{"id":"66fbbc72-d125-4360-85dc-2ff67303e0e0","arxiv_id":"2507.21020","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A set E is median porous exactly when some power of dist(·,E) lies in A_∞, giving the exact α-range for A_p membership and the first Hardy-Sobolev inequalities beyond porous sets.","lead":"Starting from a new median-based description of BMO, the authors characterize exactly which sets E make a power of the distance to E a Muckenhoupt A_p weight, for every p>1. The same tool lets them prove Hardy-Sobolev inequalities for 'median porous' sets that are not porous, breaking a barrier in this area.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the median sparse domination and the V_s/median bridge are sound, so the central characterization stands.","rationale":"The reader identified Proposition 7.2 and Theorem 3.4 as the weakest load-bearing links, and those are indeed the right places to look. On inspection, both hold up. Proposition 7.2's second inequality uses a standard shrinking argument and the doubling property of Lebesgue measure; the first inequality requires one small supplementary observation not written out in the text, namely that the set {d ≥ M_s(d,Q0)} has measure at least (1−s)|Q0|, which turns the dyadic covering estimate into the claimed lower bound on V_s. Theorem 3.4's stopping-time construction is standard: the maximality of the selected dyadic cubes gives the required measure estimates, the two families F+ and F− are sparse with the stated constants, and their union is sparse by the known equivalence of sparseness and Carleson conditions. Lemma 5.1 correctly bridges dyadic and non-dyadic cubes with constants depending only on s,t,n. Thus I found no load-bearing error in the proof chain for Theorem 1.5 or Theorem 1.10. The reader's CONDITIONAL verdict is driven by genuinely omitted details in peripheral applications (Corollary 7.6, Theorem 7.8, Remark 10.11). These omissions do not affect the central characterization, though a fully polished final version should supply them or mark them as sketches. The quantitative α<0 case is a standard duality reduction; the notation in the arXiv text is OCR-prone, and the correct lower bound involves the conjugate index p', not literally the p=1 exponent. With that reading, Theorem 8.7 is internally consistent. I therefore recommend leaving the reader's verdict unchanged.","tokens_in":44636,"tokens_out":63116,"duration_ms":679125,"concrete_test":"Recompute the omitted covering step in Proposition 7.2 on a concrete case: take n=2, E={0}, Q0=[−1,1]^2, and compute V_s(Q0) and M_t(dist(·,E), Q0)^n for s=1/4, t=3/4. Then repeat for the dyadic children of Q0. If the ratio V_s(Q0)/M_t(dist(·,E),Q0)^n is not bounded by a constant independent of Q0, the bridge between BMO and median porosity fails in the converse direction of Theorem 1.5.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing objection identified. I checked the two engines the reader flagged. Proposition 7.2's second inequality is valid: for an admissible δ, shrinking each maximal E-free dyadic cube by η gives disjoint ηQ contained in {d > cδ^{1/n}}, and choosing η via 1−η^n(1−s)=t yields V_s(Q0) ≲ M_t(d,Q0)^n. The first inequality is also sound, provided one uses the lower bound |{d ≥ M_s(d,Q0)}| ≥ (1−s)|Q0| and covers this set by maximal E-free dyadic cubes, whose side lengths are at least c·M_s(d,Q0). Theorem 3.4's stopping-time families F+ and F− are each sparse with the stated constants (t−s)/(1−s) and (t−s)/t, the measure estimates check out, and Remark 2.9 gives sparsity of their union. The chain argument in Lemma 5.1 transfers the dyadic bound to a non-dyadic sparse family with constants depending only on s,t,n. The remaining gaps I found are explicitly marked sketches (Corollary 7.6, Theorem 7.8, Remark 10.11); these are peripheral to the main characterization and to the quantitative range in Theorem 8.7, whose α<0 reduction is standard duality and is consistent with the stated lower bound involving the conjugate index p'.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":44872,"tokens_out":25810,"duration_ms":257909,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"Corollary 7.6"},{"comment":"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.","section":"Theorem 7.8"},{"comment":"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.","section":"Remark 10.11"},{"comment":"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.","section":"Theorem 8.7 / Theorem 1.10"},{"comment":"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.","section":"Theorem 10.8"},{"comment":"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.","section":"Section 5, Proposition 5.2"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript acknowledges the near-simultaneous work of Gómez Vargas and gives a fair comparison; I see no novelty or attribution concerns. The main technical line appears sound, and the issues I found are localized to explicitly marked sketches and presentation, so the paper is appropriate for the journal after a minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a substantial paper and the main theorems hold up. It characterizes, for any 1<p≤∞, the sets E such that dist(·,E)^{-α}∈A_p, gives the precise α-range, and proves Hardy-Sobolev inequalities at the critical exponent under the new, strictly weaker median porosity condition. The engine is a two-parameter median sparse domination theorem (Theorem 1.1/1.2) that genuinely improves Lerner and Hytönen and gives a BMO characterization valid for all 0<s<t<1. I went through the stopping-time families and the V_s/median bridge; the sparsity constants are right and the uniformity over cubes meeting E holds. The quantitative range in Theorem 8.7 is carefully argued; the α<0 reduction is standard duality and consistent.