A Trudinger-Moser inequality under a refined constraint in fractional dimensions and extremal functions

Jos\'e Francisco de Oliveira; Ruan Diego da Silva Paiva

arxiv: 2604.04676 · v1 · submitted 2026-04-06 · 🧮 math.AP · math.FA

A Trudinger-Moser inequality under a refined constraint in fractional dimensions and extremal functions

Ruan Diego da Silva Paiva , Jos\'e Francisco de Oliveira This is my paper

Pith reviewed 2026-05-10 19:58 UTC · model grok-4.3

classification 🧮 math.AP math.FA

keywords Trudinger-Moser inequalityTintarev constraintfractional dimensionsSobolev spacesextremal functionsmaximizerscritical regime

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The pith

A Trudinger-Moser inequality with Tintarev constraint holds and attains its supremum in fractional dimensions.

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

The paper establishes a Trudinger-Moser type inequality that incorporates a refined Tintarev-type constraint, set in fractional-dimensional spaces, and shows that the supremum is attained by extremal functions when the exponent reaches its critical value. This refines earlier versions of the inequality both for fractional dimensions and for standard Sobolev spaces. A reader would care because the inequality governs the exponential integrability of functions whose fractional Sobolev norm is controlled, which is the borderline case between Sobolev embeddings and Orlicz spaces. The existence of maximizers turns the inequality into a sharp tool that can be used directly in variational problems.

Core claim

We establish a Trudinger-Moser type inequality with a Tintarev-type constraint in fractional-dimensional spaces and prove the existence of maximizers in the critical regime. Our results provide a refinement of those in Calc. Var. 52 (2015), 125-163 in the setting of fractional-dimensional spaces, as well as of those in Ann. Global Anal. Geom. 54 (2018), 237-256 for classical Sobolev spaces.

What carries the argument

The Tintarev-type constraint, a refined restriction imposed on functions in the fractional Sobolev space that restores the compactness required for the variational problem to attain its supremum.

Load-bearing premise

The fractional-dimensional Sobolev spaces possess the embeddings and compactness properties needed under the refined constraint so that the supremum is attained.

What would settle it

A sequence of functions obeying the Tintarev constraint whose associated Trudinger-Moser functional values exceed the claimed bound, or a maximizing sequence that fails to converge strongly to any limit function.

read the original abstract

We establish a Trudinger-Moser type inequality with a Tintarev-type constraint in fractional-dimensional spaces and prove the existence of maximizers in the critical regime. Our results provide a refinement of those in (Calc. Var. 52 (2015), 125-163) in the setting of fractional-dimensional spaces, as well as of those in (Ann. Global Anal. Geom. 54 (2018), 237-256) for classical Sobolev spaces.

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 proves the Trudinger-Moser inequality under a Tintarev-type constraint in fractional dimensions and shows maximizers exist, using an adapted profile decomposition with no major gaps.

read the letter

The main thing here is that the authors establish a Trudinger-Moser type inequality with a Tintarev-type constraint in fractional-dimensional spaces and prove existence of maximizers in the critical regime. They adapt the profile decomposition to respect the constraint and recover compactness where it would otherwise be lost. The fractional Sobolev embeddings into the Orlicz space are handled directly, and the argument stays self-contained without circular reductions to the classical case. That part works cleanly and follows the logic of the setup. The paper does a competent job carrying the constraint over from the cited works while making the necessary adjustments for the fractional setting. The concentration-compactness tools are modified just enough to fit, and the existence result comes out as a consequence. On the soft side, this is an incremental refinement rather than a conceptual shift. The overall template is the standard variational one, so the advance is mostly in the specific combination of constraint and fractional dimensions. No new methods are invented, and the technical difficulties introduced by the fractional case are addressed but do not open broader directions. This is the sort of paper that specialists in Trudinger-Moser inequalities and fractional Sobolev spaces will want to see for the refined statement and the existence proof. A reader outside that niche would not get much from it. I would send it to a referee because the central claims hold up under the adapted tools and the execution looks careful enough to benefit from expert feedback.

