{"id":"548a91ca-890b-48d6-9151-8c07c7adfdc5","arxiv_id":"2607.03769","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Fractional-integral and Hardy–Sobolev inequalities hold for mixed-homogeneity distance weights to bounded median porous sets under explicit Muckenhoupt-index conditions.","lead":"The paper proves weighted bounds for fractional integrals when the weights are distances to bounded median porous sets, using mixed local/global homogeneity. This extends classical Hardy–Sobolev inequalities from porous sets to a strictly larger geometric class.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the black-box appeal to [14] as the only potential soft spot, yet that appeal is standard and the paper supplies every geometric ingredient required by the black boxes. The mixed-homogeneity construction is handled by the lattice property already recorded in §2.1; the Minkowski-gap condition (1.14) is converted into the L^{1,∞} bound for g_E by elementary covering arguments (Lemmas 2.5 and 3.1). No constant-tracking or lattice-property failure is visible for the weights under consideration. Consequently the reader’s ACCEPT verdict stands; the single verification step above merely reconfirms an already transparent estimate.","tokens_in":18705,"tokens_out":626,"duration_ms":5276,"concrete_test":"Independently recompute the weak-L^{n/Θ} norm of u^{1/q}/v^{1/p} for the model set E = ∪_j ∂B(0,j^{-γ}) ∪ {0} (weakly porous, dim_M(E) < n) with the concrete exponents of Corollary 1.4; verify that the resulting distribution function is bounded by C t^{-1} with C controlled by diam(E)^{n-s} as claimed in (3.2).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims of Theorems 1.1 and 1.7 rest on verifying that the mixed-homogeneity weights u_{γ1,γ2} and u_{δ1,δ2} satisfy the abstract hypotheses of the authors’ prior criteria [14, Thms 2.2–2.3]: A_∞ membership of the pair together with either the pointwise comparison (2.2) or the weak-L^{n/Θ} membership (2.3). Section 4 carries out exactly those verifications. A_∞ membership follows from the lattice property of A_∞ (explicitly recalled in §2.1 and applied to the local/far pieces of (1.11)) once the exponents lie in I_∞(E). The weak-L condition is reduced, via the algebraic choice (4.1), to the elementary function g_E of Lemma 3.1, whose L^{1,∞} membership is equivalent to the Minkowski-volume bound (3.1) that holds for any bounded median-porous set by Lemma 2.5 and (2.13). The critical case is likewise reduced to the elementary pointwise comparison (1.24)–(1.25). No hidden analytic gap appears; the geometric hypotheses are used precisely where needed and the reduction is self-contained.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves two-weight L^p–L^q bounds for the fractional integral I_α when the weights are mixed-homogeneity distance functions u_{γ1,γ2} and u_{δ1,δ2} built from a bounded median porous set E ⊂ R^n (Theorems 1.1 and 1.7). The subcritical case Θ_{p,q}(α)>0 is reduced to the weak-L^{n/Θ} membership of the ratio via an auxiliary function g_E (Lemma 3.1), while the critical case Θ=0 is reduced to a pointwise comparison of the weights. These operator bounds, combined with the pointwise estimate |f| ≲ I_1(|∇f|), yield corresponding Hardy–Sobolev inequalities (Corollaries 1.3–1.4, Theorems 1.6 and 1.9) that extend earlier porous-set results to the strictly larger median-porous class. A blow-up rate for the constant when the integral is restricted away from a weakly porous set is also obtained (Theorem 1.10).","tokens_in":19049,"tokens_out":832,"duration_ms":25163,"significance":"The work cleanly enlarges the geometric range of weighted Hardy–Sobolev inequalities from porous sets (positive Assouad codimension) to median porous sets (Mu_∞(E)>0), a class that properly contains non-porous examples already studied in the literature. The mixed-homogeneity construction is a natural device that separates local and far-field scaling, and the reduction of the ratio condition to the elementary weak-type function g_E is elegant and reusable. The logical chain rests on standard A_∞ lattice properties, the characterization of I_∞(E) from recent work, and the authors’ abstract two-weight criteria; once those hypotheses are verified, the geometric conclusions follow. The paper therefore supplies concrete new inequalities and a transparent method for producing further