Asymptotically sharp embedding of A_infty into A_p for flat weights and applications to Poincar\'e-Sobolev inequalities

Alejandro Claros, Ezequiel Rela

Pith reviewed 2026-05-07 13:59 UTC · model grok-4.3

classification 🧮 math.CA

keywords A_∞ weightsA_p weightsFujii-Wilson constantMuckenhoupt classesPoincaré-Sobolev inequalitiesBMO normsflat weightsweighted inequalities

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The pith

A_∞ weights with Fujii-Wilson constant near 1 embed into A_p for p near 1, yielding weighted Poincaré-Sobolev inequalities with classical exponents.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes quantitative embeddings showing that Muckenhoupt A_∞ weights with small Fujii-Wilson constant belong to A_p classes for parameters p approaching 1. Intermediate quantitative bounds on the BMO norms of log w are derived for these nearly constant weights. These estimates enable precise weighted Poincaré-Sobolev inequalities whose exponents recover the classical Sobolev value np/(n-p) in the limit as the weight constant approaches 1. The results matter for obtaining sharp constants in weighted inequalities that arise in harmonic analysis and partial differential equations.

Core claim

For weights w in A_∞ with [w]_{A_∞} close to 1, there is an explicit embedding into A_p with p depending on the distance to 1, together with bounds on the BMO norm of log w; this yields a quantitative weighted Poincaré-Sobolev inequality that recovers the unweighted classical exponent p* = np/(n-p) as [w]_{A_∞} approaches 1 from above.

What carries the argument

The Fujii-Wilson constant [w]_{A_∞} for flat weights, which quantifies near-constancy and controls the BMO norm of log w to produce the asymptotic embedding into A_p.

Load-bearing premise

The weights are required to be flat, with their A_∞ constant close to 1, so that the asymptotic sharpness and exponent recovery apply.

What would settle it

An explicit weight with [w]_{A_∞} = 1 + ε whose minimal p for membership in A_p fails to approach 1 at the predicted rate, or whose associated Sobolev exponent fails to approach np/(n-p).

read the original abstract

We provide new quantitative results on the embedding of the Muckenhoupt class $A_\infty$ into $A_p$ with the correct asymptotic behavior when the Fujii--Wilson constant $[w]_{A_\infty}$ is close to 1, namely that the parameter $p$ goes to 1 when the weight is nearly constant. As intermediate steps towards the result, we obtain quantitative estimates on the weighted and unweighted BMO norms of $\log w$ for an $A_\infty$ weight $w$. As a consequence, we show that a precise quantitative weighted Poincar\'e-Sobolev inequality can be proved for weights with small $[w]_{A_\infty}$ that recovers the classical Sobolev exponent $p^*=\frac{np}{n-p}$ when $[w]_{A_\infty}\to 1^+$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives explicit asymptotic control on embedding A_∞ weights into A_p when the Fujii-Wilson constant is near 1, plus BMO bounds on log w that feed into weighted Poincaré-Sobolev inequalities recovering the classical exponent.

read the letter

The main thing to know is that for weights with [w]_{A_∞} close to 1, the authors produce an embedding into A_p where p-1 is bounded by a multiple of [w]_{A_∞}-1, and this yields Poincaré-Sobolev inequalities whose exponent approaches the standard np/(n-p) as the weight flattens. They also supply quantitative estimates for both the weighted and unweighted BMO norms of log w derived directly from the Fujii-Wilson definition.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes quantitative embeddings of the Muckenhoupt class A_∞ into A_p for weights with Fujii-Wilson constant [w]_{A_∞} close to 1, showing that p can be taken arbitrarily close to 1 with explicit control in terms of [w]_{A_∞}-1. It derives quantitative bounds on both the weighted and unweighted BMO norms of log w directly from the A_∞ definition, and applies these to obtain a quantitative weighted Poincaré-Sobolev inequality whose exponent recovers the classical Sobolev exponent p^*=np/(n-p) in the flat limit as [w]_{A_∞}→1^+. The results are stated for flat weights and emphasize the asymptotic sharpness of the embedding.

Significance. If the derivations hold, the work supplies asymptotically sharp control on the A_p constant for nearly constant weights, which is valuable for perturbation arguments in weighted inequalities and PDEs. The direct derivation of the BMO estimates for log w from the Fujii-Wilson constant, without auxiliary parameters, and the clean recovery of the classical Sobolev exponent in the limit are notable strengths that enhance applicability.

minor comments (3)

[§2] §2, after the statement of the BMO estimates: the dependence of the implicit constants on the dimension n should be tracked explicitly throughout the proofs, as it affects the final constants in the Poincaré-Sobolev application.
[§3] §3, Theorem 3.2: the quantitative Poincaré-Sobolev statement would benefit from an explicit error term or modulus of continuity showing how the exponent approaches p^* as [w]_{A_∞}→1^+, to make the asymptotic recovery fully precise.
[Introduction] Introduction, paragraph 3: a brief comparison with prior quantitative A_∞→A_p results (e.g., those using reverse Hölder or other characterizations) would better highlight the novelty of the Fujii-Wilson-based approach.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript on the asymptotically sharp embedding of A_∞ into A_p for flat weights, the direct BMO estimates for log w, and the quantitative weighted Poincaré-Sobolev inequalities that recover the classical exponent in the flat limit. We appreciate the recognition of the work's value for perturbation arguments in weighted inequalities and PDEs. The recommendation for minor revision is noted.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from definitions

full rationale

The paper derives quantitative BMO estimates for log w directly from the Fujii-Wilson definition of [w]_{A_∞} and the standard definitions of A_p classes. These yield the embedding p-1 ≲ [w]_{A_∞}-1 for flat weights, and the weighted Poincaré-Sobolev inequality recovers the classical exponent in the limit [w]_{A_∞}→1^+ by direct substitution of the constants. No fitted parameters are renamed as predictions, no self-citations are load-bearing for the central claims, and no ansatz or uniqueness theorem is smuggled in. The chain from weight-class definitions to the asymptotic statements is independent and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard definitions and properties of Muckenhoupt A_p and A_∞ classes together with the BMO space; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)

standard math Standard properties of Muckenhoupt A_p weights, the Fujii-Wilson constant, and the BMO norm of log w
These are background facts from harmonic analysis invoked throughout the abstract.

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discussion (0)

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