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Generalization for Poincar\'e--Sobolev inequalities with local weights

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Pith's one-line read The paper proves a two-weight Poincaré–Sobolev inequality on every cube, with cube-dependent local weights and no Muckenhoupt condition, covering both p ≤ q and p > q.

desk verdict Genuine advance on local two-weight Poincaré–Sobolev inequalities, with a real gap in Theorem 3.5 and an index error in Corollary 7.6; the core theorems deserve a serious referee, not a desk reject. read the letter →

arxiv 2608.15453 v1 pith:XV2D7WW5 submitted 2026-08-16 math.CA math.FA

Naoya Hatano This is my paper · ORCID
classification math.CAmath.FA MSC 42B3546E35
keywords Poincaré–Sobolevinequalitytwo-weightinequalitieslocalweightsMuckenhouptsparsedominationMorreyspacesSobolevembeddingtheoremdyadicweightclasses
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 proves a two-weight local Poincaré–Sobolev inequality: for any cube $Q\subset \mathbb{R}^n$, any Lipschitz function $f$, and any pair of weights $(w,v)$ that may depend on $Q$, the $v$-weighted oscillation of $f$ in $L^q(Q)$ is controlled by the $w$-weighted $L^p$ norm of $\nabla f$, with the constant given by a new dyadic seminorm $[w,v]_{A^{r,\tilde r}_{p,q,\rho}(Q)}$. The result covers both $p\le q$ and $p>q$ (including $q\le 1$), and it requires no Muckenhoupt ($A_p$) or local $A_\infty$ condition on the weights. Because the weight condition is local — a sum over dyadic subcubes of $Q$ — singular or degenerate weights are admissible. A limiting double critical case ($p=q$) is shown to fail, marking the boundary of the method. From the main estimate the author derives Fefferman–Phong-type inequalities with Morrey norms, two-weight Sobolev embeddings on $\mathbb{R}^n$ and on the half-space, and weighted Gagliardo–Nirenberg interpolations that improve the classical Lorentz–Sobolev embedding.

What carries the argument

The decisive object is the dyadic local weight class $A^{r,\tilde r}_{p,q,\rho}(Q)$, a seminorm that measures, on each dyadic subcube $R$ of $Q$, the mixed averages of $w^{-p'r}$ and $v^{q\tilde r}$ scaled by $\ell_R |R|^{1/p-1/q}$, and then takes the $\ell^\rho$ sum over the dyadic tree. The proof of the main theorems runs as follows: a pointwise sparse domination lemma (Lemma 4.3) rewrites the oscillation $|f-f_Q|$ as a sum over a sparse family of dyadic cubes; an integral representation (Lemma 4.2) extracts $|\nabla f|$ from the oscillation; Hölder steps place the averages of $w^{-p'r}$ and $v^{q\tilde r}$ into the seminorm; and the $L^p$ boundedness of the local Hardy–Littlewood maximal operator (Lemma 4.1) converts the sparse sum into $\|w|\nabla f|\|_{L^p(Q)}$. The critical cases (Theorem 3.5) additionally use a Hedberg-type truncation (Lemma 4.4) on the modified sparse oscillation $\mathrm{OSC}_{S,\alpha}f$.

What would settle it

Take $n=1$, $p=2$, $q=1$, $r=1$, $\rho=1$, and the power weights $(|x|^\alpha, |x|^\beta)$ with parameters strictly inside the region allowed by Example 2.6, so the seminorm $[w,v]_{A^{1,1}_{2,1,1}(Q)}$ is finite and fixed. If a sequence of Lipschitz functions $f_j$ makes the ratio $\|v(f_j-(f_j)_Q)\|_{L^1(Q)} / \| |x|^\alpha \nabla f_j\|_{L^2(Q)}$ grow without bound, the main theorem fails. Alternatively, test Lemma 4.3 directly on a fixed nonsmooth Lipschitz function $f$; the sparse domination must hold with an absolute constant independent of $f$ and $Q$, and any example where that constant must depend on $f$ destroys the proof.

