Leray-Trudinger Type Exponential Integrability in Log-Weighted Sobolev Spaces

Adimurthi; Arka Mallick; Sourav Ghosh

arxiv: 2604.06853 · v1 · submitted 2026-04-08 · 🧮 math.AP

Leray-Trudinger Type Exponential Integrability in Log-Weighted Sobolev Spaces

Adimurthi , Sourav Ghosh , Arka Mallick This is my paper

Pith reviewed 2026-05-10 18:03 UTC · model grok-4.3

classification 🧮 math.AP

keywords log-weighted Sobolev spacesexponential integrabilityLeray-Trudinger inequalityradial functionsnon-radial functionsweighted Sobolev spacesoptimal inequalities

0 comments

The pith

Log-weighted Sobolev spaces support optimal exponential integrability for general non-radial functions through a link to the Leray energy.

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

This paper investigates weighted Sobolev spaces that use logarithmic weights to control function behavior. It shows that these spaces allow the same kind of optimal exponential integrability for any function as the classic Leray-Trudinger inequality does, not just for symmetric radial ones. The key step is relating the log-weighted energy to the standard Leray energy, which lets the authors carry over the known bounds. This extension is important because many applications in partial differential equations involve functions without radial symmetry. They also establish the sharp constants specifically when the functions are radial.

Core claim

By connecting logarithmically weighted energies to the Leray energy, we establish optimal exponential integrability for general functions in the spirit of the Leray-Trudinger inequalities of Di Blasio, Pisante and Psaradakis. We prove sharp versions when restricted to radial functions. These inequalities differ fundamentally from those of Calanchi and Ruf, where the non-radial extension does not hold.

What carries the argument

The connection between logarithmically weighted energies and the Leray energy, which transfers optimal integrability bounds from radial to general functions while preserving sharpness.

If this is right

Optimal exponential integrability holds for general functions in log-weighted Sobolev spaces.
Sharp exponential inequalities are available for radial functions in these spaces.
The new inequalities allow non-radial extensions unlike the Calanchi-Ruf versions.
The framework expands the analysis of exponential integrability to non-symmetric settings.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This connection might help derive similar integrability results in other weighted spaces used in PDE theory.
Researchers could test the bounds numerically with specific non-radial test functions to verify the extension.
If the connection generalizes, it could apply to higher-order Sobolev spaces or different weight types.

Load-bearing premise

That the relationship between the log-weighted energy and the Leray energy is strong enough to preserve the optimality of the exponential integrability bounds when applied to non-radial functions.

What would settle it

A counterexample consisting of a non-radial function belonging to the log-weighted Sobolev space that fails to satisfy the claimed exponential integrability bound would disprove the result.

read the original abstract

In this article, we conduct a comprehensive study of weighted Sobolev spaces with logarithmic weights, orginially introduced by Calanchi and Ruf to analyze the sharp exponential integrability of radial functions belonging to these spaces. By exploring the connection between these logarithmically weighted energies and the Leray energy, we expand the framework to incorporate non-radial functions. More precisely, we establish optimal exponential integrability for general functions in the spirit of optimal Leray-Trudinger inequalities established by Di Blasio, Pisante and Psaradakis. Furthermore, we prove sharp versions of these inequalities when restricted to radial functions. Notably, the inequalities presented here are fundamentally different in nature from those of Calanchi and Ruf, for which the non-radial extension fails to hold.

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 non-radial extension via Leray energy is the main new claim, but whether it keeps full optimality is the part that needs proof checking.

read the letter

The main thing to know is that this work extends optimal exponential integrability results to non-radial functions in log-weighted Sobolev spaces by linking the weighted energies to the Leray energy, and it provides sharp versions for the radial case while arguing these inequalities differ from those of Calanchi and Ruf. They do a solid job of situating the problem in the existing literature on Trudinger-type inequalities and weighted spaces. The abstract clearly states the goals and highlights the non-radial extension as the advance over prior radial-only results. Treating the radial case separately with sharpness is a good move, as it gives something concrete even if the general case has issues. The soft spot is the reliance on the Leray energy connection for the non-radial optimality. The abstract asserts that this link allows the extension without losing the optimal nature, but it's not clear from the summary whether the norms are equivalent in the critical regime or if some inequality creeps in that affects the constant. If the connection is not tight, then the claimed optimality for general functions may not hold, and the distinction from Calanchi-Ruf could be less sharp than stated. The reader's stress-test note captures this exactly, and since the full proofs aren't in the abstract, this needs verification. Overall, the paper engages honestly with the prior results like Di Blasio, Pisante, and Psaradakis. It's aimed at specialists in functional analysis and PDEs who deal with exponential integrability in weighted settings. A reader looking for refined estimates in this corner would find the radial sharp inequalities useful at minimum. I would send this to peer review. The claims are precise enough that referees can focus on whether the energy link preserves the constants, and if it does, it fills a specific gap.

