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Fractional Trudinger-Moser type inequalities with logarithmic convolution potentials

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Pith's one-line read At decay exponent γ ≥ 1 the fractional Trudinger–Moser supremum with logarithmic convolution potential is finite, with extremals for γ > 1; at γ < 1 it is infinite.

desk verdict The paper's main estimates rely on a one-dimensional radial convolution identity that is numerically false even for even functions, so Theorems 1.6 and 1.7 are currently unproven despite substantial technical work. read the letter →

arxiv 2507.20069 v1 pith:5TTAFZA7 submitted 2025-07-26 math.AP

Huxiao Luo , Shiying Wang This is my paper · ORCID
classification math.AP MSC 35J5035Q4031A10
keywords fractionalTrudinger-MoserinequalitylogarithmicconvolutionpotentialextremalfunctionscriticalgrowthexponentSobolevspaceradialsymmetrymovingplanemethodMosersequence
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 pins down the exact decay rate at which a one-dimensional fractional Trudinger–Moser inequality with a logarithmic convolution potential stays finite. The authors prove that on a bounded interval $I$, the supremum of $\Phi(u) = \int_I\int_I \log(1/|x-y|) G(u(x))G(u(y))\,dx\,dy$ over the fractional Dirichlet unit ball is finite whenever $G$ has at most $\gamma$-critical growth with $\gamma \ge 1$, and is infinite when $G$ has at least $\gamma$-critical growth with $\gamma < 1$. Under the additional subcritical growth condition $G(s)-G(0)\le c(e^{\pi s^2}-1)$, they show for $\gamma>1$ that the supremum is attained and every maximizer is, up to sign, even and radially decreasing. The same dichotomy is established in $H^{1/2}(\mathbb{R})$ for the whole-space functional $\Psi$. Since the borderline value $\gamma=1$ separates finiteness from blow-up, the result completes the one-dimensional counterpart of the planar logarithmic-convolution inequalities and identifies the sharp threshold for existence of extremals.

What carries the argument

The load-bearing identity is the radial representation $\Phi(u) = 2\int_0^1 G(u(r))\log(1/r)\int_0^r G(u(\rho))\,d\rho\,dr$, valid for even functions via Newton's theorem for the logarithmic kernel. Lemma 2.10 bounds the two-weight functional $\Phi_{\gamma_1,\gamma_2}$ whenever $\gamma_1+\gamma_2\ge 2$, converting the exponential growth of $G$ in the $\gamma\ge 1$ regime into an integrable singularity at $r=0$. For the blow-up case $\gamma<1$, a Moser-type sequence with explicitly computed fractional norm asymptotics is shown to make $\Phi(w_n)\to\infty$. Attainability rests on Proposition 3.4, where the additional growth condition (1.5) and a Chebyshev-truncation argument force $\Phi(u_n)\to\Phi(0)$ whenever $u_n\rightharpoonup 0$; the moving-plane part uses the decay bounds $u(x)=O(|x|^{-2})$ and $w(x)=O(\log|x|)$ and a Hardy–Littlewood–Sobolev estimate with the logarithmic kernel.

What would settle it

Compute the supremum of $\Phi$ on $I=(-1,1)$ for the exact $\gamma=1$ nonlinearity $G(s)=e^{\pi s^2}/(1+|s|)$; the theorem predicts $m(G)<\infty$, so any sequence $u_n$ in the unit ball with $\Phi(u_n)\to\infty$ would refute part (i) of Theorem 1.6.

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Extended reading notes

Core claim

The central discovery is a sharp exponent threshold. For the functional $\Phi(u) = \int_I\int_I \log(1/|x-y|) G(u(x))G(u(y))\,dx\,dy$ on the unit ball of $W^{1/2,2}_0(I)$ determined by $\|(-\Delta)^{1/4}u\|_{L^2}\le 1$, the supremum $m(G)$ is finite exactly when $G$ has at most $\gamma$-critical growth for some $\gamma\ge 1$, and infinite when $G$ has at least $\gamma$-critical growth for some $\gamma<1$. When $\gamma>1$ and $G(s)-G(0)\le c(e^{\pi s^2}-1)$, the supremum is attained, and every maximizer is, up to sign, an even and radially decreasing function in the constraint set. The corresponding statement on $H^{1/2}(\mathbb{R})$ holds up to translation. The positivity of maximizers and the moving-plane analysis further show that positive classical solutions of the associated Euler–Lagrange system decay like $|x|^{-2}$ and are radially symmetric.

