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Adams-Trudinger-Moser inequalities of Adimurthi-Druet type regulated by the vanishing phenomenon and its extremals

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Pith's one-line read This paper proves a sharp Adams–Trudinger–Moser inequality of Adimurthi–Druet type on the whole space R^n for every order m < n, and shows that in the critical case n = 4, m = 2 the supremum is attained for small vanishing parameters.

desk verdict Genuinely new sharp higher-order Adams–Adimurthi–Druet inequality with a clean proof, but the critical attainability proof rests on a false local estimate and needs major repair. read the letter →

arxiv 2505.24297 v1 pith:YJKRPFBG submitted 2025-05-30 math.AP math.FA

Abiel Costa Macedo , José Francisco de Oliveira , Fábio Sodré Rocha This is my paper
classification math.APmath.FA MSC 35J6035B3335J9135J3031A3026D10
keywords Trudinger-MoserinequalitiesAdamsPolyharmonicequationsHardyAdimurthi-Druettypeextremalfunctionsvanishingphenomenonblow-upanalysis
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 sharp Adams–Trudinger–Moser inequality of Adimurthi–Druet type on the whole space R^n for all higher orders 1 ≤ m < n, with a modified exponent that grows when the $L^{{n/m}}$-norm of the function dominates the norm of its m-th gradient—the 'vanishing phenomenon.' The factor ((1 + α||u||_q^q)/(1 − γα||u||_q^q))^{m/(n−m)} can become arbitrarily large along vanishing sequences, so the inequality bundles a genuine loss of compactness into the constant. The paper shows the Adams critical constant β0 is sharp, and that the supremum is attained in the subcritical range for n = 2m (for a non-empty interval of β) and, in the critical case β0, for n = 4, m = 2 and small α, γ. This supplies the first m ≥ 2 Adimurthi–Druet type inequality on an unbounded domain and the first critical extremal result for it.

What carries the argument

The central object is the vanishing-regulated exponent factor ζ(u) = ((1 + α||u||_{n/m}^{n/m})/(1 − γα||u||_{n/m}^{n/m}))^{m/(n−m)}, which multiplies β0|u|^{n/(n−m)} inside the exponential; along sequences with ||∇^m u_j||_{n/m} → 0 it approaches β0(1 + α)/(1 − γα), which can be arbitrarily large. The proof of Theorem 1.1 uses a two-step scaling (defining v and w with τ = 1 − α, μ = τ − γα) that converts the vanishing factor into the standard Adams normalization. For the extremal problems the machinery is the concentration-compactness alternative for radial sequences, a blow-up analysis with the biharmonic truncation of DelaTorre–Mancini, and a comparison in n = 4, m = 2 between the Adams functional on the Green function G of Δ² + κ0 (κ0 = 1 − α(γ + 1)) and the test function built from the explicit bubble z(x) = −1/(16π²) ln(1 + (π/√6)|x|²). The constant K0, the value at the origin of the regular part of G, marks the threshold that the test function must exceed.

What would settle it

Compute whether the limsup in Lemma 6.25 is still bounded by |B_R| $e^{{-1/3}}$/3 for a sequence u_j ∈ $H_0^{2}$(B_R) with ||Δu_j||_2 = 1, u_j ⇀ 0, and the exponent β_j ζ̃_j $u_j^{2}$, where ζ̃_j = (1 + α||u_j||$_2^{2}$)/(1 − γα||u_j||$_2^{2}$). If any sequence yields a larger limsup, the comparison with the test function value in Section 7 breaks and Theorem 1.4 would not follow from the present argument.

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Extended reading notes

Core claim

Theorem 1.1 fixes the space $W^{{m,n/m}}$(R^n) and proves that the supremum of the integral of Φ(β0 ((1 + α||u||_{n/m}^{n/m})/(1 − γα||u||_{n/m}^{n/m}))^{m/(n−m)} |u|^{n/(n−m)}) over the unit ball of ||∇^m u||_{n/m}^{n/m} + ||u||_{n/m}^{n/m} is finite for every 0 ≤ α < 1 and 0 < γ < 1/α − 1, and that β0 is the largest constant with this property. The proof scales the normalization to absorb the factors (1 − α||u||^q) and (1 − γα||u||^q), reducing the claimed inequality to the known sharp Adams inequality on R^n. The paper then studies the extremal problem: for n = 2m it proves attainability of the subcritical supremum for β in a range above a threshold involving the Gagliardo–Nirenberg constant B_GN, non-attainability for β sufficiently small, and in the critical case n = 4, m = 2 it proves that AD(4,2,β0,α,γ) is attained for small α and γ, using blow-up analysis, biharmonic truncations, and a test-function comparison against the Green function of Δ² + κ0.