Referee Report

0 major / 2 minor

Summary. The manuscript establishes a Trudinger-Moser type inequality incorporating a Tintarev-type constraint in fractional-dimensional spaces and proves the existence of maximizers in the critical regime. It refines earlier results from Calc. Var. 52 (2015) and Ann. Global Anal. Geom. 54 (2018) by adapting the inequality and compactness arguments to the fractional setting.

Significance. If the result holds, the work is significant for extending constrained Trudinger-Moser theory to fractional dimensions, where the refined constraint aids in obtaining compactness via an adapted profile decomposition. The self-contained proof using carefully modified concentration-compactness tools, without reduction to classical cases, strengthens the contribution and supports applications to fractional PDEs.

minor comments (2)

[Introduction] In the introduction, the precise range of the fractional dimension parameter could be stated explicitly when comparing to the cited works in Calc. Var. 52 (2015) and Ann. Global Anal. Geom. 54 (2018).
[Preliminaries] Notation for the Orlicz space and the refined constraint should be introduced with a dedicated preliminary subsection to improve readability for readers unfamiliar with the fractional setting.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation to accept it. We appreciate the recognition of the significance of extending the constrained Trudinger-Moser theory to fractional dimensions.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives the Trudinger-Moser inequality under the refined Tintarev-type constraint in fractional dimensions via a profile decomposition adapted to the constraint, combined with standard fractional Sobolev embeddings into Orlicz spaces and compactness arguments. These steps rely on modified concentration-compactness tools that are independent of the target result and do not reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations. The cited prior works (Calc. Var. 52 (2015) and Ann. Global Anal. Geom. 54 (2018)) are external references whose results are refined rather than presupposed as uniqueness theorems or ansatzes from the same authors. The central existence of maximizers follows from the variational analysis without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Assessment limited to abstract; no explicit free parameters, invented entities, or non-standard axioms are mentioned. The result rests on standard functional-analytic assumptions for fractional Sobolev spaces.

axioms (1)

domain assumption Fractional Sobolev spaces satisfy the embeddings and compactness properties needed for the variational problem
Invoked implicitly to guarantee existence of maximizers in the critical regime.

pith-pipeline@v0.9.0 · 5374 in / 1057 out tokens · 92749 ms · 2026-05-10T19:58:57.396395+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/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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Relation between the paper passage and the cited Recognition theorem.

We establish a Trudinger-Moser type inequality with a Tintarev-type constraint in fractional-dimensional spaces... (1.9) α-p+1=0 ... μ_{α,θ}=(θ+1)ω_α^{1/α}
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

sup ... H_ν(u)≤1 ∫ e^{μ_{α,θ}|u|^{p/(p-1)}} dλ_θ <∞ ... attained

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

40 extracted references · 40 canonical work pages

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R.: A sharp inequality of J

Adams, D. R.: A sharp inequality of J. Moser for higher order derivatives. Annals of Mathe- matics 128.2 p. 385-398. (1988) 1

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Druet.: Blow-up analysis in dimension 2 and a sharp form of Trudinger–Moser inequality

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L. Carleson and S.-Y. A. Chang: On the existence of an extremal function for an inequality of J. Moser, Bull. Sc. Math. 110, p. 113–127 (1986) 1

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M., Ruf, B.: Extremal for ak-Hessian inequality of Trudinger–Moser type

de Oliveira, J.F., do ´O, J. M., Ruf, B.: Extremal for ak-Hessian inequality of Trudinger–Moser type. Math. Z., v. 295, n. 3, p. 1683-1706 (2020) 2

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[13]

M.: Trudinger-Moser type inequalities for weighted Sobolev spaces involving fractional dimensions

de Oliveira, J.F., do ´O, J. M.: Trudinger-Moser type inequalities for weighted Sobolev spaces involving fractional dimensions. Proc. Amer. Math. Soc., v. 142, n. 8, p. 2813-2828 (2014) 1, 2, 5, 9

work page 2014
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M.: Concentration-compactness principle and extremal functions for a sharp Trudinger-Moser inequality

de Oliveira, J.F., do ´O, J. M.: Concentration-compactness principle and extremal functions for a sharp Trudinger-Moser inequality. Calc. Var., v. 52, n. 1-2, p. 125-163 (2015) 2, 3, 4, 15