Load-bearing premise

The proof assumes that every function in the constrained set may be replaced by its symmetric decreasing rearrangement, so that the radial representation and the uniform convergence on $[\delta,1]$ apply to arbitrary maximizing sequences; if rearrangement changes the weak-limit behaviour, the attainment argument collapses.

Editorial extensions

If this is right

  • For any bounded interval and any $G$ of at most $\gamma$-critical growth with $\gamma\ge 1$, the supremum $m(G)$ is finite; the inequality holds with a constant independent of $u$.
  • For $\gamma>1$ together with condition (1.5), the supremum is attained, so the constrained maximization problem has a genuine solution rather than only a maximizing sequence.
  • Every maximizer in the attaining regime is, up to sign, even and radially decreasing; on the whole line it is so up to translation as well.
  • Positive classical solutions of the Euler–Lagrange system decay polynomially and are radially symmetric, complementing the one-dimensional fractional landscape.
  • The sharp threshold at $\gamma=1$ means that adding a polynomial decay factor to the critical exponential growth is exactly the right amount to restore compactness.

Reading between the lines

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

  • The $\gamma=1$ endpoint under only at-most-critical growth remains open for attainment; the present machinery yields maximizers only for $\gamma>1$.
  • In higher dimensions or for other fractional exponents $s$, the critical decay exponent may shift because the radial weight multiplying the convolution changes with dimension.
  • A numerical test on non-even peaked functions could reveal whether the radial-rearrangement reduction is essential, by comparing $\Phi(u)$ and $\Phi(u^*)$ for unit-norm functions.
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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

4 major / 3 minor

Summary. The manuscript studies one-dimensional fractional Trudinger-Moser type inequalities with logarithmic convolution potentials. For a nonnegative even, increasing G satisfying the exponential bound (A), it claims in Theorem 1.6 that the supremum m(G) over the unit ball E of H^{1/2}_0(I) is finite when G has at most γ-critical growth with γ ≥ 1, that m(G) is attained with even decreasing maximizers when γ > 1 and (1.5) holds, and that m(G) = ∞ when G has at least γ-critical growth with γ < 1. Theorem 1.7 states the analogous dichotomy on the whole line for Ψ. The proofs use a radial representation of the logarithmic energy (Lemma 2.7 and Corollary 2.8), a one-dimensional Moser sequence, concentration-compactness arguments, and a moving-plane argument for the corresponding Euler-Lagrange system.

Significance. The problem is natural and the stated threshold γ = 1 is plausible; if established, the results would extend the Cingolani–Weth theory to the fractional one-dimensional setting, and the symmetry theorem for the nonlocal system is of independent interest. The paper contains substantial technical work, including detailed asymptotic estimates for Moser sequences and a careful decay analysis for the fractional Laplacian. However, the central radial convolution identity used throughout the paper is false, even for even functions. Since this identity underlies the bounds in Propositions 3.2, 3.4, and 4.2, the proofs of the main finiteness and attainment theorems do not go through. The manuscript therefore does not establish its headline claims in the submitted form.

major comments (4)
  1. [Section 2, Eq. (2.14)] Corollary 2.8(ii), Eq. (2.14), is false even for even functions. Taking v = 1_{(-1,1)}, the left side equals ∫_{-1}^1∫_{-1}^1 log(1/|x-y|) dx dy = 6 - 4 log 2 ≈ 3.23, whereas the right side equals 2∫_0^1 r log(1/r) dr = 1/2. The error originates in Lemma 2.7: in one dimension the logarithmic potential of an even density is not constant on the two-point set {|y| = r}, so the Newton-shell formula (2.12) does not hold. Since Proposition 3.2 uses (2.14) to reduce Φ(u) to the radial expression bounded in Lemma 2.10, Theorem 1.6(i) is not proved.
  2. [Section 2, Lemma 2.7 and Eq. (2.13)] The mixed identity (2.13) in Corollary 2.8(i), also based on Lemma 2.7, is invalid for the same reason. For v = w = 1_{(-1,1)}, the right-hand side of (2.13) gives 1/2 instead of the true value 6 - 4 log 2. This invalidates the decomposition in Proposition 3.4(ii), where the cross term IIn is identified as G(0)∫_0^1 v_n(r)(1-r) dr; the function f(r) = 1-r is not the correct one-dimensional weight. The proof of (3.10) therefore does not establish the claimed continuity at 0.
  3. [Section 3, Proposition 3.4] The proof assumes that the sequence {u_n} is radial and non-increasing after asserting u_n → 0 in L^p: the sentence 'And since u_n are radial and non-increasing in the radial coordinate' introduces an hypothesis that is not derived. A general maximizing sequence in H^{1/2}_0(I) is not radial, and the paper does not justify a symmetric-decreasing rearrangement that preserves both the energy Φ and the weak convergence property used in the argument. The uniform convergence on [δ,1] used in (3.7) and (3.11) depends on this missing reduction. The same unjustified radiality appears in the proof of Proposition 4.2.
  4. [Section 4, Proposition 4.2] The decomposition Ψ(u_n) = 2(Ψ_2(u_n) - Ψ_1(u_n)) at the start of the proof invokes Corollary 2.8 and therefore inherits the false radial identity. Even if the sequence were radial, the representation is wrong; consequently the upper semicontinuity claim (4.8) is not supported. Since attainment of m∞(G) in Theorem 1.7(ii) rests on this claim, Theorem 1.7 is also not established by the submitted proof.
minor comments (3)
  1. [Abstract and Theorem 1.6] The phrase 'even and radially decreasing' should be defined for functions on an interval; for I = (-1,1), radial decreasing should be stated explicitly as depending only on |x| and being non-increasing in |x|.
  2. [Proposition 3.3] The notation I_{1/n}(0) is used without definition; presumably it denotes the interval (-1/n, 1/n), but this should be stated.
  3. [Lemma 2.3] The equality case in Lemma 2.3(iii) is asserted without proof or precise reference; please provide a citation or a short argument.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the paper's claims are derived from stated hypotheses and external results, with no fitted inputs, self-citation chains, or target-result assumptions.

