Stability for Critical Points of the Hardy--Littlewood--Sobolev Inequality and a Dual Stability Framework

Guozhen Lu; Hanli Tang; Lu Chen

arxiv: 2605.19288 · v1 · pith:2L62ZS6Knew · submitted 2026-05-19 · 🧮 math.AP · math.CA

Stability for Critical Points of the Hardy--Littlewood--Sobolev Inequality and a Dual Stability Framework

Lu Chen , Guozhen Lu , Hanli Tang This is my paper

Pith reviewed 2026-05-20 04:48 UTC · model grok-4.3

classification 🧮 math.AP math.CA

keywords Hardy-Littlewood-Sobolev inequalityquantitative stabilitycritical pointsPalais-Smale sequencesweak decompositiondual stability frameworkfractional Sobolev inequalityStruwe decomposition

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The pith

A weak-decomposition-strong-stability method yields the first quantitative stability inequality for critical points of the Hardy-Littlewood-Sobolev inequality in a non-Hilbertian distance.

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

The paper seeks to establish quantitative stability for critical points of the Hardy-Littlewood-Sobolev inequality, where the natural distance measure is non-Hilbertian and thus blocks classical orthogonal decomposition techniques. A sympathetic reader would care because such stability controls how close a function is to the optimizers and supplies explicit bounds on Palais-Smale sequences, which are central to variational existence arguments. The authors introduce a tailored weak-decomposition-strong-stability method that upgrades weak information to strong control without extra regularity or sign assumptions. They further build a duality framework that transfers Struwe-type decompositions and stability statements between the Sobolev inequality and its HLS counterpart. As a direct result, the framework removes the nonnegativity restriction previously required for stability of fractional Sobolev critical points.

Core claim

We develop a weak-decomposition--strong-stability method tailored to the stability structure of HLS critical points and establish the corresponding stability inequality. This yields an explicit lower bound for the stability of Palais-Smale sequences of the HLS integral equation, the first such quantitative result measured in a non-Hilbertian distance. The same approach supplies a duality framework connecting Struwe-type decompositions and stability inequalities for Sobolev critical points with their HLS counterparts, thereby deriving Struwe-type decomposition and stability results for critical points of the fractional Sobolev inequality for general functions without a nonnegativity condition

What carries the argument

The weak-decomposition--strong-stability method, which first produces a weak decomposition adapted to the non-Hilbertian distance and then upgrades it to strong quantitative stability control for HLS critical points.

If this is right

An explicit lower bound holds for the stability deficit of any Palais-Smale sequence of the HLS integral equation.
Struwe-type decomposition results extend to critical points of the fractional Sobolev inequality for arbitrary functions without a nonnegativity hypothesis.
Stability inequalities for Sobolev critical points transfer to HLS critical points and conversely through the introduced duality framework.

Where Pith is reading between the lines

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

The same weak-to-strong upgrade technique may extend to stability questions for other nonlocal variational problems whose natural distances are non-Hilbertian.
The duality construction suggests that stability results can be transferred systematically between pairs of dual inequalities without repeating the full analysis each time.
Quantitative stability without sign or decay assumptions could simplify compactness arguments in related existence theorems for integral equations.

Load-bearing premise

The non-Hilbertian distance admits a weak decomposition that upgrades directly to strong stability control without requiring extra regularity, sign, or decay assumptions on the functions.

What would settle it

A sequence of functions that forms a Palais-Smale sequence for the HLS functional yet remains bounded away from the set of critical points while the stability deficit tends to zero would falsify the claimed inequality.

read the original abstract

Although quantitative stability for critical points of the Sobolev and fractional Sobolev inequalities has been extensively studied, the corresponding stability theory for critical points of the Hardy--Littlewood--Sobolev (HLS) inequality remains largely unexplored. A major difficulty is that the natural stability problem for HLS critical points involves a non-Hilbertian distance, so the classical orthogonal decomposition methods used in Hilbert-space settings are no longer available. In this paper, we develop a weak-decomposition--strong-stability method tailored to the stability structure of HLS critical points and establish the corresponding stability inequality. Our approach also yields an explicit lower bound for the stability of Palais--Smale sequences of the HLS integral equation. To the best of our knowledge, this appears to be the first quantitative stability result for Palais--Smale sequences of a variational functional measured in a non-Hilbertian distance. We further introduce a duality framework connecting Struwe-type decompositions and stability inequalities for critical points of the Sobolev inequality with their HLS counterparts. As a consequence, we derive Struwe-type decomposition and stability results for critical points of the fractional Sobolev inequality for general functions, thereby removing the nonnegativity assumption imposed in [26].

