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Medians, Oscillations, and Distance Functions

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Pith's one-line read A set E is median porous exactly when some power of dist(·,E) is an A_p weight.

desk verdict A solid paper that resolves the A_p distance-weight characterization and breaks the porosity barrier at the critical exponent; the main proof chain is sound, with a few peripheral sketches that a referee should ask to be completed or flagged. read the letter →

arxiv 2507.21020 v2 pith:XERNMN6Z submitted 2025-07-28 math.CA

Marcus Pasquariello , Ignacio Uriarte-Tuero This is my paper · ORCID
classification math.CA MSC 42B2542B3526D1046E35
keywords MuckenhouptA_pweightsmedianporosityBMOdistancefunctionsHardy-Sobolevinequalitiessparsedominationexponentscritical
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 closes a gap left by the A1 theory: it characterizes, for every 1

What carries the argument

The engine is the upper s-median M_s(f,Q), the largest λ for which the sublevel set {f<λ} has measure at most s|Q| and {f>λ} at most (1-s)|Q|. The paper's Theorem 3.4 produces, for any measurable f and any 0<s<t<1, an η-sparse family of dyadic subcubes such that |f-M_s(f,Q0)| is controlled by the median differences M_t(f,Q)-M_s(f,Q) plus median drift across generations; merging the upper and lower stopping-time families gives the sparse domination Theorem 1.1 and hence the BMO/BLO characterizations. For the set-theoretic applications, the bridge is Proposition 7.2: for d=dist(·,E) and a cube Q0 meeting E, M_s(d,Q0)^n≲V_s(Q0)≲M_t(d,Q0)^n, where V_s(Q0) is the largest scale δ such that (1-s)|Q0| can be filled by E-free dyadic subcubes of side at least δ. This comparability converts bounded median oscillation of log d into the median porosity condition and back, and the quantitative Muckenhoupt exponents Mu_p(E) are defined from the same filling scales.

What would settle it

Find a set E and a sequence of cubes Q_k meeting E with V_t(Q_k)/V_s(Q_k)→∞ while sup_Q[M_t(log dist(·,E),Q)-M_s(log dist(·,E),Q)] stays bounded; Theorem 1.5 and Proposition 7.2 say this cannot happen, so such a family would refute the characterization. A concrete family to test is E={0}∪{$2^{{-k}}$:k≥1} in R, computing dyadic filling volumes and median differences explicitly.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.5: a nonempty set E⊂R^n is median porous if and only if dist(·,E)^{-α}∈A_∞=∪_{1≤p<∞}A_p for some α>0, equivalently log dist(·,E)∈BMO. Median porosity means there exist 0<s<t≤1 and 0<δ<1 such that for every cube Q0 meeting E, dyadic cubes avoiding E can be chosen with total volume at least (1-s)|Q0| and each of volume at least δ V_t(Q0), where V_t(Q0) is the largest scale filling a (1-t)-fraction of Q0. The quantitative refinement Theorem 1.10 gives the sharp interval -(p-1)Mu_1(E)<α<Mu_p(E) for membership in A_p, with Mu_p a dimension-like exponent defined through these filling scales. These results rest on a new BMO criterion: for any measurable f and any 0<s<t<1, f∈BMO iff sup_Q[M_t(f,Q)-M_s(f,Q)]<∞, which follows from a sparse domination inequality by median differences. The same machinery yields critical Hardy–Sobolev inequalities for median porous domains and sharp necessary conditions, the first such statements that do not assume porosity.

Load-bearing premise

The argument turns on the dyadic sparse bound (Theorem 3.4) holding for every measurable function and every pair 0<s<t<1 with constants independent of f, and on the two-sided comparison of median values of dist(·,E) with E-free volume quantities being uniform over all cubes; if either uniformity fails, the equivalence between median porosity and A_p distance weights collapses.

Editorial extensions

If this is right

  • For every 1<p≤∞, whether dist(·,E)^{-α} is an A_p weight for some α>0 is settled by one geometric condition: E is median porous, equivalently log dist(·,E)∈BMO.
  • The exact interval -(p-1)Mu_1(E)<α<Mu_p(E) gives a quantitative dictionary between a set's hole structure and the admissible singularity of the distance weight, recovering the p=1 theorems at the boundary.
  • Critical-exponent Hardy–Sobolev inequalities hold for median porous domains, including sets that are neither porous nor weakly porous, and the matching necessary condition is expressed by Mu_8(E).
  • The Riesz potential method is sharp: for median porous but non-porous sets it proves the subcritical weighted inequality only when the set is porous, so subcritical results require a new strategy.
  • The median characterization of BMO holds for all 0<s<t<1 and fails for separated parameters in the dyadic setting, so the non-dyadic medians carry information that dyadic oscillations cannot see.