full rationale

The derivation chain is self-contained with respect to circularity. Theorem 1.6 is proved by combining the fractional Trudinger-Moser inequality (1.2)/(1.3) from Iula-Maalaoui-Martinazzi and Chen-Wang-Zhu with rearrangement lemmas quoted from Cingolani-Weth and concentration-compactness from Lions; Theorem 1.7 uses the same external ingredients plus the structural assumption (A1). The upper-bound arguments in Propositions 3.2 and 4.1 reduce Φ and Ψ to Lemma 2.10 and a truncated comparison function, neither of which presumes the target finiteness. The nonexistence/divergence side in Proposition 3.3 is a direct calculation on an explicit Moser-type sequence, not a hidden use of the theorem being proved. Attainment arguments in Propositions 3.5 and 4.2 go through weak compactness, concentration-compactness cases, and continuity statements; they do not assume the extremal value is attained. The moving-plane section derives radial decreasing from the Euler-Lagrange system and standard maximum-principle/HLS estimates, with no self-citation bearing mathematical weight. There is a separate mathematical concern, not a circularity: Proposition 3.4 states 'Since un are radial and non-increasing in the radial coordinate' after assuming only that un is a sequence in E, and Corollary 2.8(ii) is applied to arbitrary non-even functions in Proposition 3.2. Those are potential validity gaps about symmetry reduction or a possibly false radial convolution identity, but they are not instances of a conclusion being equivalent to its input by construction. No fitted parameters are renamed as predictions, no uniqueness theorem is imported from the present authors' prior work, and no ansatz is smuggled through self-citation. Thus the circularity score is 0.

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

The central results are derived from stated assumptions on G and from established external tools: fractional Sobolev equivalence, fractional Trudinger-Moser inequalities, concentration-compactness, rearrangements, and Hardy-Littlewood-Sobolev. No free parameters are fitted and no new entities are introduced.

assumptions (5)
  • standard math Fractional Sobolev norm equivalence: [u]_{W^{s,2}(R^N)} < infinity iff (-Delta)^{s/2}u in L^2, with [u]^2 = 2 C_{N,s}^{-1} |(-Delta)^{s/2}u|^2.
    Invoked as Proposition 1.1 from Di Nezza-Palatucci-Valdinoci [17] to connect the Gagliardo seminorm to the operator norm used in the definition of E.
  • standard math Fractional Trudinger-Moser inequality on intervals (Iula-Maalaoui-Martinazzi [21, Theorem 1.1]): sup_{||(-Delta)^{1/4}u||_2 <= 1} integral_I (e^{pi u^2} - 1) dx <= C|I|.
    Used in Lemma 2.10 and Section 3 to uniformly bound integral_I e^{pi u^2} dx over E.
  • standard math Improved fractional Trudinger-Moser inequality (Chen-Wang-Zhu [5]): sup over ||(-Delta)^{1/4}u||_2^2 - alpha ||u||_2^2 <= 1 of integral_I e^{pi u^2} dx is finite.
    Used as (1.3) in the boundedness estimates and in the convergence proof of Proposition 3.4.
  • standard math Concentration-compactness alternative (Lions [23]): a bounded maximizing sequence either has a nonzero weak limit and e^{(pi+t)u_n^2} bounded in L^1, or has weak limit zero.
    Used in Propositions 3.5 and 4.2 to split the maximizer existence proof into two cases.
  • standard math Rearrangement and Hardy-Littlewood-Sobolev inequalities (Almgren-Lieb, Baernstein, Lieb-Loss): symmetric decreasing rearrangement increases Phi and decreases Psi^-, and HLS holds for the kernel |x-y|^{-1/2} in one dimension.
    Used in Lemma 2.3, Lemma 2.4, and Lemma 5.6; the equality case in rearrangement is needed for the symmetry of maximizers.