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Extended reading notes

Core claim

The central assertion is Theorems 3.1 and 3.2: for $1<p<\infty$ and $0<q<\infty$, whenever $(w,v)$ belongs to the dyadic local class $A^{r,\tilde r}_{p,q,\rho}(Q)$ — the class whose seminorm is the $\ell^\rho$ sum over dyadic subcubes $R$ of $Q$ of $\ell_R |R|^{1/p-1/q} (\fint_R w^{-p'r})^{1/(p'r)} (\fint_R v^{q\tilde r})^{1/(q\tilde r)}$ — every Lipschitz $f$ on $Q$ satisfies $\|v(f-f_Q)\|_{L^q(Q)} \le C\, [w,v]_{A^{r,\tilde r}_{p,q,\rho}(Q)} \|w|\nabla f|\|_{L^p(Q)}$. The case $p\le q$ uses $\rho=\infty$ and $\tilde r=r$; the case $p>q$ uses $1/q=1/p+1/\rho$, with $\tilde r=1$ when $q\le 1$. No Muckenhoupt, local $A_\infty$, or Fujii–Wilson-type condition is assumed, and the weight pair is allowed to depend on the cube $Q$ itself. Corollaries convert the left-hand weight $v$ into a Morrey-space multiplier $g$, giving $\|g(f-f_Q)\|_{L^q(Q)} \le \|g\|_{M^s_r(Q)}\|\nabla f\|_{L^p(Q)}$, and from these the author obtains homogeneous and inhomogeneous two-weight Sobolev embedding theorems, including on the upper half-space.

Load-bearing premise

The proof rests on a quoted sparse-domination lemma: the oscillation $|f-f_Q|$ of any Lipschitz function is dominated pointwise on $Q$ by a sparse sum of dyadic average oscillations, with a constant independent of $f$ and $Q$; if that lemma cannot be applied to arbitrary Lipschitz functions with the constant compatible with the dyadic $A$-seminorm, the main theorems do not follow.

Editorial extensions

If this is right

  • Weighted oscillation estimates now hold for singular or degenerate pairs $(w,v)$ as long as the dyadic $A^{r,\tilde r}$ sum is finite, going beyond the Muckenhoupt framework.
  • The range $p>q$ is included, with the parameter $\rho$ linking $p$ and $q$; previous local two-weight results covered only $1\le p\le q$ and assumed a Fujii–Wilson-type local $A_\infty$ condition.
  • Letting the cube expand to $\mathbb{R}^n$ yields two-weight Sobolev embeddings: homogeneous ones under a global $A^{r,\tilde r}$ condition and inhomogeneous ones under the new local condition (cubes of size at most one), both also on the half-space.
  • The Fefferman–Phong-type corollaries give $\|g(f-f_Q)\|_{L^q(Q)} \lesssim \|g\|_{M^s_r(Q)}\|\nabla f\|_{L^p(Q)}$; the case $q=1$ embeds Bourgain–Morrey spaces into negative Sobolev spaces, improving the Lorentz–Sobolev embedding $L^{p,p^*}\hookrightarrow \dot{W}^{-1,p^*}$.
  • The double critical case $p=q$ in Theorem 3.5 is shown to fail (Theorem 3.6), marking the boundary of what the dyadic method can prove.

Reading between the lines

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

  • The conditions in Example 2.6 look necessary as well as sufficient for power weights: at the stated boundary the dyadic sum diverges while the inequality should fail, so the $A^{r,\tilde r}$ seminorm likely characterizes the estimate quantitatively, although the paper does not prove necessity.
  • The same sparse-domination route should transfer to Riesz potentials of fractional order, giving fractional (two-weight) Poincaré–Sobolev inequalities with an analogous dyadic class; the parameter $\alpha$ in Lemma 4.4 already anticipates this extension.
  • Since the weight condition is local to each cube, the inequality is a natural tool for Moser–Harnack arguments in degenerate elliptic equations whose coefficients satisfy the dyadic local condition rather than uniform ellipticity; the paper does not make this connection.
  • The new inhomogeneous class $A^{r,\tilde r,\mathrm{loc}}_{p,q,\rho}(\mathbb{R}^n)$ should provide two-weight local Sobolev embeddings on bounded domains where the typical $A_\infty$ hypothesis is replaced by a finite dyadic sum over cubes of size at most one.
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Editorial analysis