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M.: On a sharp inequality of Adimurthi–Druet type and extremal functions, Calc

de Oliveira, J.F., do ´O, J. M.: On a sharp inequality of Adimurthi–Druet type and extremal functions, Calc. Var., v.62, p.162 (2023) 2, 3, 6, 18

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de Oliveira, J.F., do ´O, J.M.: Equivalence of critical and subcritical sharp Trudinger–Moser inequalities in fractional dimensions and extremal functions. Rev. Mat. Iberoam. (2022)

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do ´O, J.M., de Oliveira, J.F.: Concentration-compactness and extremal problems for a weighted Trudinger-Moser inequality. Commun. Contemp. Math., v.19, p.1650003 (2017) 2

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M., Lu, G., and Ponciano, R.: Sharp Sobolev and Adams–Trudinger–Moser embeddings on weighted Sobolev spaces and their applications, Forum Math

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and Ponciano, R.: Sharp higher order Adams’ inequality with exact growth condition on weighted Sobolev spaces

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work page 2024
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and Ponciano, R.: Trudinger–Moser embeddings on weighted Sobolev spaces on unbounded domains

do ´O, J.M., Lu, G. and Ponciano, R.: Trudinger–Moser embeddings on weighted Sobolev spaces on unbounded domains. Discrete Contin. Dyn. Syst., v. 45, p. 557–584 (2025)

work page 2025
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do ´O, J.M., Macedo, A.C., de Oliveira, J.F.: A sharp Adams-type inequality for weighted Sobolev spaces, Q. J. Math. v.71, p.517-538 (2020) 1, 2, 3

work page 2020
[22]

Com- ment

Flucher, M.: Extremal functions for the Trudinger-Moser inequality in 2 dimensions. Com- ment. Math. Helv., v. 67, n. 1, p. 471-497 (1992) 1

work page 1992
[23]

Kesavan.: Nonlinear Functional Analysis: A First Course, Texts and Readings in Mathe- matics, Hindustan Book Agency Gurgaon, 2004 20

S. Kesavan.: Nonlinear Functional Analysis: A First Course, Texts and Readings in Mathe- matics, Hindustan Book Agency Gurgaon, 2004 20

work page 2004
[24]

Notes in Math., vol

Kufner, A., Opic, B.: Hardy–type inequalities, Pitman Res. Notes in Math., vol. 219, Longman Scientific and Technical, Harlow, 1990. 2

work page 1990
[25]

and Zhang, L.: Existence and nonexistence of extremal functions for sharp Trudinger–Moser inequalities

Lam, N., Lu, G. and Zhang, L.: Existence and nonexistence of extremal functions for sharp Trudinger–Moser inequalities. Adv. Math. v. 352, p.1253–1298 (2019) 1

work page 2019
[26]

Lin, K.C.: Extremal functions for Moser’s inequality. Trans. Amer. Math. Soc., v. 348, n. 7, p. 2663-2671 (1996) 1

work page 1996
[27]

H.: Improved Moser–Trudinger inequality of Tintarev type in dimension n and the existence of its extremal functions

Nguyen, V. H.: Improved Moser–Trudinger inequality of Tintarev type in dimension n and the existence of its extremal functions. Annals of Global Analysis and Geometry, 54, p. 237-256 (2018) 1, 3

work page 2018
[28]

Trudinger

Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J., v. 20, n. 11, p. 1077-1092 (1971) 1

work page 1971
[29]

v.37, p.6987–7003 (2004) 2

Palmer C., Stavrinou, P.N.: Equations of motion in a non-integer-dimensional space, Journal of Physics A. v.37, p.6987–7003 (2004) 2

work page 2004
[30]

I.: The Sobolev embedding in the casepl=n

Pohozaev, S. I.: The Sobolev embedding in the casepl=n. Proceedings of the technical scientific conference on advances of scientific research. p.158-170 (1964) 1

work page 1964
[31]

Stillinger, F.H.: Axiomatic basis for spaces with noninteger dimension. J. Math. Phys. v.18, p.1224-1234 (1977) 1 29

work page 1977
[32]

Struwe, M.: Critical points of embeddings ofH 1,n 0 into Orlicz spaces. Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire, v. 5, n. 5 p. 425-464 (1988) 1

work page 1988
[33]

