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One-sided median porous sets and one-sided Muckenhoupt distance functions

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Pith's one-line read One-sided median porosity of E is necessary and sufficient for d_E^{-α} to be a one-sided Muckenhoupt A_p weight for some α>0 and 1<p<∞.

desk verdict The paper defines one-sided median porosity and shows it exactly characterizes when d_E^{-α} sits in a one-sided A_p class for the right range of α, plus new median characterizations of A_p and BMO. read the letter →

arxiv 2607.01167 v1 pith:NW7HRNOI submitted 2026-07-01 math.CA

Alptekin Can Goksan , Ignacio Uriarte-Tuero This is my paper · ORCID
classification math.CA
keywords one-sidedmedianporosityMuckenhouptA_pweightsdistanceBMOporoussetsreallineweightedinequalities
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

This paper defines one-sided median porosity for subsets of the real line. It shows this geometric property is exactly equivalent to the distance weight d_E raised to a negative power belonging to a one-sided A_p class for appropriate α and p. The work also yields median-based characterizations of one-sided A_p weights and BMO functions. It determines the exact range of α for both p=1 and p>1 cases, and distinguishes one-sided median porosity from related notions with an example. Readers care because these conditions determine when distance weights can be used in one-sided weighted inequalities in real analysis.

What carries the argument

one-sided median porosity, a median-based condition on subsets E of the real line that controls the distribution of E and its complement in intervals

What would settle it

A set E that is one-sided median porous but for which d_E^{-α} fails to satisfy the one-sided A_p condition for every α>0, or conversely a set where the weight condition holds but the set is not one-sided median porous.

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Extended reading notes

Core claim

We introduce the notion of one-sided median porosity for subsets E of R. We prove that this condition is necessary and sufficient for the distance weight d_E^{-α} to belong to a one-sided Muckenhoupt A_p class for some α>0 and 1<p<∞. As part of the proof, we obtain new characterizations of one-sided A_p weights and one-sided BMO functions, in terms of medians. We find the precise range of exponents α>0 such that d_E^{-α} belongs to a one-sided A_p class, both for p=1 and for 1<p<∞. In addition, we show that E is median porous if and only if it is both left and right median porous, and we give an example of a one-sided median porous set which is neither median porous nor one-sided weakly poro

Load-bearing premise

The definitions of one-sided median porosity and the one-sided Muckenhoupt classes are consistent with the median-based characterizations of A_p weights and BMO functions introduced in the paper.

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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

Summary. The paper introduces one-sided median porosity for subsets E of R. It proves this condition is necessary and sufficient for the distance weight d_E^{-α} to belong to a one-sided Muckenhoupt A_p class for some α>0 and 1<p<∞. As part of the argument, new median-based characterizations of one-sided A_p weights and BMO functions are obtained. The precise range of α is determined for both the p=1 case (building on prior weak porosity results) and for 1<p<∞. It is also shown that E is median porous if and only if it is both left and right median porous, and an explicit example is given of a one-sided median porous set that is neither median porous nor one-sided weakly porous.

Significance. If the equivalences hold, the work supplies a geometric characterization linking one-sided median porosity to membership of distance weights in one-sided A_p classes, extending the recent A_1/weak-porosity equivalence. The median characterizations of A_p and BMO provide potentially useful alternative tools in one-sided harmonic analysis. The range of α and the distinguishing example clarify the relationships among porosity notions. These are solid contributions to the study of weights and function spaces on the line.

minor comments (3)
  1. [Abstract] The abstract states that new median characterizations are derived but does not indicate their precise form (e.g., the median condition replacing the usual integral or supremum). Adding one sentence summarizing the characterization would improve readability for readers scanning the abstract.
  2. [Introduction] Notation for the one-sided median operator and the precise definition of one-sided median porosity should be introduced with a displayed equation or numbered definition in the introduction or §2 to avoid any ambiguity when the reader reaches the main theorems.
  3. In the example distinguishing the notions, confirm that the set is constructed so that the median porosity constant is positive while the weak porosity constant is zero; a short calculation or reference to the relevant inequality would strengthen the claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the paper, the clear summary of its contributions, and the recommendation for minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper introduces the new notion of one-sided median porosity and derives its equivalence to d_E^{-α} belonging to one-sided A_p (1<p<∞) as a fresh result, while obtaining median characterizations of A_p and BMO during the proof. The cited prior equivalence for A_1 and weak porosity serves only as background contrast and does not enter the new derivations by construction; no step reduces a claimed prediction or uniqueness theorem to a fitted parameter, self-citation chain, or renamed input. The argument is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 2 assumptions · 1 invented entities

The central claim rests on standard properties of medians and distance functions in R together with the prior definition of one-sided Muckenhoupt classes; the new porosity notion is introduced without independent evidence beyond the claimed equivalence.

assumptions (2)
  • standard math Standard properties of the median operator on intervals in R hold and interact with the distance function d_E in the expected way.
    Invoked implicitly when defining median porosity and relating it to A_p via medians.
  • domain assumption The one-sided Muckenhoupt A_p classes are defined via the usual one-sided maximal function or averaging operators.
    Background definition from prior literature used to state the target class.
invented entities (1)
  • one-sided median porosity
    purpose: New condition on subsets E of R that exactly characterizes membership of d_E^{-α} in one-sided A_p.
    Introduced in the paper as the load-bearing new notion; no independent evidence outside the claimed equivalence is given in the abstract.

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Pith. "Pith review of One-sided median porous sets and one-sided Muckenhoupt distance functions." pith.science (2026). https://pith.science/paper/NW7HRNOI

@misc{pith2026260701167,
  author       = {Pith},
  title        = {Pith review of: One-sided median porous sets and one-sided Muckenhoupt distance functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NW7HRNOI}},
  note         = {Machine review of arXiv:2607.01167}
}
abstract

We introduce the notion of one-sided median porosity for subsets $E$ of $\mathbb{R}$. We prove that this condition is necessary and sufficient for the distance weight $d_E^{-\alpha}$ to belong to a one-sided Muckenhoupt $A_p$ class for some $\alpha>0$ and $1<p<\infty$. As part of the proof, we obtain new characterizations of one-sided $A_p$ weights and one-sided $\mathrm{BMO}$ functions, in terms of medians. It was recently shown that $d_E^{-\alpha}$ is a one-sided Muckenhoupt $A_1$ weight for some $\alpha>0$ if and only if $E$ is one-sided weakly porous. In this paper, we find the precise range of exponents $\alpha>0$ such that $d_E^{-\alpha}$ belongs to a one-sided $A_p$ class, both for $p=1$ and for $1<p<\infty$. In addition, we show that $E$ is median porous if and only if it is both left and right median porous, and we give an example of a one-sided median porous set which is neither median porous nor one-sided weakly porous.

Discussion (0). Continue with ORCID to comment.

Reference graph

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