Load-bearing premise

The critical attainability proof assumes that the Lu–Yang upper bound on concentrating $H_0^{2}$(B_R) sequences remains valid with the same constant after the exponent β_j $u_j^{2}$ is replaced by β_j ζ̃_j $u_j^{2}$, a step that is cited rather than demonstrated.

Editorial extensions

If this is right

  • A sharp higher-order Adams–Adimurthi–Druet inequality holds on the entire space R^n for every 1 ≤ m < n, with the same critical constant β0 as the classical Adams inequality and no loss at the critical exponent.
  • In even dimensions n = 2m, the subcritical supremum AD(2m, m, β, α, γ) is attained for α, γ in the stated range and β close to β0, and is not attained for small β, so the extremal landscape has a threshold behaviour.
  • In the critical case n = 4, m = 2, β = β0, the supremum is attained for all sufficiently small α and γ, giving the first critical extremal result for an m ≥ 2 Adimurthi–Druet type inequality.
  • The inequality is governed by the vanishing phenomenon: along normalized sequences with vanishing gradient part the effective exponent grows, and the full Sobolev normalization in Theorem 1.1 is exactly what restores finiteness at the critical constant.

Reading between the lines

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

  • The scaling reduction in Theorem 1.1 suggests a family of interpolated Adimurthi–Druet inequalities obtained by replacing the vanishing factor's ||u||^q with ||u||_r^q for other r, provided the known Adams inequality holds in the interpolated norm; the sharp constant and the extremal structure would presumably change accordingly.
  • The test-function comparison in Section 7, which for α → 0 reduces to the Chen–Lu–Zhu constant, suggests that the critical attainability set in (α, γ) might be much larger than the small-α interval stated in Theorem 1.4; one could numerically explore the sign of the difference in (7.1) for larger α.
  • The non-attainment result of Theorem 1.3 points to a phase transition in β: there should exist a threshold β*(α, γ) below which no maximizer exists and above which it does; locating this threshold would be a natural next problem.
  • Because Lemma 6.25 cites rather than derives the transfer of the Lu–Yang bound to the vanishing-regulated exponent, the critical attainability theorem is more fragile than the inequalities in Theorem 1.1; a direct proof of that transfer would settle Theorem 1.4 on solid ground.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proves an Adams-Trudinger-Moser inequality of Adimurthi-Druet type on the whole space R^n: for 1≤m<n, parameters α,γ as stated, and the critical Adams constant β0, the supremum of ∫ Φ(β0((1+α||u||^q)/(1-γα||u||^q))^{m/(n-m)} |u|^{n/(n-m)}) dx over the full Sobolev ball is finite, and β0 is claimed to be sharp. It then studies extremals: for n=2m it proves attainability for β in a subcritical interval (Theorem 1.2) and non-attainability for small β (Theorem 1.3), and for n=4,m=2 it claims attainability at β=β0 for small α,γ (Theorem 1.4). The arguments combine a scaling reduction of the finite-part inequality to known Adams bounds, a concentration-compactness-vanishing analysis, and a blow-up/truncation argument in the critical case.

Significance. If correct, Theorem 1.1 would give the first higher-order Adimurthi-Druet type sharp inequality on the whole space, and Theorem 1.4 would give a first critical extremal result for that inequality; the subcritical Theorems 1.2 and 1.3 would also be new. The scaling proof of the finite part of Theorem 1.1 is clean and self-contained modulo the external sharp Adams bound of Fontana-Morpurgo, and the paper gives credit where earlier machinery is imported (DelaTorre-Mancini, Chen-Lu-Zhu, Lu-Yang). The manuscript is ambitious and the overall strategy is coherent. However, the critical part contains a load-bearing estimate whose proof is not merely incomplete but inconsistent with the cited inequality, and the advertised sharpness of β0 is never proved.

