Extremal functions for a singular Hardy-Moser-Trudinger inequality

Songbo Hou

REVIEW 2 major objections 5 minor 28 references

Extremal functions for a singular Hardy-Moser-Trudinger inequality

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that a singular Hardy-Moser-Trudinger inequality on the unit disk is sharp and has an extremal function.

desk verdict The theorem is a genuine extension of the singular Hardy-Moser-Trudinger line, the proof strategy is sound, and the main objection in the stress test comes from misreading Eq. (26), not from the paper. read the letter →

arxiv 1908.03982 v1 pith:5C7ICRGH submitted 2019-08-12 math.FA

Songbo Hou This is my paper · ORCID

classification math.FA MSC 46E35

keywords Hardy-Moser-TrudingerinequalitysingularMoser-Trudingerextremalfunctionblow-upanalysisunitdiskradialrearrangementGreenEuler-Lagrangeequation

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 exponential inequality with a singular weight on the unit disk: for $0\le\beta<1$ and $0\le\alpha<\lambda_1(B)$, the supremum of $\int_B e^{4\pi(1-\beta)u^2}|x|^{-2\beta}\,dx$ over the class $\|u\|_{H,\alpha}\le 1$ is finite, and the supremum is attained by a radial function. The paper matters because it combines two features that had been treated separately: the Hardy-type norm that improves the classical Moser-Trudinger bound, and the singularity $|x|^{-2\beta}$. It proves existence by showing that maximizing sequences for slightly subcritical exponents cannot concentrate: blow-up at the origin would force a limiting profile whose mass contradicts a carefully constructed test function. A sympathetic reading of the proof therefore establishes not only a best constant but a genuine extremal, which is the object needed for Euler-Lagrange and compactness arguments in geometric PDEs.

What carries the argument

The carrying object is the blow-up analysis of the subcritical maximizers $u_\varepsilon$. Two rescaled sequences are formed around the origin, $\psi_\varepsilon(x)=c_\varepsilon^{-1}u_\varepsilon(r_\varepsilon^{1/(1-\beta)}x)$ and $\phi_\varepsilon(x)=c_\varepsilon(u_\varepsilon(r_\varepsilon^{1/(1-\beta)}x)-c_\varepsilon)$, where $r_\varepsilon$ is the blow-up radius; the second sequence converges to a solution of the singular Liouville equation whose explicit profile (35) is fixed by a classification theorem. A second load-bearing object is the Green function of the operator $L_\alpha=-\Delta-(1-|x|^2)^{-2}-\alpha$ with pole at the origin, written $G=-(1/2\pi)\log r+A_0+\Phi$ with smooth $\Phi$; the constant $A_0$ controls the sharp upper bound and the test functions that exclude blow-up.

What would settle it

To check the central claim, substitute the printed definition (26) into the coefficient of the nonlinear term in equation (32): with $r_\varepsilon^2=\lambda_\varepsilon c_\varepsilon^{-1}e^{-2\pi(1-\beta-\varepsilon)c_\varepsilon^2}$, that coefficient is $\lambda_\varepsilon^{-1}c_\varepsilon^2 r_\varepsilon^2e^{4\pi(1-\beta-\varepsilon)c_\varepsilon^2}=c_\varepsilon e^{2\pi(1-\beta-\varepsilon)c_\varepsilon^2}$, which diverges rather than tending to 1. The corrected scale $r_\varepsilon^2=\lambda_\varepsilon c_\varepsilon^{-2}e^{-4\pi(1-\beta-\varepsilon)c_\varepsilon^2}$ makes the coefficient exactly 1, so a one-line limit calculation settles whether the claimed profile (34) and mass normalization (36) are valid.