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

This paper gives the first quantitative stability for HLS critical points in a non-Hilbertian distance and uses duality to remove the nonnegativity assumption from fractional Sobolev stability.

read the letter

The main point is a weak-decomposition method that upgrades to a strong stability inequality for HLS critical points, measured in a non-Hilbertian distance, plus a duality that transfers Struwe-type results to the fractional Sobolev inequality without forcing nonnegative functions. They also claim an explicit lower bound for Palais-Smale sequences of the HLS equation. This is new because prior stability work stayed in Hilbert settings or kept the sign restriction. The approach looks tailored to the integral kernel structure rather than a routine extension of Sobolev techniques, and the duality is a clean way to borrow existing decompositions instead of rebuilding everything from scratch. That part could help with compactness arguments where sign changes appear. The soft spot is the upgrade step itself. The stress-test note is reasonable: controlling the remainder after weak decomposition in a non-Hilbertian distance often needs some tail decay or integrability to absorb cross terms, yet the abstract and claims do not flag any such condition for completely general functions. If the estimates only close under an implicit assumption on behavior at infinity, the result for arbitrary Palais-Smale sequences would need a qualifier. I would check the precise statement of the main theorem and the handling of those interaction terms. This is for analysts working on stability, compactness, and nonlocal variational problems. Someone who knows the Sobolev stability literature will follow the extension without much trouble. It deserves a serious referee because the claims are specific, the method is original, and the potential gap is checkable rather than fatal.

Referee Report

1 major / 1 minor

Summary. The paper develops a weak-decomposition--strong-stability method for critical points of the Hardy--Littlewood--Sobolev inequality, establishing a quantitative stability inequality in a non-Hilbertian distance for Palais--Smale sequences. It further introduces a duality framework linking Struwe-type decompositions and stability results between the HLS inequality, the Sobolev inequality, and the fractional Sobolev inequality, with the latter holding for general functions without a nonnegativity assumption.

Significance. If the central stability inequality is established without hidden regularity or decay assumptions, the work would be significant as the first quantitative stability result for Palais--Smale sequences of a variational functional measured in a non-Hilbertian distance. The duality framework that removes the nonnegativity hypothesis from prior fractional Sobolev results is a clear strength, and the explicit lower bound for stability of such sequences adds concrete value.

major comments (1)

[Abstract / main stability theorem] Abstract and the statement of the main stability result: the upgrade from weak decomposition to strong stability control in the non-Hilbertian distance is asserted to hold for arbitrary functions without regularity, sign, or decay assumptions, yet the control of the remainder term after decomposition is not shown to close uniformly; typical concentration-compactness arguments for integral equations require either pointwise decay at infinity or L^p tail integrability to absorb cross terms, and it is unclear whether the estimates in the relevant theorem impose an implicit tail condition that would restrict the claim to general Palais--Smale sequences.

minor comments (1)

[Introduction] The notation for the non-Hilbertian distance induced by the HLS kernel should be introduced with an explicit formula in the introduction to improve readability for readers unfamiliar with the dual formulation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The major concern regarding potential implicit tail conditions in the main stability result is addressed point-by-point below. We maintain that the tailored weak-decomposition method establishes the claimed uniformity without additional assumptions, but we will revise the manuscript to clarify the relevant estimates.

read point-by-point responses

Referee: [Abstract / main stability theorem] Abstract and the statement of the main stability result: the upgrade from weak decomposition to strong stability control in the non-Hilbertian distance is asserted to hold for arbitrary functions without regularity, sign, or decay assumptions, yet the control of the remainder term after decomposition is not shown to close uniformly; typical concentration-compactness arguments for integral equations require either pointwise decay at infinity or L^p tail integrability to absorb cross terms, and it is unclear whether the estimates in the relevant theorem impose an implicit tail condition that would restrict the claim to general Palais--Smale sequences.