Reading between the lines

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

  • The sparse median argument in Theorem 3.4 is written for arbitrary locally finite measures, so the BMO-by-medians characterization, and hence the median-porosity dictionary for distance weights, should transfer to doubling metric measure spaces.
  • The Muckenhoupt exponents Mu_p(E) are defined through dyadic volume ratios, so for explicit or computable fractals one could estimate them numerically and certify membership of dist(·,E)^{-α} in A_p.
  • The family E_γ={±m^γ} provides a one-parameter interpolation between porosity and weak porosity; testing fractional Hardy or Poincaré inequalities on these sets could reveal exactly where the critical exponent is needed.
  • Since the paper also characterizes Hölder continuous w with log w∈BMO by the same median-porosity condition, the method may extend to weight classes whose zero sets are Hölder regular but not distance sets.
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Editorial analysis

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

Referee Report

0 major / 6 minor

Summary. The paper develops a new two-parameter median sparse domination theorem and derives from it a median-value characterization of BMO and BLO. It then applies this machinery to distance functions: a set E is called median porous and is shown to be characterized by dist(·,E)^{-α} ∈ A_∞ = ∪_{p≥1} A_p for some α>0, equivalently by log dist(·,E) ∈ BMO. The paper also gives the exact range of exponents α for which dist(·,E)^{-α} ∈ A_p in terms of newly defined p-Muckenhoupt exponents, and uses this to prove weighted Hardy-Sobolev and related inequalities for median porous sets in the critical case, with examples of median porous sets that are not weakly porous and a discussion of the sharpness of the A_p and Riesz-potential methods.

Significance. If the main results are correct, the paper substantially advances the geometric theory of Muckenhoupt distance weights: it removes the porosity side condition that was present in all prior work for p>1 and gives the first exact exponent range for A_p. The central line from Theorem 1.1 through Theorem 1.5 to the quantitative Theorem 8.7 is presented with detailed proofs, including the stopping-time estimates, the dyadic-to-nondyadic sparse argument, and the volume-to-median comparison in Proposition 7.2. The paper also provides concrete examples, explicit exponent computations, and a discussion of the limits of the two main proof methods. These are substantial contributions regardless of the peripheral sketches.

minor comments (6)
  1. [Corollary 7.6] The proof of Corollary 7.6, presented as a second proof of the ALMV24 A_1 characterization, leaves the final details to the reader; since this is a standalone proof of a previously known result, it would be better either to complete the argument or to label it explicitly as a sketch rather than presenting it as a full proof.
  2. [Theorem 7.8] The proof of Theorem 7.8 for Hölder continuous functions is omitted with the statement that it is analogous to the distance-function case; to make the claim verifiable, the authors should provide at least the analogue of Proposition 7.2 and the key steps where the Hölder modulus is used.
  3. [Remark 10.11] The claimed improvements to distance-weighted Poincaré inequalities are stated with 'we leave the details to the reader'; because these are applications of the main method, the relevant hypotheses and the checking of the A_∞-type conditions should be included or the statements should be explicitly marked as sketches.
  4. [Theorem 8.7 / Theorem 1.10] In the displayed statements, the lower endpoint of the A_p exponent interval should read -(p-1)Mu_{p'}(E), with the subscript denoting the conjugate index; the proof's duality step for α<0 produces exactly this conjugate-index quantity, and the notation in the text is currently ambiguous.
  5. [Theorem 10.8] The text 'Combining Theorem 10.9, Lemma 10.6, and Lemma 10.7' refers to a non-existent Theorem 10.9; this should presumably be Theorem 10.4, and the cross-reference should be corrected.
  6. [Section 5, Proposition 5.2] In the definition of the stopping-time family B_{Q1}, maximality is used implicitly to ensure disjointness; adding a sentence that maximal dyadic subcubes are pairwise disjoint would make the sparsity estimate in Proposition 5.2 easier to follow.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation chain is self-contained from the sparse domination theorem through the median/BMO characterization to the median-porosity and quantitative exponent results.