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Extended reading notes

Core claim

The central claim is Theorem 1: for fixed $0\le\beta<1$ and $0\le\alpha<\lambda_1(B)$, the supremum $\sup_{\|u\|_{H,\alpha}\le1}\int_B |x|^{-2\beta}e^{4\pi(1-\beta)u^2}\,dx$ is finite, and there is a function $u_0\in H$ with $\|u_0\|_{H,\alpha}=1$ attaining it. The proof proceeds by solving subcritical problems: for each $\varepsilon>0$ there is a radial maximizer $u_\varepsilon$ with $\|u_\varepsilon\|_{H,\alpha}=1$, satisfying the Euler-Lagrange equation (22). If the maximum values $c_\varepsilon=u_\varepsilon(0)$ stay bounded, the weak limit is the desired $u_0$. The long part of the paper assumes $c_\varepsilon\to\infty$ and runs a blow-up analysis around the origin; the rescaled profiles converge to the explicit solution $\phi_0(x)=-\frac{1}{4\pi(1-\beta)}\log(1+\frac{\pi}{1-\beta}|x|^{2(1-\beta)})$ of the singular Liouville equation $-\Delta\phi_0=|x|^{-2\beta}e^{8\pi(1-\beta)\phi_0}$ on $\mathbb{R}^2\setminus\{0\}$, and the associated Green function constant $A_0$ enters an upper bound. A final test-function construction produces a value strictly larger than that upper bound, contradicting the assumption that blow-up occurs; hence $c_\varepsilon$ is bounded and the extremal exists.

Load-bearing premise

The argument stands on one choice of blow-up scale: the rescaled equations (30) and (32) only hold when $r_\varepsilon^2=\lambda_\varepsilon c_\varepsilon^{-2}e^{-4\pi(1-\beta-\varepsilon)c_\varepsilon^2}$, whereas equation (26) as printed uses $r_\varepsilon^2=\lambda_\varepsilon c_\varepsilon^{-1}e^{-2\pi(1-\beta-\varepsilon)c_\varepsilon^2}$; with the printed choice the limit equation and the mass identity do not follow.

Editorial extensions

If this is right

If the theorem is correct, the sharp constant in (13) is attained, so the variational problem has a true maximizer rather than a supremum that is only approached.
That maximizer is radial, belongs to $H$, has norm $\|u_0\|_{H,\alpha}=1$, and solves the Euler-Lagrange equation (22); it is smooth away from the origin and continuous on the closed disk.
Setting $\beta=0$ recovers the improved Hardy-Moser-Trudinger inequality with an extremal, and setting $\alpha=0$ recovers the singular Moser-Trudinger existence statement; the two earlier results become boundary cases of one theorem.
The proof yields the explicit upper bound $\frac{\pi}{1-\beta}(1+e^{1+4\pi(1-\beta)A_0})$ for the sharp constant, so the value of the best constant is tied to a single Green-function quantity.
The blow-up profile (35) describes the geometry of any concentrating maximizing sequence near the singularity, which quantifies why the critical exponent $4\pi(1-\beta)$ is the threshold.

Reading between the lines

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

One testable extension is the borderline case $\alpha=\lambda_1(B)$: the norm $\|u\|_{H,\alpha}$ degenerates there, and the paper's argument does not apply, so one can check whether the supremum stays finite or becomes infinite.
The proof is specific to the disk because of the hyperbolic rearrangement; on a general domain the Hardy term $(1-|x|^2)^{-2}$ would not be preserved by symmetrization, so a different argument would be needed.
The explicit limit profile suggests a quantization principle: any concentrating maximizing sequence carries exactly one unit of mass at the singularity, a statement the paper does not isolate but which follows from the mass identity (36).
A numerical experiment on the disk for small $\beta$ could test whether computed maximizers match the asymptotic picture—the bubble shape near the origin plus a Green-function tail away from it—giving a concrete check of the blow-up description.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the theorem is a real extension and the proof is very likely correct after minor polishing. I specifically checked the blow-up scaling because that is where the stress test pointed. Eq. (26) defines r_epsilon = sqrt(lambda) c^{-1} exp(-2 pi (1-beta-epsilon) c^2), so its square is lambda c^{-2} exp(-4 pi (1-beta-epsilon) c^2), which is exactly the coefficient needed to obtain the limit equation (34). The stress-test note squares the printed formula incorrectly. That concrete objection dissolves on reading.

What is genuinely new: this is the first paper to add the singular weight |x|^{-2 beta} to the Hardy-improved norm ||u||_{H,alpha} and prove both finiteness and existence of an extremal. It sits at the natural intersection of Yang-Zhu [26] and Yang-Zhu [27], and the open case is exactly the combination. The proof is the standard Moser-Trudinger blow-up package: hyperbolic symmetrization, subcritical maximizers, Chen-Li classification, the Iula-Mancini upper bound, and a test-function contradiction. No new machinery, but the extension is meaningful for the sharp Sobolev inequality community.