Authors: We appreciate this detailed observation on the uniformity of the remainder control. In the proof of the main stability theorem (Theorem 1.2), the weak decomposition for Palais-Smale sequences is constructed using a profile decomposition at the level of measures induced by the HLS kernel. This approach does not rely on pointwise decay or a priori L^p tail integrability of the functions. The remainder term after bubble extraction is controlled uniformly by combining the quantitative stability inequality in the non-Hilbertian distance with the boundedness and energy convergence implied by the Palais-Smale condition. Cross terms between profiles are absorbed directly via the HLS inequality and the decay properties of the kernel at large distances, without imposing extra integrability on the sequence tails; the non-local interactions are handled through the duality framework developed in Section 4, which transfers estimates from the Sobolev side. Consequently, no implicit tail condition restricts the result, and the estimates hold for general Palais-Smale sequences in the natural space. To eliminate any ambiguity, we will insert a clarifying remark immediately following the theorem statement that explicitly notes the absence of regularity, sign, or decay hypotheses and outlines why the standard concentration-compactness requirements are circumvented by the method. revision: partial

Circularity Check

0 steps flagged

No circularity: new weak-decomposition method connects to external priors without self-referential reduction

full rationale

The paper introduces a weak-decomposition--strong-stability method for HLS critical points and derives stability inequalities for Palais-Smale sequences in a non-Hilbertian distance. This is presented as novel, with the duality framework explicitly linking to external results such as [26] to remove nonnegativity assumptions on fractional Sobolev critical points. No load-bearing step reduces by the paper's own equations or self-citations to a fitted input or definitional tautology; the central claims rest on the new construction rather than renaming or importing uniqueness from prior self-work. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard functional-analytic background for HLS and Sobolev inequalities; no free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

axioms (1)

standard math Standard embedding and inequality properties of Sobolev and fractional Sobolev spaces hold.
Invoked implicitly when transferring stability via duality.

pith-pipeline@v0.9.0 · 5750 in / 1243 out tokens · 37982 ms · 2026-05-20T04:48:30.741261+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.

We develop a weak-decomposition–strong-stability method tailored to the stability structure of HLS critical points... first quantitative stability result for Palais–Smale sequences... measured in a non-Hilbertian distance.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Theorem 1.1... stability inequality... d(f,M_HLS) = inf ... ||f-h||_{L^{2n/(n+2s)}}

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

30 extracted references · 30 canonical work pages

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T. Bartsch, T. Weth and M. Willem,A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator. Calc. Var. Partial Differential Equations 18(2003), 253-268

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L. Chen, G. Lu and H. Tang,Stability of Hardy-Littlewood-Sobolev inequalities with explicit lower bounds, Adv. Math.450(2024), 109778

work page 2024
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L. Chen, G. Lu and H. Tang,Optimal asymptotic lower bound for stability of fractional Sobolev inequality and the stability of Log-Sobolev inequality on the sphere, Adv. Math.479(2025), 110438

work page 2025
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L. Chen, G. Lu and H. Tang,Optimal stability of Hardy-Littlewood-Sobolev and Sobolev inequalities of arbitrary orders with dimension-dependent constants, Math. Ann. 394 (2026), no. 4, 46 pp

work page 2026
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B. Deng, L. Sun, and J. Wei,Sharp quantitative estimates of Struwe’s decomposition. Duke. Math. J., 2025

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Dolbeault, M

J. Dolbeault, M. Esteban, A. Figalli, R. Frank and M. Loss,Sharp stability for Sobolev and log- Sobolev inequalities, with optimal dimensional dependence. Camb. J. Math. 13(2), 359–430 (2025)

work page 2025
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Fang and M

Y. Fang and M. d. M. Gonz´ alez,Asymptotic behavior of Palais-Smale sequences associated with fractional Yamabe-type equations. Pacific J. Math. 278 (2015), 369-405

work page 2015
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Figalli and F

A. Figalli and F. Glaudo,On the sharp stability of critical points of the Sobolev inequality. Arch. Ration. Mech. Anal. , 237(1):201-258, 2020

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G´ erard,Description of the lack of compactness for the Sobolev imbedding

P. G´ erard,Description of the lack of compactness for the Sobolev imbedding. ESAIM Control Optim. Calc. Var. 3, (1998) 213-233

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Gross,Logarithmic Sobolev inequalities

L. Gross,Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975), no. 4, 1061-1083

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E. H. Lieb,Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118(1983), 349-374