major comments (3)
  1. [§6.5, Lemma 6.25] The transfer from the cited Lu-Yang estimate is not valid, and the displayed bound is inconsistent with the cited inequality. With A0=(8π²)^{-1}ln R − (16π²)^{-1}, the intermediate expression attributed to [42, Ineq. (5.23)] equals π²/6 e^{5/3} + 4 ln R − 2, while the claimed |B_R| e^{-1/3}/3 equals π² R^4 e^{-1/3}/6; these are not equal, and for small R the former even becomes negative although it is asserted as a limsup of nonnegative integrals. The factor ζ̃_j→1 also occurs at an uncontrolled rate, so replacing β_j by β_j ζ̃_j is not covered by the cited inequality. Since Lemma 6.26 uses Lemma 6.25 to exclude concentration, and Section 7 constructs test functions whose value is strictly above π²/6 e^{5/3}+32π²K0, the proof of Theorem 1.4 depends entirely on this unsupported upper bound. A correct proof or a corrected citation of the concentration estimate is needed before the critical attainability claim can be assessed.
  2. [§1.4 and §3.1] The sharpness of β0 in Theorem 1.1 is asserted but never proved. The proof in §3.1 establishes only finiteness for β=β0, stopping after the reduction to the known Adams bound (3.2). No sequence (u_j) with ||∇^m u_j||^q + ||u_j||^q ≤ 1 is constructed for which the integral diverges when β>β0. Sharpness for the standard Adams inequality does not automatically transfer to the modified coefficient ((1+α||u||^q)/(1-γα||u||^q))^{m/(n-m)} with the full Sobolev normalization, so a separate argument is required; as written, the central claim of sharpness is unsupported.
  3. [§4, Proposition 4.3] Formula (4.10) and the expansion (4.14) use (n/m−1)! and t^{n/m} for general n>m, but these expressions are undefined or are not the correct truncation terms when n/m is not an integer. For non-integer q=n/m, the lowest surviving term in Φ is t^{j_{m,n}−1}/(j_{m,n}−1)! with j_{m,n}=⌈q⌉, not t^{q−1}/(q−1)!. The proposition should be restricted to n/m∈N (or at least the proof must be reworked for non-integer ratios); as stated, it overclaims beyond what the expansion proves.
minor comments (4)
  1. [§2, equation (2.1)] The functional is written for u∈W^{m,2}(R^n), which is inconsistent with the spaces W^{m,n/m}(R^n) used throughout the paper; this should be corrected.
  2. [§6.2, Lemma 6.3] In the statement and proof, 'u_j→0 in L^2(R^2)' appears; the ambient dimension is 4, so this should read L^2(R^4).
  3. [§4.1, Lemma 4.1] The proof uses the notation ũ_j before defining it clearly, and the displayed estimate containing 'e^{C(n,m,R) 1/R ζ_j ||u_j||_{n/m}}' is garbled; the argument needs to be rewritten with exact definitions and exponents.
  4. [General presentation] Several displayed formulas are misnumbered or repeated (for instance, the displayed '(1.4)' in the introduction appears with different content), and some inline formulas in the statements of Theorems 1.2 and 1.4 are typeset in a compressed way that makes the hypotheses hard to read; these should be cleaned up.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central inequalities and extremal results are derived from external sharp estimates, and the few self-citations are not load-bearing.

full rationale

The paper's main inequality (Theorem 1.1) is proved by an explicit scaling reduction to the known finite sharp constant K_{1/\mu} from Fontana-Morpurgo, displayed in equations (3.1)-(3.3); no fitted parameter is later relabeled as a prediction. The subcritical extremal results (Theorems 1.2-1.3) use Lemma 4.2 and Proposition 4.3, which are internal derivations from the Adams/Tarsi estimates and from the Gagliardo-Nirenberg constant estimate (2.5), not from the conclusions they establish. The critical attainability result (Theorem 1.4) rests on the DelaTorre-Mancini truncation argument, Chen-Lu-Zhu ideas, and the Lu-Yang concentration estimate [42, Ineq. (5.23)]; these are external works by different authors and are invoked as mathematical facts, so the chain is not a self-citation loop. The paper does cite works by members of the present team, namely [12] and [13], but those are used only for a Taylor expansion of a Gamma ratio and as a literature reference to concentration levels; neither carries an unverified load-bearing premise. The proof of Lemma 6.25 is very terse and the asserted arithmetic is not transparent, so there is a potential correctness risk in transferring the Lu-Yang estimate to the modified exponent \beta_j\tilde{\zeta}_j; however, an unsupported or even erroneous citation transfer is not circularity: the target bound is not identical by construction to its input. No step defines a quantity in terms of the very result it is supposed to yield, and no fitted number is renamed as a prediction. Score 1 rather than 0 only because the paper contains repeated self-citations for background and technical lemmas; none of them is load-bearing.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims rest on several imported sharp inequalities rather than on fitted parameters. No numerical constants are fit to data. The most load-bearing inputs are the weighted Adams inequality, the Fourier rearrangement reduction, Tarsi's bounded-domain Adams bound, and Lu-Yang's local concentration upper bound; these are prior results, not free parameters.

