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

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Pith's one-line read 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.

desk verdict Solid, correctly executed extension of weighted Hardy–Sobolev inequalities from porous to bounded median-porous sets via mixed-homogeneity distance weights. read the letter →

arxiv 2607.03769 v1 pith:RSAWX2ZI submitted 2026-07-04 math.CA

Diego Maldonado , Javier Soria This is my paper · ORCID
classification math.CA MSC 42B2542B37
keywords MuckenhouptweightsfractionalintegralsHardy–SobolevinequalitiesmedianporoussetsdistancemixedhomogeneityAssouaddimensionMinkowski
verification ladder T0 review T1 audit T2 compute T3 formal

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The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

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Referee Report

0 major / 5 minor

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).

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.

minor comments (5)
  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. Minor typographical and stylistic points: hyphenation of “Hardy–Sobolev” is inconsistent; “median porous” versus “median-porous”; and the abstract capitalizes “Distances” unnecessarily.

Circularity Check

1 steps flagged · score 2.0 of 10

Mild load-bearing self-citation to the authors' own abstract two-weight criteria [14]; the geometric verification for mixed-homogeneity distance weights is independent and non-circular.

  1. self citation load bearing [Section 2.3 (Theorems 2.2–2.3) and opening of Section 4]
    "The proofs of Theorems 1.1, 1.6, 1.7, and 1.9 will rely on Theorems 2.2 and 2.3 as well as on the pointwise estimate … Theorem 2.2 (Theorem 1.1 from [14]) … Theorem 2.3 (Theorem 1.2 from [14])."

    The operator-theoretic estimates that convert A_∞ membership plus either (2.2) or (2.3) into the desired I_α bounds are taken entirely from the authors' own prior preprint [14]. While the geometric verification that the mixed weights satisfy those abstract hypotheses is independent and non-circular, the load-bearing analytic step itself reduces to a self-citation.

full rationale

The derivation chain of Theorems 1.1, 1.6, 1.7 and 1.9 consists of (i) placing the mixed-homogeneity weights u_{γ1,γ2} and u_{δ1,δ2} into A_∞ via the lattice property once the exponents lie in I_∞(E) (independent characterizations from [1,16,8]), and (ii) algebraically arranging the exponents so that either the pointwise comparison (2.2) or the weak-L^{n/Θ} membership (2.3) holds, reducing the latter to the elementary function g_E of Lemma 3.1 whose L^{1,∞} bound follows from the Minkowski-volume estimate that is automatic for bounded median-porous sets (Lemmas 2.5 and 2.7). Both reductions are explicit, parameter-free and non-tautological. The only self-citation that is load-bearing is the invocation of the abstract operator criteria themselves (restated as Theorems 2.2–2.3 from the authors' submitted preprint [14]). Those criteria are general and independent of median porosity; the present paper's novelty is precisely the construction that verifies their hypotheses for the new weights. No equation forces a conclusion by definition, no fitted parameter is renamed a prediction, and no uniqueness theorem is imported to forbid alternatives. Score 2 reflects the single self-citation while recognizing that the central geometric content stands independently.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

Pure-analysis paper: no fitted numerical parameters. Load-bearing background consists of standard Muckenhoupt/RH theory, the abstract two-weight theorems of the authors’ prior work [14], the median-porosity characterizations of [16], and elementary dimension comparisons. The only entities invented here are the mixed-homogeneity weights and the auxiliary function g_E used to check the weak-type condition.

assumptions (5)
  • standard math A_∞ is a lattice: max/min of two A_∞ weights remain A_∞ (used to place u_{γ1,γ2} in A_∞).
    Invoked in §2.1 and at the start of the proof of Thm 1.1; classical fact from [11].
  • domain assumption d(·,E)^{-β} ∈ A_∞(R^n) ⇔ β ∈ I_∞(E) = (ℓ_∞(E), Mu_∞(E)), with Mu_∞(E) > 0 precisely when E is median porous ([16]).
    Central geometric input; quoted as (2.12) and used throughout §4 to guarantee weight membership.
  • domain assumption Abstract two-weight criteria for I_α: if u, v^{-p′/p} ∈ A_∞ and either u^{1/q} ≤ C v^{1/p} (critical) or u^{1/q}/v^{1/p} ∈ L^{n/Θ,∞} (subcritical), then the weighted I_α bound holds ([14, Thms 2.2–2.3]).
    Black-box engine of all main theorems; invoked explicitly in §4.
  • standard math Pointwise bound |f(x)| ≤ (1/(n ω_n)) I_1(|∇f|)(x) for f ∈ C_0^1 (Gilbarg–Trudinger).
    Converts I_1 estimates into Hardy–Sobolev inequalities; (2.4).
  • ad hoc to paper Mu_∞(E) ≤ n − dim_M(E) for bounded nonempty E (Lemma 2.7).
    Proved in the paper from the integral identity (2.14); guarantees dim_M(E) < n so that exponents satisfying (1.14) exist (Remark 1.2).
invented entities (2)
  • Mixed-homogeneity distance weights u_{γ1,γ2}
    purpose: Allow different power behavior of d(·,E) inside and outside the diam(E)-neighborhood, enabling inequalities when pure powers fail for non-porous sets.
    Defined in (1.10)–(1.11); the central new object of the paper. No independent experimental handle; purely analytic construction.
  • Auxiliary function g_E
    purpose: Model the ratio u^{1/q}/v^{1/p} so that membership in weak L^1 is equivalent to a Minkowski-content bound on neighborhoods of E.
    Introduced in Lemma 3.1; used only as a technical device inside the proofs of the main theorems.

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Pith. "Pith review of Weighted estimates for fractional integrals with Distances to Bounded Median Porous Sets and applications to Hardy--Sobolev Inequalities." pith.science (2026). https://pith.science/paper/RSAWX2ZI

@misc{pith2026260703769,
  author       = {Pith},
  title        = {Pith review of: Weighted estimates for fractional integrals with Distances to Bounded Median Porous Sets and applications to Hardy--Sobolev Inequalities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RSAWX2ZI}},
  note         = {Machine review of arXiv:2607.03769}
}
abstract

Weighted estimates for the fractional integral operator $I_\alpha$ are established and subsequently applied to derive corresponding Hardy--Sobolev inequalities. The weights are constructed from distance functions to bounded median porous sets and possess mixed homogeneity, which enables us to extend earlier results obtained for porous sets to a significantly broader class of geometries.

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