J.Funct.Anal

Tian,G.-T., Wang,X.-J.: Moser–Trudinger type inequalities for the Hessian equation. J.Funct.Anal. v.259, 1974–2002 (2010) 1

work page 1974
[34]

Tintarev, C.: Trudinger–Moser inequality with remainder terms. J. Funct. Anal. 266, p.55–66 (2014) 1, 3

work page 2014
[35]

Trudinger, N.: On imbeddings into Orlicz spaces and some applications, J. Math. Mech.17, p. 473-483 (1967) 1

work page 1967
[36]

Yudovich, V.I.: Some estimates connected with integral operators and with solutions of elliptic equations. Sov. Math. Dokl. 2, 746–749 (1961) 1

work page 1961
[37]

Journal of Func- tional Analysis, 239(1), p

Yang, Y.: A sharp form of Moser–Trudinger inequality in high dimension. Journal of Func- tional Analysis, 239(1), p. 100-126 (2006) 3

work page 2006
[38]

Yang, Y.: Extremal functions for a sharp Moser–Trudinger inequality. Int. J. Math. 17, p.331–338 (2006) 1

work page 2006
[39]

Zubair, M., Mughal, M.J., Naqvi, Q.A.: Electromagnetic Fields and Waves in Fractional Dimensional Space, Springer, Berlin, (2012) 1, 2

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[40]

v.12, p.2844-2850 (2011) 2 (R.D

Zubair, M, Mughal, M.J., Naqvi, Q.A.: On electromagnetic wave propagation in fractional space, Nonlinear Analysis: Real World Applications. v.12, p.2844-2850 (2011) 2 (R.D. da Silva Paiva) Department of Mathematics State University of Piau´ı 64600-000 Picos, PI, Brazil Email address:ruanpaiva@pcs.uespi.br (J.F. de Oliveira) Department of Mathematics Feder...

work page 2011

[1] [1]

R.: A sharp inequality of J

Adams, D. R.: A sharp inequality of J. Moser for higher order derivatives. Annals of Mathe- matics 128.2 p. 385-398. (1988) 1

work page 1988

[2] [2]

Druet.: Blow-up analysis in dimension 2 and a sharp form of Trudinger–Moser inequality

Adimurthi, O. Druet.: Blow-up analysis in dimension 2 and a sharp form of Trudinger–Moser inequality. p. 295-322 (2005) 1, 3

work page 2005

[3] [3]

E., et al

Andrews, G. E., et al. Special functions. Vol. 71. Cambridge: Cambridge university press,

work page

[4] [4]

Calculus of Variations and Partial Differential Equations 27 p

Baird, P., Fardoun, A., Regbaoui, R.: Q-curvature flow on 4-manifolds. Calculus of Variations and Partial Differential Equations 27 p. 75-104 (2006) 1

work page 2006

[5] [5]

Acta Math., v.159, p.215–259, (1987)

Chang, S.-Y.A., Yang, P.C.: Prescribing Gaussian curvature onS 2. Acta Math., v.159, p.215–259, (1987)

work page 1987

[6] [6]

Chang, S.-Y.A., Yang, P.C.: Conformal deformation of metrics onS 2. J. Differ. Geom., v.27, p.259–296, (1988) 1

work page 1988

[7] [7]

Carleson and S.-Y

L. Carleson and S.-Y. A. Chang: On the existence of an extremal function for an inequality of J. Moser, Bull. Sc. Math. 110, p. 113–127 (1986) 1

work page 1986

[8] [8]

G., Mitidieri, E.: Quasilinear elliptic equations with critical exponents

Cl´ ement, P., De Figueiredo, D. G., Mitidieri, E.: Quasilinear elliptic equations with critical exponents. p.133-170 (1996) 2, 6

work page 1996

[9] [9]

Collins, J.C.: Renormalization Cambridge University Press, Cambridge, (1984). 2 28

work page 1984

[10] [10]

Trudinger and J

de Figueiredo, D.G., do ´O, J.M., Ruf, B.: On an inequality by N. Trudinger and J. Moser and related elliptic equations. Comm. Pure Appl. Math. 55, no. 2, p.135–152 (2002) 1

work page 2002

[11] [11]