full rationale

The paper's central derivation is internally consistent and does not reduce to its inputs. Theorem 1.1/3.3 (median sparse domination) is proved from first principles in Section 5, and Theorem 1.2/3.6 (median characterization of BMO and BLO) is derived from it by integration over sparse cubes. The bridge Proposition 7.2, comparing upper medians of the distance function with the geometric quantities V_s(Q0), is an honest proved estimate, not a definitional identification: the direction M_s(d,Q0)^n ≲ V_s(Q0) follows by covering the superlevel set {d ≥ λ} with maximal E-free dyadic cubes, and the reverse direction uses shrinking cubes and the doubling property of Lebesgue measure. Median porosity (Definition 7.3) is thereby a genuine geometric condition, equivalent to the BMO condition on log dist(·,E) only after the nontrivial Proposition 7.2 and Theorem 1.2 are applied. Theorem 1.5 then follows directly rather than by renaming. The quantitative Theorem 8.7 similarly derives the range of admissible α from the Ap definition plus Proposition 7.2; the definition of Mup(E) is not engineered to be the Ap condition itself but is a separate cubical measure estimate, and the proof supplies both necessity and sufficiency. No parameter is fitted to data and then renamed a prediction. The paper invokes external standard equivalences (A∞ = ∪ Ap, log w ∈ BMO iff w^α ∈ Ap for some α), which is legitimate use of known theorems. The only external theorem used at a proof step is Theorem 7.4 from [ALMV24], used to dispose of the weakly porous case in Theorem 1.5; that theorem is not authored by the present authors, and the paper explicitly provides a second, independent proof in Corollary 7.6, making the manuscript self-contained. No self-citation is load-bearing, and no uniqueness theorem is imported to forbid alternatives. The remainder of Section 10 consists of applications that inherit the established characterization rather than assuming it. Accordingly, the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

The central theory is built on standard harmonic analysis infrastructure and standard cited theorems; no free parameters are fitted to make the theorems true. The paper's new definitions are explicit and are characterized by independent external conditions, so they are not invented entities without evidence.

assumptions (6)
  • standard math Standard Lebesgue measure and dyadic cube structure on R^n.
    Used throughout, e.g. Definitions 7.1 and 7.3 for E-free cubes and V_s(Q0); no modification is introduced.
  • standard math Properties of upper s-medians, including the limiting behavior M_s(f,Q)→f(x) a.e.
    Proposition 2.5, with parts taken from Poelhuis-Torchinsky [JP12], is used in Proposition 5.2 and Theorem 1.2.
  • standard math Standard equivalences between A_p, A_∞, BMO, BLO and log weights.
    Proposition 2.1, cited to García-Cuerva and Rubio de Francia, converts log dist∈BMO into dist^{-α}∈A_∞ and uses A_∞=∪A_p.
  • standard math The weak porosity/A_1 characterization of ALMV24 and Vasin for n=1.
    Used for comparison and in one branch of Theorem 1.5; Corollary 7.6 gives a self-contained proof, so this is not load-bearing.
  • standard math Muckenhoupt-Wheeden and Pérez-Wheeden T1-type conditions for weighted Riesz potentials.
    Used in Section 10 to convert the testing condition (10.10) into weighted Riesz potential bounds and hence Hardy-Sobolev inequalities.
  • domain assumption E is a nonempty proper subset of R^n; dist(·,E) may be replaced by dist(·,cl E).
    Assumed throughout the distance-function sections; harmless for distance weights and standard in the literature.
invented entities (2)
  • Median porous sets independent evidence
    purpose: Geometric class of sets whose distance functions generate A_p weights and support Hardy-Sobolev inequalities.
    Characterized by Theorem 1.5 via dist^{-α}∈A_∞ and constructed explicitly in Section 9, so the notion has an external falsifiable handle.
  • p-Muckenhoupt exponents Mu_p(E) and Mu_∞(E) independent evidence
    purpose: Dimension-like quantities giving the exact α-range for dist^{-α}∈A_p.
    Defined in Definition 8.6 and tied to actual A_p membership in Theorem 8.7; computed for the Section 9 example family.

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Pith. "Pith review of Medians, Oscillations, and Distance Functions." pith.science (2026). https://pith.science/paper/XERNMN6Z

@misc{pith2026250721020,
  author       = {Pith},
  title        = {Pith review of: Medians, Oscillations, and Distance Functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XERNMN6Z}},
  note         = {Machine review of arXiv:2507.21020}
}
abstract