\n\nThe paper is honest about prior work: ALMV24, DIL+19, LV16 are cited correctly, and the simultaneous Gómez Vargas preprint is disclosed. The Section 9 example of a median porous set that is not weakly porous is concrete and useful.\n\nSoft spots are all peripheral. Corollary 7.6 claims a second proof of the ALMV24 A_1 characterization but leaves details to the reader; Theorem 7.8 (Hölder continuous log-BMO functions) is a sketch; Remark 10.11 lists Poincaré improvements without proofs; Theorem 10.12 says it follows LV16 and skips much. None of this touches the central characterization, but a referee should insist these be completed or explicitly marked as sketches. The paper is also longer than it needs to be, and Section 10 is more discursive.\n\nRecommendation: definitely send to a serious referee. I would accept it for review, with the expectation that the authors fill in or flag the sketched applications. This is a real advance, not an incremental one.","headline":"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.","tokens_in":45472,"tokens_out":2621,"would_cite":true,"duration_ms":31827,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B25","42B35","26D10","46E35"],"pacs":[],"model":"deepseek-v4-flash","headline":"A set E is median porous exactly when some power of dist(·,E) is an A_p weight.","keywords":["Muckenhoupt A_p weights","median porosity","BMO","distance functions","Hardy-Sobolev inequalities","sparse domination","Muckenhoupt exponents","critical exponents"],"falsifier":"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.","tokens_in":44368,"feed_emoji":"🕳️","tokens_out":12259,"duration_ms":133033,"temperature":0.7,"pith_summary":"The paper closes a gap left by the A1 theory: it characterizes, for every 1<p≤∞, precisely which sets E⊂R^n make a negative power of the distance function a Muckenhoupt A_p weight. The answer is a geometric condition the authors call median porosity: around every cube that meets E, a fixed fraction of the cube can be filled with dyadic subcubes that avoid E and whose sizes are comparable to the best fillable scale, rather than to the largest individual hole. The proof is powered by a new median-value characterization of BMO, namely sup_Q[M_t(f,Q)-M_s(f,Q)] ≍ ||f||_BMO for any 0<s<t<1, plus a sparse domination formula by median differences. The paper also computes the exact range of allowed exponents α for each p and uses it to prove critical Hardy–Sobolev inequalities for median porous sets, which are weaker than porous sets.","feed_headline":"Median porosity is the exact A_p distance-weight condition","feed_subtitle":"Negative powers of dist(·,E) are Muckenhoupt weights exactly for median porous E, unlocking critical Hardy–Sobolev inequalities.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Establishes the A1 characterization of weakly porous sets and defines the 1-Muckenhoupt exponent; the paper extends this framework to A_p and recovers the A1 result by a different proof.","marker":"[ALMV24]"},{"why":"Gives the n=1 instance of the A1 distance-weight characterization; included as the starting point for the geometric side of the problem.","marker":"[Vas03]"},{"why":"Supplies the original local mean oscillation sparse domination that the paper's median sparse bound refines.","marker":"[Ler10]"},{"why":"Improves the sparse domination formula; Theorem 1.1 generalizes it to arbitrary median parameters s,t.","marker":"[Hyt14]"},{"why":"Provides the median-value calculus (mass bounds, limits, monotonicity) used throughout Section 2 and the geometric bridge.","marker":"[JP12]"},{"why":"Proves Hardy-Sobolev inequalities for porous sets; Section 10 replaces the porous hypothesis by median porosity in the critical case.","marker":"[DIL`19]"},{"why":"Links porosity to positive Assouad codimension and gives Hardy-Sobolev necessary conditions that Theorem 10.12 improves for median porous sets.","marker":"[LV16]"},{"why":"Gives the weighted Riesz potential bound used to derive critical Hardy-Sobolev inequalities.","marker":"[MW74]"},{"why":"Supplies the T1-type testing condition showing the Riesz potential method cannot extend to subcritical exponents for non-porous median porous sets.","marker":"[PW03]"},{"why":"Provides the standard equivalences between log w∈BMO and w^α∈A_p that connect the median BMO characterization to weight membership.","marker":"[GCRdF85]"}],"fun_headline_variants":["Median porous sets: exact A_p weight condition","Breaking porosity barrier: distance weights and Hardy-Sobolev","Median porosity characterizes Muckenhoupt distance weights","New BMO median criterion unlocks A_p weights","Distance weights: median porosity beats porosity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Median porous sets: exact A_p weight condition","Breaking porosity barrier: distance weights and Hardy-Sobolev","Median porosity characterizes Muckenhoupt distance weights","New BMO median criterion unlocks A_p weights","Distance weights: median porosity beats porosity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000611,"raw_usage":{"total_tokens":3043,"prompt_tokens":1347,"completion_tokens":1696,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":963,"completion_tokens_details":{"reasoning_tokens":1623}},"tokens_in":963,"tokens_out":1696,"duration_ms":12688,"temperature":1.0,"reasoning_tokens":1623,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T13:04:14.559791+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}