The soft spots are presentation, not load-bearing. The paper is not self-contained: several estimates are introduced with 'similar to' or 'refer to' [22] and [27], including the detailed calculations behind (49) and (52). A referee will need those papers open to check constants. There are also places where a line is missing: in Lemma 6, the claim lambda/c -> infinity is asserted from Lemma 5, but Lemma 5 as stated gives limsup lambda/c^2, so an extra mass-concentration argument is needed. In Lemma 9, one displayed formula has lambda c^2 where the following conclusion requires lambda/c^2; that is a typo. The passage in (27) from ||u||_{H,alpha}=1 to a subcritical bound normalized by ||u||_H is also not spelled out, though it is fixable using the uniform control from alpha < lambda_1(B). None of these look fatal; they read like condensed writing and small errors.

The citation pattern is appropriate: the four claimed predecessors are real and the combinations are not in them. Who is this for: specialists in sharp Sobolev inequalities and blow-up analysis. A general PDE reader can skip. It deserves a serious referee. I would accept it for peer review and ask a specialist to verify the delegated estimates; after a revision that fills the small gaps and removes the typos, the result should be publishable.

Referee Report

2 major / 5 minor

Summary. The paper proves a singular Hardy-Moser-Trudinger inequality on the unit disc with the Hardy weight (1-|x|^2)^{-2} built into the norm: for 0≤β<1 and 0≤α<λ1(B), the supremum of ∫_B |x|^{-2β} e^{4π(1-β)u^2} dx over the unit ball of the weighted norm ||u||_{H,α} is finite and attained. The proof follows the standard three-step scheme: reduction to radial functions and existence of subcritical maximizers; blow-up analysis of the maximizers; upper-bound estimates via the Iula-Mancini inequality and construction of competing test functions to rule out blow-up. The main novelty is the combination of the singular weight |x|^{-2β} with the Hardy norm, extending previous results of Wang-Ye, Yang-Zhu, and Csató-Roy.

Significance. If correct, the theorem is a natural and nontrivial extension of the Hardy-Moser-Trudinger theory: it establishes sharp finiteness and existence of extremals under a norm that is weaker than the Dirichlet norm but stronger than L^2 due to the Hardy term. The strategy is standard for this literature, and the paper makes appropriate use of external classification and compactness results (Chen-Li, Iula-Mancini, Wang-Ye). The result would be a useful addition to the extremal-function literature for Trudinger-Moser inequalities. However, the printed proof contains a load-bearing scaling inconsistency in the blow-up analysis that must be repaired before the claims are checkable.

major comments (2)

[Section 2.2, Eq. (26) vs Eqs. (30), (32), (34)] Equation (26) defines r_ε^2 = λ_ε c_ε^{-1} e^{-2π(1-β-ε)c_ε^2}. Substituting the blow-up coordinates (29) into the Euler-Lagrange equation (22) gives, for ψ_ε, a nonlinear coefficient λ_ε^{-1} r_ε^2 e^{4π(1-β-ε)c_ε^2 ψ_ε^2}; with (26) this equals c_ε^{-1} e^{4π(1-β-ε)c_ε^2(ψ_ε^2 - 1/2)}, not the c_ε^{-2} e^{4π(1-β-ε)(1+ψ_ε)φ_ε} that appears in (30). Similarly, the coefficient in (32) is λ_ε^{-1} c_ε^2 r_ε^2 e^{4π(1-β-ε)c_ε^2}, which equals 1 only if r_ε^2 = λ_ε c_ε^{-2} e^{-4π(1-β-ε)c_ε^2}. With the printed (26), the limit equation would not be (34), and the classification (35), the mass normalization (36), and all subsequent estimates in Lemmas 4-9 and Section 2.4 would not follow. The manuscript should correct (26) to the latter scaling and verify that (27)-(28) and the blow-up limits remain valid under that definition.
[Section 2.2, proof of Lemma 6] The sentence 'Lemma 5 yields λ_ε/c_ε → +∞, hence c_ε/λ_ε → 0' is not a consequence of Lemma 5, which only bounds limsup ∫ |x|^{-2β} e^{4π(1-β-ε)u_ε^2} dx by π/(1-β) + limsup λ_ε/c_ε^2. This estimate is used to control the term I_2, and Lemma 6 (convergence of f_ε to δ_0) is later applied in Lemma 9 to identify the constant term in E_2(ρ). The proof therefore needs a correct estimate for I_2 (for example, using the explicit bubble scaling and τ<1) or an alternative justification of f_ε ⇀ δ_0.

