the A1+ inequality from Proposition 2.2 controls the measure of points in I^- that are within 2γρ(I^+) of E, forcing most of I^- to be covered by sufficiently long E-free components; this is a direct consequence of the weight condition, not a restatement of the definition. In II⇒I, Lemmas 4.2 and 4.3 are proved from the geometric porosity hypothesis and used to bound the average of d(x,E)^{-α}; the contraction |F(β1ε)|≤β2|F(ε)| is derived from the pore structure, not assumed. The only imported load-bearing result is Lemma 3.6, taken from the external paper [2] by Anderson et al., and it is used only for the auxiliary equivalence in Theorem 3.7, not for the main characterization. The self-citation [1] by the present authors is used for analogy and context, not as evidence for any theorem. No fitted parameters, data, or uniqueness claims are involved. The reader's flagged estimate in Lemma 4.3, |J_i^j \ E(ε')| ≥ 1/2|J_i^j|, is terse but follows from the fact that each pore has length at least 4ε', so its midpoint strip of length at least half the pore lies outside E(ε'); this is a missing detail in exposition, not a circular reduction. No prediction or theorem reduces to its own input by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

Pure mathematics paper; no empirical fitting occurs. The constants α, σ, and γ are existential parameters of the theorems, not fitted values. The main Theorem 4.5 is self-contained apart from standard measure theory; Theorem 3.7 additionally borrows Lemma 3.6 from [2], and Corollary 4.4 borrows the Minkowski dimension formula from [8].

assumptions (4)

standard math Standard Lebesgue measure theory, the Lebesgue Differentiation Theorem, and basic properties of open intervals.
Used throughout Sections 2 and 4 as background for truncation arguments, density estimates, and pore decompositions.
domain assumption Lemma 3.6 of [2]: for a (σ,γ)-weakly porous set E, if J ⊂ I and |I| = 2|J| then ρ(I) ≤ Φ ρ(J).
Borrowed, not re-proved; it is load-bearing for Theorem 3.7, which derives one-sided weak porosity from ordinary weak porosity.
domain assumption Known characterization Ap(R) = Ap-(R) ∩ Ap+(R) from [7, Theorem 4].
Used in Remark 2.1 to position one-sided classes relative to the two-sided class; not needed for Theorem 4.5.
domain assumption Minkowski dimension characterization dim_M Ω = sup{s>0 : limsup_{ε→0} |Ω(ε)|/ε^{1-s} > 0} from [8].
Used in Corollary 4.4 to convert the neighborhood decay estimate into a bound on Hausdorff dimension.

invented entities (2)

E-free maximal hole function ρ_E(I)
purpose: Local geometric quantity measuring the radius of the largest E-free open interval contained in I; central to both notions of weak porosity and to the proofs in Section 4.
Introduced in Definition 3.1. It is a mathematical tool, not an empirical entity, and it carries no observable prediction outside the theory.
One-sided right- and left-weakly porous sets
purpose: Geometric one-sided analogues of weak porosity; intended to characterize which sets E make d(·,E)^(-α) an A1+ or A1- weight.
New mathematical definitions from Definitions 3.3 and 3.4. Their justification is the equivalence theorems proved here and the analogy with the weak porosity notion from [2].

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Cite this review

Pith. "Pith review of One-sided Muckenhoupt weights and one-sided weakly porous sets in $\mathbb{R}$." pith.science (2026). https://pith.science/paper/MQBHJDPI

@misc{pith2026241119856,
  author       = {Pith},
  title        = {Pith review of: One-sided Muckenhoupt weights and one-sided weakly porous sets in $\mathbbR$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MQBHJDPI}},
  note         = {Machine review of arXiv:2411.19856}
}

abstract

In this work, we introduce the geometric concept of one-sided weakly porous sets in the real line and show that a set $E\subset\mathbb{R}$ satisfies $d(\cdot,E)^{-\alpha}\in A_1^+(\mathbb{R})\cap L^1_\textrm{loc}(\mathbb{R})$ for some $\alpha>0$ if and only if $E$ is right-sided weakly porous. Furthermore, we find that the property of being both left-sided and right-sided weakly porous is equivalent to the recent weakly porous condition discussed in the bibliography, which, in turn, was previously found to be intimately related to the usual class of Muckenhoupt weights $A_1$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 12 canonical work pages

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Hugo Aimar, Ivana G´ omez, and Ignacio G´ omez Vargas. Weakly porous sets andA1 Muckenhoupt weights in spaces of homogeneous type, 2024. arXiv 2406.14369

arXiv 2024
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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Weakly porous sets andA1 Muckenhoupt weights in spaces of homogeneous type, 2024

Hugo Aimar, Ivana G´ omez, and Ignacio G´ omez Vargas. Weakly porous sets andA1 Muckenhoupt weights in spaces of homogeneous type, 2024. arXiv 2406.14369

arXiv 2024

[2] [2]

Anderson, Juha Lehrb¨ ack, Carlos Mudarra, and Antti V

Theresa C. Anderson, Juha Lehrb¨ ack, Carlos Mudarra, and Antti V. V¨ ah¨ akangas. Weakly porous sets and Muckenhoupt Ap distance functions. J. Funct. Anal. , 287(8):Paper No. 110558, 34, 2024

work page 2024

[3] [3]

Dur´ an and Fernando L´ opez Garc ´ ıa

Ricardo G. Dur´ an and Fernando L´ opez Garc ´ ıa. Solutions of the divergence and analysis of the Stokes equations in planar H¨ older-α domains. Math. Models Methods Appl. Sci. , 20(1):95–120, 2010

work page 2010

[4] [4]

V¨ ah¨ akangas

Bart lomiej Dyda, Lizaveta Ihnatsyeva, Juha Lehrb¨ ack, Heli Tuominen, and Antti V. V¨ ah¨ akangas. Muckenhoupt Ap-properties of distance functions and applications to Hardy-Sobolev–type inequal- ities. Potential Anal., 50(1):83–105, 2019

work page 2019

[5] [5]

Fractal geometry

Kenneth Falconer. Fractal geometry. John Wiley & Sons, Ltd., Chichester, third edition, 2014. Mathematical foundations and applications

work page 2014

[6] [6]

Rubio de Francia.Weighted norm inequalities and related topics , volume 116 of North-Holland Mathematics Studies

Jos´ e Garc ´ ıa-Cuerva and Jos´ e L. Rubio de Francia.Weighted norm inequalities and related topics , volume 116 of North-Holland Mathematics Studies . North-Holland Publishing Co., Amsterdam,

work page

[7] [7]

F. J. Mart ´ ın-Reyes, P. Ortega Salvador, and A. de la Torre. Weighted inequalities for one-sided maximal functions. Trans. Amer. Math. Soc. , 319(2):517–534, 1990

work page 1990

[8] [8]

Geometry of sets and measures in Euclidean spaces , volume 44 of Cambridge Studies in Advanced Mathematics

Pertti Mattila. Geometry of sets and measures in Euclidean spaces , volume 44 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1995. Fractals and rectifiability

work page 1995