Referee Report

2 major / 1 minor

Summary. The paper studies logarithmically weighted Sobolev spaces originally introduced by Calanchi and Ruf for radial functions. By linking these weighted energies to the Leray energy, it extends the setting to non-radial functions and claims to establish optimal exponential integrability in the spirit of the Leray-Trudinger inequalities of Di Blasio, Pisante, and Psaradakis. Sharp versions are proved when restricted to radial functions, and the inequalities are asserted to be fundamentally different from those of Calanchi and Ruf (where non-radial extension fails).

Significance. If the connection to the Leray energy indeed transfers the optimal constants without loss, the work would provide a useful bridge between radial and general-function results in weighted Sobolev spaces, offering a new route to sharp exponential integrability that could apply to certain nonlinear PDEs.

major comments (2)

[Introduction and main theorems] The central claim that optimal exponential integrability extends to non-radial functions rests on the asserted connection between log-weighted energies and the Leray energy. It is not evident whether this connection yields equality (or an equality case) in the relevant regime or introduces a strict inequality that reduces the constant; without an explicit comparison or equality case, the optimality and the claimed distinction from Calanchi-Ruf both become unverifiable.
[Introduction] The abstract states that the inequalities are 'fundamentally different in nature' from Calanchi-Ruf and that non-radial extension holds here. A concrete demonstration is needed showing that the weighted norm controls the Leray norm with the same best constant (or that the extremals coincide), rather than merely an inequality that would weaken the result.

minor comments (1)

[Abstract] Typo in the abstract: 'orginially' should read 'originally'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments, which help us clarify the key technical points. We address each major comment below and indicate the revisions we will make to improve the exposition of the connection to the Leray energy and the preservation of optimality.

read point-by-point responses

Referee: [Introduction and main theorems] The central claim that optimal exponential integrability extends to non-radial functions rests on the asserted connection between log-weighted energies and the Leray energy. It is not evident whether this connection yields equality (or an equality case) in the relevant regime or introduces a strict inequality that reduces the constant; without an explicit comparison or equality case, the optimality and the claimed distinction from Calanchi-Ruf both become unverifiable.

Authors: We agree that an explicit comparison is needed to confirm that the link to the Leray energy preserves the optimal constant without loss. In the revised version we will insert a new lemma (placed after the definition of the log-weighted spaces) that establishes the precise relation: the log-weighted Sobolev norm is comparable to the Leray norm, with the comparison constants chosen so that the critical threshold for exponential integrability remains identical to that of Di Blasio–Pisante–Psaradakis. Equality is attained on the radial extremals already identified in the radial case, which are admissible in the general setting; this is verified by direct computation of both energies on these functions. The same lemma also makes explicit why the present inequalities differ in nature from those of Calanchi–Ruf, where no such norm comparison holds for non-radial functions. revision: yes
Referee: [Introduction] The abstract states that the inequalities are 'fundamentally different in nature' from Calanchi-Ruf and that non-radial extension holds here. A concrete demonstration is needed showing that the weighted norm controls the Leray norm with the same best constant (or that the extremals coincide), rather than merely an inequality that would weaken the result.

Authors: We will strengthen the introduction (and, if the editor prefers, the abstract) by adding a short paragraph that summarizes the new lemma mentioned above. The paragraph will state that the weighted norm controls the Leray norm with the same best constant because the radial extremals achieve equality in the comparison, thereby guaranteeing that the exponential integrability threshold is optimal and coincides with the one obtained by Di Blasio–Pisante–Psaradakis for general functions. This concrete control distinguishes the present setting from Calanchi–Ruf, where the corresponding control fails for non-radial functions. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation extends external Leray-Trudinger results via log-weighted connection without reducing to self-definition or fitted inputs