Vasin (for $n=1$) and Anderson, Lehrb\"ack, Mudarra, and V\"ah\"akangas (arXiv:2209.06284) (for $n>1$) provided a geometric characterization of the sets $E \subset \mathbb{R}^n$ so that $w = \text{dist}(\cdot, E)^{-\alpha}$ is a Muckenhoupt $A_1$ weight for some $\alpha > 0$. In this paper, we provide a geometric characterization of the sets $E \subset \mathbb{R}^n$ (which we call median porous sets) so that $w = \text{dist}(\cdot, E)^{-\alpha}$ is a Muckenhoupt $A_p$ weight for some $\alpha > 0$ (given any $1 < p \leq \infty$). Given $1 < p \leq \infty$, we also find the precise range of exponents $\alpha$ so that $w = \text{dist}(\cdot, E)^{-\alpha} \in A_p$, in analogy to the $p=1$ case done in arXiv:2209.06284. With our characterization we prove that $\mathbb{R}^n \setminus E$ supports a Hardy-Sobolev inequality if $E$ is an appropriate median porous set. All previous such results that we are aware of make the strictly stronger assumption that the set $E$ is porous, e.g. arXiv:1705.01360, arXiv:1502.01190. As far as we know, this is the first instance in the literature that the ``porosity barrier" is broken in this context. Examples of such appropriate median porous (but not porous) sets were known. We provide further such examples, additional applications to weighted Poincar\'e inequalities, and a geometric characterization of the nonnegative H\"older continuous functions $w$ such that $\log (w) \in BMO$. We prove that two of the methods we use ($A_p$ and Riesz potential methods) are sharp, i.e. they cannot be improved beyond the results we obtain. The proofs rely on a new median-value characterization of $BMO$: For a real-valued measurable function on $\mathbb{R}^n$ and constants $0 < s < t < 1$, \[\|f\|_{BMO} \approx_{s, t, n} \sup_{Q}[M_t(f, Q) - M_s(f, Q)]\] where $M_s(f, Q)$ denotes the $s$-median value of $f$ on $Q$.

Figures

Figures reproduced from arXiv: 2507.21020 by the authors.

Figure 1
Figure 1. A set E ⊂ Q0 and the procedure to compute VspQ0q for s “ 3 4 and s “ 3 8 . The grey cubes are E-free dyadic subcubes of Q0. Therefore V3 4 pQ0q “ 1 16 |Q0| and V3 8 pQ0q “ 1 64 |Q0|. The key step in applying Theorem 1.2 to prove Theorem 1.5 is a general statement relating the volume quantities VspQ0q with the median values of the distance function (see Proposition 7.2). This proposition also allows us to interpret t… view at source ↗
Figure 2
Figure 2. The set Eγ “ t˘mγ : m ∈ Nu for γ “ 0.25. Remark 9.2. Let f be an even function on R. To show that f ∈ BMO, it suffies to show that sup I⊂r0,8q ´ ż I |f ´ fI | “ C ă 8. Indeed, suppose this supremum is finite. Given an arbitrary interval I, write I “ I ´ ∪ I ` where I ´ ⊂ p´8, 0s and I ` ⊂ r0, 8q. Without loss of generality, assume ´I ´ ⊂ I `. Then since f is even, ż I |f ´ xfyI` | dx ď 2 ż I` |f ´ xfyI` | dx ď 2C|I … view at source ↗
Figure 3
Figure 3. (i) A good interval I0 “ pa, bq where c “ a ` p1 ´ sqpb ´ aq. (ii) The choice of intervals tIku so that ř |Ik| ě p1 ´ sq|I0|. The next lemma says that if we restrict ourselves to considering intervals that contain a uniformly bounded number of points of E, then the median porosity condition (9.1) is trivial. Lemma 9.5. Fix a natural number m ě 1. Then if I0 “ pa, bq contains m points of E we have Lr1pI0q Àγ,m Lr2 3 … view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The choice of intervals tIku when I0 contains m “ 5 points. Finally, if I0 contains a large number of points of E, we can approximate I0 by good intervals and apply Lemma 9.4. Lemma 9.6. Let 0 ă s ă 1. Then for any ϵ ą 0, there is some large N ě 10 so that if I0 “ pa, …
Figure 5
Figure 5. Figure 5: The intervals Ji “ pai , biq with the points ai , ci ∈ E where c1 “ a1 ` p1 ´ sq|J1| and c2 “ a2 ` p1 ´ p s`1 2 qq|J2|. Proof of Theorem 9.1. We start by showing that E is not weakly porous. Suppose to the contrary that E is weakly porous with constants 0 ă s, δ ă 1. L…

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Weighted estimates for fractional integrals with Distances to Bounded Median Porous Sets and applications to Hardy--Sobolev Inequalities

    math.CA 2026-07 accept novelty 6.0 of 10

    Fractional-integral and Hardy–Sobolev inequalities hold for mixed-homogeneity distance weights to bounded median porous sets under explicit Muckenhoupt-index conditions.

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