assumptions (6)
  • domain assumption Weighted Adams-type inequality: sup over ||grad^m u||^q + tau||u||^q <= 1 of Integral Phi(beta0 |u|^{n/(n-m)}) is finite, as in Fontana-Morpurgo [20, Theorem 1-(b)].
    Used in Eq (3.1) to start the scaling chain for Theorem 1.1; if the weighted form failed, (1.13) would not follow.
  • domain assumption Fourier rearrangement of Lenzmann-Sok [34] permits the reduction H to H_rad in (2.2).
    Invoked in Section 2 to take maximizing sequences radial; needed in Theorems 1.2 and 1.4.
  • domain assumption Tarsi's Adams inequality for W^{m,n/m}_N(B_R) holds with sharp constant beta0, as in Theorem 4 of [58].
    Used in Lemma 4.1 and Lemma 6.3 to control local integrals; a quantitative failure would break the vanishing and concentration estimates.
  • ad hoc to paper Sharp local concentration estimate of Lu-Yang [42, Ineq. (5.23)] transfers to e^{beta_j zeta_tilde_j u_j^2} with bound |B_R| e^{-1/3}/3, as stated in Lemma 6.25.
    This is the upper threshold that Section 7's test function must beat; the paper does not derive the transfer. It is the most fragile imported premise.
  • standard math Standard elliptic regularity, Pizzetti's formula, and Martinazzi's classification of biharmonic entire solutions are applied as black boxes, as in Lemmas 6.7 through 6.10, 6.12 and 6.13.
    Used throughout Section 6 to justify C^3 convergence and Green function asymptotics; these are standard in the field.
  • ad hoc to paper The degenerate case 1 - mu_j ||u_j||_2^2 = 0 in Eq (6.12) does not occur for large j.
    This is asserted between (6.11) and (6.12) with proof deferred to the convergence ||u_j||_2 to 0 that Lemma 6.3 itself needs; without it lambda_j is undefined.

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Pith. "Pith review of Adams-Trudinger-Moser inequalities of Adimurthi-Druet type regulated by the vanishing phenomenon and its extremals." pith.science (2026). https://pith.science/paper/YJKRPFBG

@misc{pith2026250524297,
  author       = {Pith},
  title        = {Pith review of: Adams-Trudinger-Moser inequalities of Adimurthi-Druet type regulated by the vanishing phenomenon and its extremals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YJKRPFBG}},
  note         = {Machine review of arXiv:2505.24297}
}
abstract

Let $W^{m,\frac{n}{m}}(\mathbb{R}^n)$ with $1\le m < n$ be the standard higher order derivative Sobolev space in the critical exponential growth threshold. We investigate a new Adams-Adimurthi-Druet type inequality on the whole space $\mathbb{R}^n$ which is strongly influenced by the vanishing phenomenon. Specifically, we prove \begin{equation}\nonumber \sup_{\underset{\|\nabla^{m} u\|_{\frac{n}{m}}^{^{\frac{n}{m}}}+\|u\|_{\frac{n}{m}}^{\frac{n}{m}} \leq 1}{u\in W^{m,\frac{n}{m}}(\mathbb{R}^n)}} \int_{\mathbb{R}^n}\Phi\left(\beta \left(\frac{1+\alpha\|u\|_{\frac{n}{m}}^{\frac{n}{m}}}{1-\gamma\alpha\|u\|_{\frac{n}{m}}^{\frac{n}{m}}}\right)^{\frac{m}{n-m}}|u|^{\frac{n}{n-m}}\right) \mathrm{d}x<+\infty. \end{equation} where $0\le \alpha<1$, $0<\gamma<\frac{1}{\alpha}-1$ for $\alpha>0$, $\nabla^{m} u$ is the $m$-th order gradient for $u$, $0\le\beta\le \beta_0$, with $\beta_0$ being the Adams critical constant, and $\Phi(t) = \operatorname{e}^{t}-\sum_{j=0}^{j_{m,n}-2}\frac{t^{j}}{j!}$ with $j_{m,n}=\min\{j\in\mathbb{N}\;:\: j\ge n/m\}$. In addition, we prove that the constant $\beta_0$ is sharp. In the subcritical case $\beta<\beta_0$, the existence and non-existence of extremal function are investigated for $n=2m$ and attainability is proven for $n=4$ and $m=2$ in the critical case $\beta=\beta_0$. Our method to analyze the extremal problem is based on blow-up analysis, a truncation argument recently introduced by DelaTorre-Mancini \cite{DelaTorre} and some ideas by Chen-Lu-Zhu \cite{luluzhu20}, who studied the critical Adams inequality in $\mathbb{R}^4$.

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Pith tools

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