examples.","major_comments":[],"minor_comments":[{"comment":"Notation for the distance function is inconsistent: the bulk of the paper uses d(x,E), while Theorem 1.10, Corollary 5.2 and the surrounding text switch to d_E(x). Standardize throughout.","section":null},{"comment":"The equivalent definitions (1.10) (max/min) and (1.11) (piecewise) are both given; presenting only the clearer piecewise form (1.11) from the outset would improve readability and avoid the case distinction on the relative size of γ1 and γ2.","section":null},{"comment":"The companion paper [14] is cited as “submitted” (arXiv:2606.19126). A brief parenthetical remark that Theorems 2.2–2.3 are self-contained abstract criteria (independent of median porosity) would help the reader assess the logical dependence.","section":null},{"comment":"Section 2.1 invokes the lattice property of A_∞ for max and min. While this follows at once from the nested structure A_p ⊂ A_∞, a one-line reference or reminder would make the argument fully self-contained.","section":null},{"comment":"Minor typographical and stylistic points: hyphenation of “Hardy–Sobolev” is inconsistent; “median porous” versus “median-porous”; and the abstract capitalizes “Distances” unnecessarily.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is tightly coupled to the authors’ concurrent submission [14]. The geometric novelty (mixed-homogeneity weights for median porous sets) is genuine and the reduction is clean, so the paper stands on its own once the abstract criteria are granted. I see no reason to delay acceptance pending the other paper, but the editor may wish to ensure consistent notation and constant dependence across the two works."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper cleanly pushes the weighted Hardy–Sobolev program past the porosity barrier. The new objects are the mixed-homogeneity weights u_{γ1,γ2} built from distance to a bounded median-porous set E (local exponent γ1 near E, far-field exponent γ2 outside diam(E)). Theorems 1.1 and 1.7 give two-weight bounds for I_α in the subcritical and critical regimes; the Hardy–Sobolev corollaries (1.3–1.4, 1.6, 1.9) follow at once from the pointwise estimate |f| ≲ I_1(|∇f|). Corollary 1.8 recovers the pure-power critical result of Pasquariello–Uriarte-Tuero for bounded sets, so the paper sits cleanly on top of the existing literature rather than reinventing it.\n\nWhat works: the reduction is short and transparent. Once the exponents lie in I_∞(E), the lattice property of A_∞ puts both weights in A_∞. The ratio condition of the authors’ earlier abstract criteria [14] is then rewritten, via an elementary algebraic choice of exponents, as membership of the auxiliary function g_E in L^{1,∞}. Lemma 3.1 shows that this is equivalent to a Minkowski-volume bound that holds for any bounded median-porous set (by the relation Mu_∞(E) ≤ n − dim_M(E)). No hidden analytic work is required; the geometry is used exactly where it is needed.\n\nSoft spots are minor and proportional. The paper treats the two-weight criteria of [14] as black boxes; that is standard once those criteria are general operator-theoretic statements, but a reader who has not seen [14] must take the constants on faith. Everything is restricted to bounded E; the unbounded case is left open. The blow-up rate in Theorem 1.10 is a nice extra but not essential.\n\nThis is for people who already work on weighted inequalities, Muckenhoupt distance weights, or geometric measure theory of porous sets. The math is solid, the citations are honest, and the novelty is real though incremental. I would send it to a serious referee without hesitation; the claims hold up under the stress test.","headline":"Solid, correctly executed extension of weighted Hardy–Sobolev inequalities from porous to bounded median-porous sets via mixed-homogeneity distance weights.","tokens_in":19655,"tokens_out":565,"would_cite":true,"duration_ms":4983,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B25","42B37"],"pacs":[],"model":"grok-4.5","headline":"Mixed-homogeneity distance weights to bounded median porous sets give two-weight fractional-integral bounds, and therefore Hardy–Sobolev inequalities, past the classical porosity barrier.","keywords":["Muckenhoupt weights","fractional integrals","Hardy–Sobolev inequalities","median porous sets","distance weights","mixed homogeneity","Assouad dimension","Minkowski