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E. H. Lieb and M. Loss,Analysis, 2nd ed. Graduate studies in Mathematics 14, Providence, RI: American Mathematical Sociery, 2001

work page 2001
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Lu and J

G. Lu and J. Wei,On a Sobolev inequality with remainder terms. Proc. Amer. Math. Soc.128 (1999), 75-84. 30 LU CHEN, GUOZHEN LU, AND HANLI TANG

work page 1999
[26]

de Nitti and T

N. de Nitti and T. K¨ onig,Stability with explicit constants of the critical points of the fractional Sobolev inequality and applications to fast diffusion. J. Funct. Anal. 285 (2023), Article No. 110093, 30 pp

work page 2023
[27]

Palatucci and A

G. Palatucci and A. Pisante,A Global Compactness type result for Palais-Smale sequences in frac- tional Sobolev spaces. Nonlinear Anal. 117 (2015), 1-7

work page 2015
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Rosen,Minimum value forcin the Sobolev inequality∥ϕ 3∥ ≤c∥∇ϕ∥ 3

G. Rosen,Minimum value forcin the Sobolev inequality∥ϕ 3∥ ≤c∥∇ϕ∥ 3. SIAM J. Appl. Math.21 (1971), 30-32

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[29]

Struwe,A globle compactness result for elliptic equation involving the critical Sobolev exponent, Math

M. Struwe,A globle compactness result for elliptic equation involving the critical Sobolev exponent, Math. Z. 187(4)(1984), 511-517

work page 1984
[30]

Talenti,Best constants in Sobolev inequality

G. Talenti,Best constants in Sobolev inequality. Ann. Mat. Pura Appl.110(1976), 353-372. (Lu Chen)Key Laboratory of Algebraic Lie Theory and Analysis of Ministry of Edu- cation, School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, PR China Email address:chenlu5818804@163.com (Guozhen Lu)Department of Mathematics, Universi...

work page 1976

[1] [1]

Aubin,Probl` emes isoperim´ etriques et espaces de Sobolev

T. Aubin,Probl` emes isoperim´ etriques et espaces de Sobolev. J. Differ. Geometry11(1976), 573-598

work page 1976

[2] [2]

Bahouri, A

H. Bahouri, A. Cohen and G. Koch,A general wavelet-based profile decomposition in the critical embedding of function spaces. Confluentes Math.3(2011), no. 3, 387-411

work page 2011

[3] [3]

Bartsch, T

T. Bartsch, T. Weth and M. Willem,A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator. Calc. Var. Partial Differential Equations 18(2003), 253-268

work page 2003

[4] [4]

Beckner,Sobolev inequalities, the Poisson semigroup, and analysis on the sphereS n

W. Beckner,Sobolev inequalities, the Poisson semigroup, and analysis on the sphereS n. Proc. Nat. Acad. Sci. U.S.A.89(1992), no. 11, 4816-4819

work page 1992

[5] [5]

Beckner,Logarithmic Sobolev inequalities and the existence of singular integrals

W. Beckner,Logarithmic Sobolev inequalities and the existence of singular integrals. Forum Math. (3)9(1997), 303-323

work page 1997

[6] [6]

Bianchi and H

G. Bianchi and H. Egnell,A note on the Sobolev inequality. J. Funct. Anal.100(1991), no. 1, 18-24

work page 1991

[7] [7]

Brezis and E

H. Brezis and E. Lieb,Sobolev inequalities with remainder terms. J. Funct. Anal.62(1985),73-86

work page 1985

[8] [8]

Carlen,Duality and stability for functional inequalities

E. Carlen,Duality and stability for functional inequalities. Ann. Fac. Sci. Toulouse Math.26(2017), 319-350

work page 2017

[9] [9]

Ciraolo, A

G. Ciraolo, A. Figalli and F. Maggi,A quantitative analysis of metrics onR n with almost constant positive scalar curvature, with application to fast diffusion flows, Int. Math. Res. Not. 2018 (21), 6780-6797

work page 2018

[10] [10]

S. Chen, R. Frank and T. Weth,Remainder terms in the fractional Sobolev inequality. Indiana Univ. Math. J.62(2013), no. 4, 1381–1397

work page 2013

[11] [11]

W. Chen, C. Li and B. Ou,Classification of solutions for an integral equation. Commun. Pure Appl. Math. 59 (3) (2006) 330–343

work page 2006

[12] [12]

L. Chen, G. Lu and H. Tang,Sharp Stability of log-Sobolev and Moser-Onofri inequalities on the Sphere, J. Funct. Anal.285(2023), no. 5, 110022

work page 2023

[13] [13]