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Referee Report

4 major / 5 minor

Summary. The paper introduces a local two-weight class A^{r,~r}_{p,q,ρ}(Q) defined by a dyadic sum over subcubes of Q, and proves local weighted Poincaré–Sobolev inequalities for Lipschitz functions on every cube Q, with the constant given by the new seminorm. Theorem 3.1 covers p ≤ q; Theorem 3.2 covers p > q, with q > 1 using r̃ = r and q ≤ 1 using r̃ = 1. The paper then derives Fefferman–Phong-type weighted inequalities in local Morrey spaces (Theorems 3.3–3.5), states a negative result (Theorem 3.6), and applies the inequalities to two-weighted Sobolev embeddings and Gagliardo–Nirenberg inequalities (Section 7). Section 6 explains how known results of Fabes–Kenig–Serapioni, Chanillo–Wheeden, and Pérez–Rela are recovered by bounding the new seminorm.

Significance. If all claims were proven, the paper would provide a genuinely local and Muckenhoupt-free framework for weighted Poincaré–Sobolev inequalities. The explicit dyadic seminorm gives quantitative constants, the p > q range is not covered by previous two-weight results, and the reduction of known theorems to a single seminorm estimate is conceptually clean. The paper also contains explicit power-weight examples and several applications. However, the validation of the critical cases and of the Sobolev-embedding corollaries is incomplete, as detailed below, so the contribution is not yet established in its full stated form.

major comments (4)
  1. [§5.3, Lemma 4.4] The proof of Theorem 3.5 hinges on the pointwise bound osc_S f(x) ≲ |R_0|^α OSC_{S,α}f(x) + Σ_{R⊋R_0} ... . This requires χ_R(x) ≲ χ_{E_R}(x) for the cubes in the sparse family, but the sparse condition |R| ≤ 2|E_R| does not imply that containment. For x ∈ R \ E_R the term χ_R(x) ffl_R |f-f_R| can be positive while OSC_{S,α}f(x) vanishes. Lemma 4.4 is proved by integrating over the disjoint sets E_R, so the displayed argument cannot be repaired by merely replacing χ_{E_R} with χ_R, since the cubes R overlap and the sequence Hölder step would break. Consequently, the proof of Theorem 3.5, and the critical-case applications that depend on the Hedberg reduction, are not established by the manuscript as written.
  2. [Corollary 7.6 and §7.2] The embeddings stated in Corollary 7.6 have the wrong indices. The inequality derived in §7.2 is ||g f||_{L^1} ≲ ||g||_{M^{(p*)'}_{1,p'}} ||∇f||_{L^p}, which gives the embeddings M^{(p*)'}_{1,p'} ⊂ \dot{W}^{-1,p'} and m^{(p*)'}_{1,p'} ⊂ W^{-1,p'}. Corollary 7.6 instead states M^p_{1,p*} ⊂ \dot{W}^{-1,p*} and m^p_{1,p*} ⊂ W^{-1,p*}, swapping the upper and lower indices and using the wrong target space. The claimed improvement of the Lorentz–Sobolev embedding is therefore not justified as stated.
  3. [§3, Theorems 3.3–3.4] Theorems 3.3 and 3.4 are stated as corollaries of Theorems 3.1 and 3.2, but no proof is given. To apply Theorems 3.1 and 3.2 with w = 1 and v = g, one must bound the A-seminorm [1,g]_{A^{r,~r}_{p,q,ρ}(Q)} by the corresponding Morrey norm ||g||_{M^s_{r,ρ}(Q)}; this is not immediate because the seminorm contains L^{q~r} averages while the Morrey norms use L^r (or L^q) integrability, and the parameter matching changes between the cases p ≤ q and p > q. The reduction should be written out rather than left to the reader.
  4. [§5.4, Theorem 3.6] The proof of Theorem 3.6 appeals to 'the argument in Section 7 below' to pass from the local estimate to the global inequality ||g f||_{L^p} ≲ ||g||_{M^n_p} ||∇f||_{L^p}. The limiting argument is not given: one needs to justify the behaviour of f_Q and of the seminorm as |Q| → ∞ for an arbitrary increasing family of cubes, and to show that the constant in the assumed local inequality is uniform. As written, the negative result is not fully demonstrated.
minor comments (5)
  1. [Abstract and throughout] There are numerous typos that should be corrected, including 'Lipchitz', 'Sobolv', 'fmaily', 'amlgam norm', 'disscusion', and 'inequailties'.
  2. [§1, displayed inequality] In the displayed Poincaré–Sobolev inequality in the introduction, the right-hand side is written as (1/|Q| ∫_Q |∇f(x)| dx)^{1/p}; the exponent p on the integrand is missing and should be |∇f(x)|^p dx.
  3. [§7.2, displayed norm] The displayed definition of ||g||_{M^p_{q,r}} in Section 7.2 contains both a supremum over Q ∈ D and a sum over Q ∈ D, which are not equal; this should be aligned with Definition 2.5.
  4. [Corollary 7.4(1)] In Corollary 7.4(1), the assumption uses A^{r,~r,loc}_{p,q,ρ}(R^n_+) but the conclusion uses the non-local seminorm [w,v]_{A^{r,~r}_{p,q,ρ}(R^n_+)}; presumably the assumption should be A^{r,~r}_{p,q,ρ}(R^n_+).
  5. [Theorem 3.5] The sentence 'the double critical case p = q = r does not hold' refers to an undefined variable r; either r should be introduced in the statement or the sentence should be rephrased.