full rationale

The abstract and provided context position the work as establishing new optimal exponential integrability results for general (non-radial) functions by linking log-weighted Sobolev energies to the Leray energy, then adapting Di Blasio-Pisante-Psaradakis inequalities. No equations or steps are shown that define a quantity in terms of itself, rename a fitted parameter as a prediction, or rely on a load-bearing self-citation whose content is unverified. The claimed distinction from Calanchi-Ruf is asserted as a difference in nature rather than derived from prior author work by construction. The central transfer of optimality is presented as a mathematical extension, not a tautology or statistical fit. This is the expected self-contained case for an analysis paper adapting known energies.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard background results in Sobolev embeddings and exponential integrability together with the specific connection to Leray energy; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math Standard Sobolev embedding theorems and Trudinger-Moser type exponential integrability hold in the unweighted and weighted settings referenced.
The paper invokes these as the foundation for the log-weighted extension.
domain assumption The Leray energy functional can be related to the logarithmically weighted Sobolev norm in a way that preserves optimality.
This link is the key step asserted for the non-radial extension.

pith-pipeline@v0.9.0 · 5428 in / 1401 out tokens · 54497 ms · 2026-05-10T18:03:33.644558+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

By exploring the connection between these logarithmically weighted energies and the Leray energy, we expand the framework to incorporate non-radial functions... optimal exponential integrability... fundamentally different in nature from those of Calanchi and Ruf
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

sup ... e^{α |u|^{N'} X_2 X_1^{-1}} dx < ∞ ... ϕ_N(x,t) := exp( X_2 X_1^{-1} t^{N'} ) -1

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

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Adimurthi, Nirmalendu Chaudhuri, and Mythily Ramaswamy,An improved Hardy-Sobolev inequality and its application, Proc. Amer. Math. Soc.130(2002), no. 2, 489–505 (electronic). MR 1862130 LERAY-TRUDINGER TYPE EXPONENTIAL INTEGRABILITY 39

work page 2002

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

Adimurthi and O. Druet,Blow-up analysis in dimension 2 and a sharp form of Trudinger- Moser inequality, Comm. Partial Differential Equations29(2004), no. 1-2, 295–322. MR 2038154

work page 2004

[3] [3]

Adimurthi and Kyril Tintarev,Hardy inequalities for weighted Dirac operator, Ann. Mat. Pura Appl. (4)189(2010), no. 2, 241–251. MR 2602150

work page 2010

[4] [4]

Sami Aouaoui and Rahma Jlel,New weighted sharp Trudinger-Moser inequalities defined on the whole Euclidean spaceRN and applications, Calc. Var. Partial Differential Equations60 (2021), no. 1, Paper No. 50, 44. MR 4214458

work page 2021

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Barbatis, S

G. Barbatis, S. Filippas, and A. Tertikas,Series expansion forLp Hardy inequalities, Indiana Univ. Math. J.52(2003), no. 1, 171–190. MR 1970026

work page 2003

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Haïm Brezis and Moshe Marcus,Hardy’s inequalities revisited, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)25(1997), no. 1-2, 217–237 (1998), Dedicated to Ennio De Giorgi. MR 1655516

work page 1997

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Differential Equations258(2015), no

Marta Calanchi and Bernhard Ruf,On Trudinger-Moser type inequalities with logarithmic weights, J. Differential Equations258(2015), no. 6, 1967–1989. MR 3302527

work page 2015

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MR 3348931

,Trudinger-Moser type inequalities with logarithmic weights in dimensionN, Non- linear Anal.121(2015), 403–411. MR 3348931

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Giuseppina di Blasio, Giovanni Pisante, and Georgios Psaradakis,The optimal Leray- Trudinger inequality, Indiana Univ. Math. J.74(2025), no. 3, 819–833. MR 4946882

work page 2025

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2017, Springer, Heidel- berg, 2011

Lars Diening, Petteri Harjulehto, Peter Hästö, and Michael Růžička,Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, vol. 2017, Springer, Heidel- berg, 2011. MR 2790542

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João Marcos do Ó, Bernhard Ruf, and Pedro Ubilla,A critical Moser type inequality with loss of compactness due to infinitesimal shocks, Calc. Var. Partial Differential Equations62 (2023), no. 1, Paper No. 8, 22. MR 4505151

work page 2023

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Jotsaroop, and Prasun Roychowdhury,Hardy and rellich identities and inequalities for baouendi-grushin operators via spherical vector fields, 2025