dimension"],"falsifier":"Exhibit a bounded median porous set E and a choice of exponents inside I_∞(E) that satisfy the stated dimension-gap and balance conditions, yet for which the corresponding mixed-homogeneity pair fails either A_∞ membership or the weak-L^{n/Θ} bound, so that the two-weight estimate for I_α collapses.","tokens_in":19598,"feed_emoji":"📐","tokens_out":765,"duration_ms":5323,"temperature":0.7,"pith_summary":"Classical Hardy–Sobolev inequalities with distance weights require the target set to be porous (positive Assouad codimension). Many natural sets fail this test yet still satisfy a weaker median-porosity condition that keeps certain powers of the distance function inside the Muckenhoupt A_∞ class. This paper constructs two-weight estimates for the fractional integral I_α by using distance weights that switch homogeneity: one exponent near the set, another far away. The mixed-homogeneity pair satisfies the abstract A_∞-plus-weak-Lebesgue criteria that control I_α, both in the subcritical and critical regimes. As a consequence one obtains Hardy–Sobolev inequalities on a strictly larger class of geometries, including sets whose Assouad dimension equals the ambient dimension. The same construction also yields an explicit blow-up rate for the constant when the integral is cut off at a fixed distance from a weakly porous set.","feed_headline":"Distance weights beat the porosity barrier for Hardy–Sobolev","feed_subtitle":"Mixed-homogeneity weights to median porous sets give two-weight bounds for fractional integrals past classical porosity.","key_machinery":"The mixed-homogeneity weights u_{γ1,γ2} (and their duals u_{δ1,δ2}) together with the auxiliary function g_E that realises the weak-L^{n/Θ} norm of the ratio u^{1/q}/v^{1/p}; these objects convert the geometric membership −γ_j, δ_j p′/p ∈ I_∞(E) into the abstract hypotheses of the authors’ earlier two-weight theorems for I_α.","core_discovery":"For any bounded median porous set E, and for admissible mixed-homogeneity exponents γ_j, δ_j belonging to the Muckenhoupt interval I_∞(E), the two-weight estimate ∥I_α f∥_{L^q(u_{γ1,γ2})} ≤ C ∥f∥_{L^p(u_{δ1,δ2})} holds whenever the far-field balance condition and (in the subcritical case) the Minkowski-dimension gap condition are satisfied; the resulting inequalities remain valid when porosity fails.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Median-porous distance weights extend fractional integral bounds past porosity","Mixed-homogeneity weights on median porous sets give two-weight I_α estimates","Hardy–Sobolev inequalities hold for bounded median porous distance weights","Far-field and dimension-gap conditions yield I_α bounds beyond classical porosity","Distance weights to median porous sets remove porosity barrier for Hardy–Sobolev"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"Everything rests on the claim that the mixed-homogeneity distance weights stay inside A_∞ and that their ratio meets a weak-Lebesgue condition; both claims are inherited from earlier abstract criteria that are used as black boxes.","fun_headline_variants_meta":{"raw":{"variants":["Median-porous distance weights extend fractional integral bounds past porosity","Mixed-homogeneity weights on median porous sets give two-weight I_α estimates","Hardy–Sobolev inequalities hold for bounded median porous distance weights","Far-field and dimension-gap conditions yield I_α bounds beyond classical porosity","Distance weights to median porous sets remove porosity barrier for Hardy–Sobolev"]},"model":"grok-4.5","effort":"low","cost_usd":0.005578,"raw_usage":{"total_tokens":1417,"prompt_tokens":635,"num_sources_used":0,"completion_tokens":99,"cost_in_usd_ticks":55780000,"prompt_tokens_details":{"text_tokens":635,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":683,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":635,"tokens_out":99,"duration_ms":5343,"temperature":1.0,"reasoning_tokens":683,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T00:04:10.600242+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a bounded median porous set E and a choice of exponents inside I_∞(E) that satisfy the stated dimension-gap and balance conditions, yet for which the corresponding mixed-homogeneity pair fails either A_∞ membership or the weak-L^{n/Θ} bound, so that the two-weight estimate for I_α collapses.","supporting_citations":[],"review_version":1}