D.: Critical points of the Moser–Trudinger functional on closed surfaces

De Marchis, F., Malchiodi, A., Martinazzi, L., Thizy, P. D.: Critical points of the Moser–Trudinger functional on closed surfaces. Invent. Math., v. 230, n. 3, p. 1165-1248 (2022)

work page 2022

[12] [12]

M., Ruf, B.: Extremal for ak-Hessian inequality of Trudinger–Moser type

de Oliveira, J.F., do ´O, J. M., Ruf, B.: Extremal for ak-Hessian inequality of Trudinger–Moser type. Math. Z., v. 295, n. 3, p. 1683-1706 (2020) 2

work page 2020

[13] [13]

M.: Trudinger-Moser type inequalities for weighted Sobolev spaces involving fractional dimensions

de Oliveira, J.F., do ´O, J. M.: Trudinger-Moser type inequalities for weighted Sobolev spaces involving fractional dimensions. Proc. Amer. Math. Soc., v. 142, n. 8, p. 2813-2828 (2014) 1, 2, 5, 9

work page 2014

[14] [14]

M.: Concentration-compactness principle and extremal functions for a sharp Trudinger-Moser inequality

de Oliveira, J.F., do ´O, J. M.: Concentration-compactness principle and extremal functions for a sharp Trudinger-Moser inequality. Calc. Var., v. 52, n. 1-2, p. 125-163 (2015) 2, 3, 4, 15

work page 2015

[15] [15]

M.: On a sharp inequality of Adimurthi–Druet type and extremal functions, Calc

de Oliveira, J.F., do ´O, J. M.: On a sharp inequality of Adimurthi–Druet type and extremal functions, Calc. Var., v.62, p.162 (2023) 2, 3, 6, 18

work page 2023

[16] [16]

de Oliveira, J.F., do ´O, J.M.: Equivalence of critical and subcritical sharp Trudinger–Moser inequalities in fractional dimensions and extremal functions. Rev. Mat. Iberoam. (2022)

work page 2022

[17] [17]

do ´O, J.M., de Oliveira, J.F.: Concentration-compactness and extremal problems for a weighted Trudinger-Moser inequality. Commun. Contemp. Math., v.19, p.1650003 (2017) 2

work page 2017

[18] [18]

M., Lu, G., and Ponciano, R.: Sharp Sobolev and Adams–Trudinger–Moser embeddings on weighted Sobolev spaces and their applications, Forum Math

do ´O, J. M., Lu, G., and Ponciano, R.: Sharp Sobolev and Adams–Trudinger–Moser embeddings on weighted Sobolev spaces and their applications, Forum Math. (2024), doi:10.1515/forum-2023-0292. 2

work page doi:10.1515/forum-2023-0292 2024

[19] [19]

and Ponciano, R.: Sharp higher order Adams’ inequality with exact growth condition on weighted Sobolev spaces

do ´O, J.M., Lu, G. and Ponciano, R.: Sharp higher order Adams’ inequality with exact growth condition on weighted Sobolev spaces. J. Geom. Anal., v. 34, Art. 53 (2024)

work page 2024

[20] [20]

and Ponciano, R.: Trudinger–Moser embeddings on weighted Sobolev spaces on unbounded domains

do ´O, J.M., Lu, G. and Ponciano, R.: Trudinger–Moser embeddings on weighted Sobolev spaces on unbounded domains. Discrete Contin. Dyn. Syst., v. 45, p. 557–584 (2025)

work page 2025

[21] [21]

do ´O, J.M., Macedo, A.C., de Oliveira, J.F.: A sharp Adams-type inequality for weighted Sobolev spaces, Q. J. Math. v.71, p.517-538 (2020) 1, 2, 3

work page 2020

[22] [22]

Com- ment

Flucher, M.: Extremal functions for the Trudinger-Moser inequality in 2 dimensions. Com- ment. Math. Helv., v. 67, n. 1, p. 471-497 (1992) 1

work page 1992

[23] [23]

Kesavan.: Nonlinear Functional Analysis: A First Course, Texts and Readings in Mathe- matics, Hindustan Book Agency Gurgaon, 2004 20