L. Chen, G. Lu and H. Tang,Stability of Hardy-Littlewood-Sobolev inequalities with explicit lower bounds, Adv. Math.450(2024), 109778

work page 2024

[14] [14]

L. Chen, G. Lu and H. Tang,Optimal asymptotic lower bound for stability of fractional Sobolev inequality and the stability of Log-Sobolev inequality on the sphere, Adv. Math.479(2025), 110438

work page 2025

[15] [15]

L. Chen, G. Lu and H. Tang,Optimal stability of Hardy-Littlewood-Sobolev and Sobolev inequalities of arbitrary orders with dimension-dependent constants, Math. Ann. 394 (2026), no. 4, 46 pp

work page 2026

[16] [16]

B. Deng, L. Sun, and J. Wei,Sharp quantitative estimates of Struwe’s decomposition. Duke. Math. J., 2025

work page 2025

[17] [17]

Dolbeault, M

J. Dolbeault, M. Esteban, A. Figalli, R. Frank and M. Loss,Sharp stability for Sobolev and log- Sobolev inequalities, with optimal dimensional dependence. Camb. J. Math. 13(2), 359–430 (2025)

work page 2025

[18] [18]

Fang and M

Y. Fang and M. d. M. Gonz´ alez,Asymptotic behavior of Palais-Smale sequences associated with fractional Yamabe-type equations. Pacific J. Math. 278 (2015), 369-405

work page 2015

[19] [19]

Figalli and F

A. Figalli and F. Glaudo,On the sharp stability of critical points of the Sobolev inequality. Arch. Ration. Mech. Anal. , 237(1):201-258, 2020

work page 2020

[20] [20]

G´ erard,Description of the lack of compactness for the Sobolev imbedding

P. G´ erard,Description of the lack of compactness for the Sobolev imbedding. ESAIM Control Optim. Calc. Var. 3, (1998) 213-233

work page 1998

[21] [21]

Gross,Logarithmic Sobolev inequalities

L. Gross,Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975), no. 4, 1061-1083

work page 1975

[22] [22]

Li,Remark on some conformally invariant integral equations: the method of moving spheres

Y.Y. Li,Remark on some conformally invariant integral equations: the method of moving spheres. J. Eur. Math. Soc. 6 (2) (2004) 153–180

work page 2004

[23] [23]

E. H. Lieb,Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118(1983), 349-374

work page 1983

[24] [24]

E. H. Lieb and M. Loss,Analysis, 2nd ed. Graduate studies in Mathematics 14, Providence, RI: American Mathematical Sociery, 2001

work page 2001

[25] [25]

Lu and J

G. Lu and J. Wei,On a Sobolev inequality with remainder terms. Proc. Amer. Math. Soc.128 (1999), 75-84. 30 LU CHEN, GUOZHEN LU, AND HANLI TANG

work page 1999

[26] [26]

de Nitti and T

N. de Nitti and T. K¨ onig,Stability with explicit constants of the critical points of the fractional Sobolev inequality and applications to fast diffusion. J. Funct. Anal. 285 (2023), Article No. 110093, 30 pp

work page 2023

[27] [27]

Palatucci and A

G. Palatucci and A. Pisante,A Global Compactness type result for Palais-Smale sequences in frac- tional Sobolev spaces. Nonlinear Anal. 117 (2015), 1-7

work page 2015

[28] [28]

Rosen,Minimum value forcin the Sobolev inequality∥ϕ 3∥ ≤c∥∇ϕ∥ 3

G. Rosen,Minimum value forcin the Sobolev inequality∥ϕ 3∥ ≤c∥∇ϕ∥ 3. SIAM J. Appl. Math.21 (1971), 30-32

work page 1971

[29] [29]

Struwe,A globle compactness result for elliptic equation involving the critical Sobolev exponent, Math

M. Struwe,A globle compactness result for elliptic equation involving the critical Sobolev exponent, Math. Z. 187(4)(1984), 511-517

work page 1984

[30] [30]

Talenti,Best constants in Sobolev inequality

G. Talenti,Best constants in Sobolev inequality. Ann. Mat. Pura Appl.110(1976), 353-372. (Lu Chen)Key Laboratory of Algebraic Lie Theory and Analysis of Ministry of Edu- cation, School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, PR China Email address:chenlu5818804@163.com (Guozhen Lu)Department of Mathematics, Universi...

work page 1976