Circularity Check

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No circularity: the A-seminorm is an assumed sufficient condition, not a fitted or derived quantity, and the sparse-domination and maximal-function lemmas are external, parameter-free tools.

full rationale

The derivation of Theorems 3.1 and 3.2 starts from the definition of the dyadic A-seminorm [w,v]_{A^{r,r}_{p,q}(Q)} as a finite assumed quantity and proves the Poincaré–Sobolev bound via Lemma 4.3 (sparse domination, quoted from Lerner–Ombrosi–Rivera-Ríos), Lemma 4.2 (Riesz potential estimate), and the local maximal bound Lemma 4.1. The target norms appear only on the output side; the seminorm is not fitted to those norms. Known results by Fabes–Kenig–Serapioni, Chanillo–Wheeden, and Pérez–Rela are recovered by upper-bounding the new seminorm, which is the opposite of circularity. The only self-citation is [9], used in Section 7 to identify the Bourgain–Morrey embedding underlying an application; that embedding is a previously published theorem and is not needed to force the central inequality. The skeptical concern about Theorem 3.5 (χ_R versus χ_{E_R} in the Hedberg argument) is a potential proof gap regarding pointwise domination, but it is a correctness issue, not a circularity of definition or fit, and does not alter the circularity analysis.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new particles, forces, or fitted constants. It relies on standard maximal function bounds, a sparse domination lemma, and a classical oscillation estimate. The main novel object is the local A^{r,\tilde r} seminorm, which is a weight-class assumption rather than an invented entity.

assumptions (3)
  • standard math L^p boundedness of the local Hardy-Littlewood maximal operator for p > 1.
    Used in Lemma 4.1 to control the maximal function terms that appear throughout the proof of the main theorems.
  • standard math Sparse domination of the oscillation |f - f_Q| by a sparse family of dyadic subcubes.
    Lemma 4.3 is quoted from Lerner-Ombrosi-Rivera-Rios and is a load-bearing pointwise estimate in the proof of Theorems 3.1 and 3.2.
  • standard math Pointwise Riesz-potential estimate |f(x)-f_Q| bounded by the integral of |\nabla f| against |x-y|^{1-n}.
    Lemma 4.2, taken from Jost's PDE book, converts oscillation into gradient terms and is used repeatedly in the proofs and in the Hedberg method.

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Pith. "Pith review of Generalization for Poincar\'e--Sobolev inequalities with local weights." pith.science (2026). https://pith.science/paper/XV2D7WW5

@misc{pith2026260815453,
  author       = {Pith},
  title        = {Pith review of: Generalization for Poincar\'e--Sobolev inequalities with local weights},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XV2D7WW5}},
  note         = {Machine review of arXiv:2608.15453}
}
read the original abstract

It is well known the local integral inequality which is called Poincar\'e--Sobolev inequality on each domain cube (or ball). After that many authors investigated the generalization for this inequality with Muckenhoupt-type weights or global weights which are independent of the domain cube. In this paper, we investigated similar weighted generalization for this inequality with the local weights which are depending on the domain cube without assuming the Muckenhoupt condition. Moreover, we considered the weighted generalization for the homogeneous and inhomogeneous-type Sobolev embedding theorems as some applications.

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