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Pith. "Pith review of Fractional Trudinger-Moser type inequalities with logarithmic convolution potentials." pith.science (2026). https://pith.science/paper/5TTAFZA7

@misc{pith2026250720069,
  author       = {Pith},
  title        = {Pith review of: Fractional Trudinger-Moser type inequalities with logarithmic convolution potentials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5TTAFZA7}},
  note         = {Machine review of arXiv:2507.20069}
}
abstract

We establish the following fractional Trudinger-Moser type inequality with logarithmic convolution potential $$ \sup_{u\in W^{\frac{1}{2},2}_0(I),\|u\|_{W_0^{\frac{1}{2},2}}\leq1}\int_{I} \int_{I} \log \frac{1}{|x-y|} G(u(x))G(u(y)) \, dx \, dy<+\infty,$$ where $G(s)\leq C\frac{e^{\pi s^{2}}}{(1 + |s|)^{\gamma}}~ \forall s\in\mathbb{R}$ with some constant $C>0,\gamma\geq1$, the domain $I\subset\mathbb{R}$ is a bounded interval. This type of inequality in the entire space $\mathbb{R}$ is also considered. Moreover, we study the existence of corresponding extremal functions. In addition, by the moving plane method, we obtain the radial symmetry and radial decreasing property of positive solutions to the corresponding Euler-Lagrange equation.

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Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [9]

    Cingolani and T

    S. Cingolani and T. Weth, Trudinger-Moser type inequality with logarithmic convolution potentials, Journal of London Mathematical Society, 105(3) (2022), 1897-1935. 2, 6, 8, 9, 13

    work page 2022

  2. [1]

    F. J. Almgren and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc., 2 (1989), 683773. 10

    work page 1989

  3. [2]

    Baernstein, A unified approach to symmetrization, Partial Differential equations of elliptic type, eds, Symposia matematica 35, Cambridge University Press 1995, 47-91

    A. Baernstein, A unified approach to symmetrization, Partial Differential equations of elliptic type, eds, Symposia matematica 35, Cambridge University Press 1995, 47-91. 10

    work page 1995

  4. [3]

    J. E. Brothers and W. P. Ziemer, Minimal rearrangements of Sobolev functions, J. Reine Angew. Math., 384 (1988), 153-179. 10 FRACTIONAL TRUDINGER-MOSER TYPE INEQUALITIES WITH LOGARITHMIC POTENTIALS 45

    work page 1988

  5. [4]

    A sharp threshold for Trudinger-Moser type inequalities with logarithmic kernels in dimension N

    A. Cannone and S. Cingolani, A sharp threshold for Trudinger-Moser type inequalities with logarithmic kernels in dimension N, https://arxiv.org/html/2410.10013v1. 2

    work page Pith review arXiv

  6. [5]

    L. Chen, B. Wang and M. Zhu, Improved fractional Trudinger-Moser inequalities on bounded intervals and the existence of their extremals, Advanced Nonlinear Studies, 23 (2023), 20220067. 3, 11

    work page 2023

  7. [6]

    Calanchi and B

    M. Calanchi and B. Ruf, On Trudinger-Moser type inequalities with logarithmic weights, J. Differ. Equations, 258(6) (2015), 1967-1989. 2

    work page 2015

  8. [7]

    Calanchi and B

    M. Calanchi and B. Ruf, Trudinger-Moser type inequalities with logarithmic weights in dimension N, Nonlinear Anal., 121 (2015), 403-411. 2

    work page 2015

Show all 34 references
  1. [8]

    Caffarelli and L

    L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. 2

    2007

  2. [10]

    Cingolani, T

    S. Cingolani, T. Weth and M. Yu, Extremal functions for the critical Trudinger-Moser inequality with logarithmic kernels, ESAIM Control Optim, Calc. Var. 30 (2024), Paper No. 75, 25 pp. 2

    2024

  3. [11]

    Caglioti, P

    E. Caglioti, P. L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description. Part II, Comm. Math. Phys., 174 (1995), 229-260. 2

    1995

  4. [12]