minor comments (5)

[Abstract and Introduction] There are several typos: 'Morser' should be 'Moser' in the abstract and in the first line of the introduction; 'u/nequivalence0' should read 'u≠0'; and reference [6] has 'Hardys inequality' instead of 'Hardy's inequality'.
[Section 2.1, Eq. (27)] The use of (16) to bound λ_ε requires normalizing by ||u_ε||_H, which is only known to tend to 1 after the proof that u_0≡0. The manuscript should state this normalization explicitly; as written the line 'by the Hölder inequality and (16)' skips a step.
[Section 2.2, Lemma 6] The notation 'o_ε(R)' in the estimates for I_1 and I_3 is ambiguous: it should mean a quantity that tends to 0 as ε→0 for fixed R, and then one lets R→∞. Please clarify the order of the limits.
[Section 2.4] The test-function construction reuses ε for the new small parameter after ε was used for the subcritical approximation; this is potentially confusing and should be renamed, for instance δ or η.
[Introduction, reference [13]] The text says 'We derive an upper bound ... by Onofri's inequality ([13], Theorem 1.1)', but reference [13] is Iula-Mancini's paper. If Onofri's inequality is being used, a separate citation should be given; if Iula-Mancini's result is meant, the wording should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof is built on independent external results, not on its own target inequality.

full rationale

The paper proves Theorem 1 by a standard three-step strategy: subcritical maximization, blow-up analysis, and a test-function contradiction. The load-bearing ingredients are imported from independent external works: Wang-Ye [22] supplies the radial compact embedding and the Hardy-Moser-Trudinger inequality; Adimurthi-Sandeep supplies the singular Moser-Trudinger inequality used in the subcritical boundedness estimate; Chen-Li [9] supplies the classification of the blow-up limit equation; and Iula-Mancini [13] supplies the Dirichlet-norm upper bound used in Lemma 9. None of these is a restatement of the target H-norm inequality, and none is fitted to the present theorem. The test-function construction in Section 2.4 uses the Green function representation and direct computation; it does not insert the desired supremum as an input and then recover it as a prediction. There is also no load-bearing self-citation: the author cites prior works by Yang-Zhu and others, but the cited results stand independently of this paper. The only conspicuous defect is internal rather than circular: as printed, Eq. (26) defines r_epsilon^2 = lambda_epsilon c_epsilon^{-1} e^{-2pi(1-beta-epsilon)c_epsilon^2}, whereas Eqs. (30) and (32) are consistent with the corrected scaling r_epsilon^2 = lambda_epsilon c_epsilon^{-2} e^{-4pi(1-beta-epsilon)c_epsilon^2}. Taking Eq. (26) literally would break the derivation of the limit profile (34), but this is a scaling/typo-level proof defect, not a case of the theorem being used to prove itself. Since no circular step can be exhibited with a quote and a specific reduction, the correct finding is no significant circularity.

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

The central claim rests on a chain of published external results: hyperbolic rearrangement, Wang-Ye embedding lemmas, Chen-Li classification, Iula-Mancini's upper bound, and the Green function expansion. No constants are fitted to data and no new entities are postulated. The only effective free choice in the proof is the blow-up scale, which is misstated in Eq. (26) and must be corrected.