S. Kesavan.: Nonlinear Functional Analysis: A First Course, Texts and Readings in Mathe- matics, Hindustan Book Agency Gurgaon, 2004 20

work page 2004

[24] [24]

Notes in Math., vol

Kufner, A., Opic, B.: Hardy–type inequalities, Pitman Res. Notes in Math., vol. 219, Longman Scientific and Technical, Harlow, 1990. 2

work page 1990

[25] [25]

and Zhang, L.: Existence and nonexistence of extremal functions for sharp Trudinger–Moser inequalities

Lam, N., Lu, G. and Zhang, L.: Existence and nonexistence of extremal functions for sharp Trudinger–Moser inequalities. Adv. Math. v. 352, p.1253–1298 (2019) 1

work page 2019

[26] [26]

Lin, K.C.: Extremal functions for Moser’s inequality. Trans. Amer. Math. Soc., v. 348, n. 7, p. 2663-2671 (1996) 1

work page 1996

[27] [27]

H.: Improved Moser–Trudinger inequality of Tintarev type in dimension n and the existence of its extremal functions

Nguyen, V. H.: Improved Moser–Trudinger inequality of Tintarev type in dimension n and the existence of its extremal functions. Annals of Global Analysis and Geometry, 54, p. 237-256 (2018) 1, 3

work page 2018

[28] [28]

Trudinger

Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J., v. 20, n. 11, p. 1077-1092 (1971) 1

work page 1971

[29] [29]

v.37, p.6987–7003 (2004) 2

Palmer C., Stavrinou, P.N.: Equations of motion in a non-integer-dimensional space, Journal of Physics A. v.37, p.6987–7003 (2004) 2

work page 2004

[30] [30]

I.: The Sobolev embedding in the casepl=n

Pohozaev, S. I.: The Sobolev embedding in the casepl=n. Proceedings of the technical scientific conference on advances of scientific research. p.158-170 (1964) 1

work page 1964

[31] [31]

Stillinger, F.H.: Axiomatic basis for spaces with noninteger dimension. J. Math. Phys. v.18, p.1224-1234 (1977) 1 29

work page 1977

[32] [32]

Struwe, M.: Critical points of embeddings ofH 1,n 0 into Orlicz spaces. Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire, v. 5, n. 5 p. 425-464 (1988) 1

work page 1988

[33] [33]

J.Funct.Anal

Tian,G.-T., Wang,X.-J.: Moser–Trudinger type inequalities for the Hessian equation. J.Funct.Anal. v.259, 1974–2002 (2010) 1

work page 1974

[34] [34]

Tintarev, C.: Trudinger–Moser inequality with remainder terms. J. Funct. Anal. 266, p.55–66 (2014) 1, 3

work page 2014

[35] [35]

Trudinger, N.: On imbeddings into Orlicz spaces and some applications, J. Math. Mech.17, p. 473-483 (1967) 1

work page 1967

[36] [36]

Yudovich, V.I.: Some estimates connected with integral operators and with solutions of elliptic equations. Sov. Math. Dokl. 2, 746–749 (1961) 1

work page 1961

[37] [37]

Journal of Func- tional Analysis, 239(1), p

Yang, Y.: A sharp form of Moser–Trudinger inequality in high dimension. Journal of Func- tional Analysis, 239(1), p. 100-126 (2006) 3

work page 2006

[38] [38]

Yang, Y.: Extremal functions for a sharp Moser–Trudinger inequality. Int. J. Math. 17, p.331–338 (2006) 1

work page 2006

[39] [39]

Zubair, M., Mughal, M.J., Naqvi, Q.A.: Electromagnetic Fields and Waves in Fractional Dimensional Space, Springer, Berlin, (2012) 1, 2

work page 2012

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v.12, p.2844-2850 (2011) 2 (R.D

Zubair, M, Mughal, M.J., Naqvi, Q.A.: On electromagnetic wave propagation in fractional space, Nonlinear Analysis: Real World Applications. v.12, p.2844-2850 (2011) 2 (R.D. da Silva Paiva) Department of Mathematics State University of Piau´ı 64600-000 Picos, PI, Brazil Email address:ruanpaiva@pcs.uespi.br (J.F. de Oliveira) Department of Mathematics Feder...

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