    J. F. de Oliveira and J. M. do ´O, Trudinger-Moser type inequalities for weighted Sobolev spaces involving fractional dimensions, Proc. Am. Math. Soc., 142(8) (2014), 2813-2828. 2

    2014

  5. [13]

    J. F. de Oliveira and J. M. do ´O, Equivalence of critical and subcritical sharp Trudinger-Moser inequalities in fractional dimensions and extremal functions, Rev. Mat. Iberoam., 39(3) (2023), 1073-1096. 2

    2023

  6. [14]

    J. M. do ´O, G. Lu and R. Ponciano, Sharp Sobolev and Adams-Trudinger-Moser embeddings on weighted Sobolev spaces and their applications, Forum Math., 36(5) (2024), 1279-1320. 2

    2024

  7. [15]

    J. M. do ´O, G. Lu and R. C. Ponciano, Trudinger-Moser embeddings on weighted Sobolev spaces on unbounded domains, Discrete Contin. Dyn. Syst., 45(2) (2025), 557-584. 2

    2025

  8. [16]

    Dolbeault and B

    J. Dolbeault and B. Perthame, Optimal critical mass in the two dimensional Keller-Segel model in R2, C. R. Acad. Sci. Paris, Ser. I 339 (2004), 611-616. 2

    2004

  9. [17]

    Di Nezza, G

    E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521–573. 2, 3

    2012

  10. [18]

    Felmer, A

    P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schr¨ odinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262. 36

    2012

  11. [19]

    J. A. Hempel, G. R. Morris and N. S. Trudinger, On the sharpness of a limiting case of the Sobolev imbedding theorem, Bull. Aust. Math. Soc., 3 (1970), 369-373. 2

    1970

  12. [20]

    Hyder, Moser functions and fractional Moser-Trudinger type inequalities, Nonlinear Anal., 146 (2016) 185-210

    A. Hyder, Moser functions and fractional Moser-Trudinger type inequalities, Nonlinear Anal., 146 (2016) 185-210. 3

    2016

  13. [21]

    S. Iula, A. Maalaoui and L. Martinazzi, A fractional Moser-Trudinger type inequality in one dimension and its critical points, Differ. Integr. Equ., 29 (2016), 455–492. 3, 11

    2016

  14. [22]

    E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. 13, 40

    2001

  15. [23]

    P. L. Lions, The concentration-compactness principle in the calculus of variations, The limit case, part 1., Riv.Mat. Iberoam., 1 (1985), 145-201. 27, 31

    1985

  16. [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. 1

    1970

  17. [25]

    V. H. Nguyen, Remarks on the Moser-Trudinger type inequality with logarithmic weights in dimension N, Proc. Am. Math. Soc., 147(12) (2019), 5183-5193. 2

    2019

  18. [26]

    Polya and G

    G. Polya and G. Szeg¨ o, Inequalities for the capacity of a condenser, Amer. J. Math., 67 (1945), 1-32. 10

    1945

  19. [27]

    Polya and G

    G. Polya and G. Szeg¨ o, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies. Princeton, N. J. Princeton University Press (1945). 10 46 H. LUO AND S. W ANG

    1945

  20. [28]

    S. I. Pohoˇ zaev, On the eigenfunctions of the equation ∆u +λf(u) = 0, (Russian) Dokl. Akad. Nauk SSSR 165 (1965) 36-39. 1

    1965

  21. [29]

    Roy, Extremal function for Moser-Trudinger type inequality with logarithmic weight, Nonlinear Anal., 135 (2016), 194-204

    P. Roy, Extremal function for Moser-Trudinger type inequality with logarithmic weight, Nonlinear Anal., 135 (2016), 194-204. 2

    2016

  22. [30]

    Roy, On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions, Discrete Contin

    P. Roy, On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions, Discrete Contin. Dyn. Syst., 39(9) (2019), 5207-5222. 2

    2019

  23. [31]

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

    1967

  24. [32]

    Wolansky, On steady distributions of self-attracting clusters under friction and fluctuations, Arch

    G. Wolansky, On steady distributions of self-attracting clusters under friction and fluctuations, Arch. Rational Mech. Anal., 119(4) (1992), 355-391. 2

    1992

  25. [33]

    J. Xue, C. Zhang and M. Zhu, Trudinger-Moser type inequalities with logarithmic weights in fractional dimensions, Advanced Nonlinear Studies, 25(1) (2025), 152-170. 2

    2025

  26. [34]

    V. I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, (Russian) Dokl. Akad. Nauk SSSR 138 (1961) 805-808. 1 (H. Luo) Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang, 321004, China Email address : luoh...

    1961

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