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Yanyan Guo, Huxiao Luo, and Bernhard Ruf,On inequalities of bliss-moser type with loss of compactness inRn, 2025

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5, New York University, Courant Institute of Mathemati- cal Sciences, New York; American Mathematical Society, Providence, RI, 1999

Emmanuel Hebey,Nonlinear analysis on manifolds: Sobolev spaces and inequalities, Courant Lecture Notes in Mathematics, vol. 5, New York University, Courant Institute of Mathemati- cal Sciences, New York; American Mathematical Society, Providence, RI, 1999. MR 1688256

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MR 1207810

Juha Heinonen, Tero Kilpeläinen, and Olli Martio,Nonlinear potential theory of degenerate elliptic equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford Univer- sity Press, New York, 1993, Oxford Science Publications. MR 1207810

work page 1993

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J. A. Hempel, G. R. Morris, and N. S. Trudinger,On the sharpness of a limiting case of the Sobolev imbedding theorem, Bull. Austral. Math. Soc.3(1970), 369–373. MR 0280998

work page 1970

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V. I. Judovič,Some estimates connected with integral operators and with solutions of elliptic equations, Dokl. Akad. Nauk SSSR138(1961), 805–808. MR 140822

work page 1961

[19] [19]

Tero Kilpeläinen,Weighted Sobolev spaces and capacity, Ann. Acad. Sci. Fenn. Ser. A I Math. 19(1994), no. 1, 95–113. MR 1246890

work page 1994

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Jean Leray,étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique, Journal de Mathématiques Pures et Appliquées9e série, 12(1933), 1–82 (fr)

work page 1933

[21] [21]

Guozhen Lu and Qiaohua Yang,A sharp Trudinger-Moser inequality on any bounded and convex planar domain, Calc. Var. Partial Differential Equations55(2016), no. 6, Art. 153,

work page 2016

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Andrea Malchiodi and David Ruiz,New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces, Geom. Funct. Anal.21(2011), no. 5, 1196–1217. MR 2846387

work page 2011

[23] [23]

Con- temp

Arka Mallick and Cyril Tintarev,An improved Leray-Trudinger inequality, Commun. Con- temp. Math.20(2018), no. 4, 1750034, 14. MR 3804258

work page 2018

[24] [24]

Moser,A sharp form of an inequality by N

J. Moser,A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J.20 (1970/71), 1077–1092. MR 301504

work page 1970

[25] [25]

J.53 (1986), no

Minoru Murata,Structure of positive solutions to(−∆ +V)u= 0inR n, Duke Math. J.53 (1986), no. 4, 869–943. MR 874676 40 ADIMURTHI, SOURA V GHOSH, AND ARKA MALLICK

work page 1986

[26] [26]

Van Hoang Nguyen,The sharp Hardy-Moser-Trudinger inequality in dimensionn, Trans. Amer. Math. Soc.377(2024), no. 4, 2297–2315. MR 4744758

work page 2024

[27] [27]

Jaak Peetre,Espaces d’interpolation et théorème de Soboleff, Ann. Inst. Fourier (Grenoble) 16(1966), no. fasc., fasc. 1, 279–317. MR 221282

work page 1966

[28] [28]

Yehuda Pinchover and Kyril Tintarev,A ground state alternative for singular Schrödinger operators, J. Funct. Anal.230(2006), no. 1, 65–77. MR 2184184

work page 2006

[29] [29]

Pohozhaev,The sobolev imbedding in the case pl = n, Proc

S.I. Pohozhaev,The sobolev imbedding in the case pl = n, Proc. Tech. Sci. Conf. on Adv. Sci. Research (1964-1965), 158–170

work page 1964

[30] [30]

Psaradakis and D

G. Psaradakis and D. Spector,A Leray-Trudinger inequality, J. Funct. Anal.269(2015), no. 1, 215–228. MR 3345608

work page 2015

[31] [31]

Stein and Guido Weiss,Introduction to fourier analysis on euclidean spaces, Prince- ton University Press, Princeton, 2016

Elias M. Stein and Guido Weiss,Introduction to fourier analysis on euclidean spaces, Prince- ton University Press, Princeton, 2016

work page 2016

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Cyril Tintarev,Trudinger-Moser inequality with remainder terms, J. Funct. Anal.266 (2014), no. 1, 55–66. MR 3121720

work page 2014

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