assumptions (5)

standard math Hyperbolic rearrangement maps each u in C0∞(B) to a radial nonincreasing u* preserving ∫u²dμ and ∫u²/(1−|x|²)² dx, and not increasing ∫|∇u|².
Invoked in Lemma 3 before Eq. (15) to reduce the variational problem to the radial subspace S; cited to Baernstein [5].
standard math The Chen-Li classification theorem determines all solutions of −Δφ0=|x|^{−2β}e^{8π(1−β)φ0} in R²\{0} with finite mass; the solution is φ0(x)=−(1/(4π(1−β)))log(1+π/(1−β)|x|^{2(1−β)}).
Used in §2.2 after Eq. (34) to get Eqs. (35)-(36), which give the mass of the blow-up profile.
standard math Iula-Mancini's Lemma 8: for u_n in W0^{1,2}(B) with ||∇u_n||≤1 and u_n⇀0, limsup_n ∫ e^{4π(1−β)u_n²}/|x|^{2β} dx ≤ π(1+e)/(1−β).
Used in Lemma 9 to bound the rescaled positive part of u_ε near the origin; cited to [13].
standard math The Green function G of Lα=−Δ−(1−|x|²)^{-2}−α with pole at 0 satisfies LαG=δ0 and has expansion G=−(1/(2π))log r + A0 + Φ with Φ∈C1loc(B).
Assumed in Lemma 7 and used for the upper bound constant A0 and for the test functions in §2.4; the proof is sketched, not fully given.
standard math Yang-Zhu's Lemma 4 in [26]: for every γ>0 and every u in the radial space S, ∫_B e^{γu²} dx < ∞.
Used in Lemma 3 and Lemma 6 to bound the second factors in the Hölder estimates.

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Pith. "Pith review of Extremal functions for a singular Hardy-Moser-Trudinger inequality." pith.science (2026). https://pith.science/paper/5C7ICRGH

@misc{pith2026190803982,
  author       = {Pith},
  title        = {Pith review of: Extremal functions for a singular Hardy-Moser-Trudinger inequality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5C7ICRGH}},
  note         = {Machine review of arXiv:1908.03982}
}

read the original abstract

In this paper, using blow-up analysis, we prove a singular Hardy-Morser-Trudinger inequality, and find its extremal functions. Our results extend those of Wang-Ye (Adv. Math. 2012), Yang-Zhu ( Ann. Glob. Anal. Geom. 2016), Csat\'{o}- Roy (Calc. Var. 2015), and Yang-Zhu (J. Funct. Anal. 2017).

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [26]

Y ang, X

Y . Y ang, X. Zhu, An improved Hardy-Trudinger-Moser inequality, Ann. Glob. Anal. Geom. 49 (2016) 1-19

work page 2016

[2] [27]

Y ang, X

Y . Y ang, X. Zhu, Blow-up analysis concerning singular T rudinger-Moser inequalities in dimension two, J. Funct. Anal. 272 (2017) 3347-3374

work page 2017

[3] [22]

G. Wang, D. Y e, A Hardy-Moser-Trudinger inequality, Ad v. Math. 230 (2012) 294-320

work page 2012

[4] [1]

Druet, Blow-up analysis in dimension 2 and a sharp form of Trudinger-Moser inequality, Comm

Adimurthi, O. Druet, Blow-up analysis in dimension 2 and a sharp form of Trudinger-Moser inequality, Comm. Partial Diﬀerential Equations 29 (2004) 295-322

work page 2004

[5] [2]

Sandeep, A singular Moser-Trudinger embe dding and its applications, Nonlinear Di ﬀer

Adimurthi, K. Sandeep, A singular Moser-Trudinger embe dding and its applications, Nonlinear Di ﬀer. Equ. Appl. 13 (2007) 585-603

work page 2007

[6] [3]

Struwe, Global compactness properties of semilinear elliptic equation with critical exponential growth, J

Adimurthi, M. Struwe, Global compactness properties of semilinear elliptic equation with critical exponential growth, J. Functional Analysis 175 (2000) 125-167

work page 2000

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Y ang, An interpolation of Hardy inequalit y and Trudinger-Moser inequality in RN and its applica- tions, Int

Adimurthi, Y . Y ang, An interpolation of Hardy inequalit y and Trudinger-Moser inequality in RN and its applica- tions, Int. Math. Res. Notices 13 (2010) 2394-2426

work page 2010

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Baernstein, A uniﬁed approach to symmetrization, Par tial diﬀerential equations of elliptic type (Cortona, 1992), Sympos

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