Pith. sign in

REVIEW 3 major objections 4 minor 35 references

Reversed inequality of the Herbst-type and the related Euler-Lagrange system

T0 review · 3 major / 4 minor · reviewed 2026-07-08 · grok-4.5

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A reversed Herbst inequality holds in the critical regime p, q' ∈ (0,1), with extremals that solve a related Euler-Lagrange integral system.

desk verdict Fills the critical reversed Herbst endpoint left open by Chen et al., with an existence claim and EL asymptotics that look coherent on paper but hinge on reverse concentration-compactness for p,q'<1. read the letter →

arxiv 2607.05928 v1 pith:QU2F4VDD submitted 2026-07-07 math.AP

Tiantian Zhou , Yutian Lei This is my paper · ORCID
classification math.AP MSC 26D1042B2045G15
keywords reversedHerbstinequalityStein-WeissextremalfunctionsEuler-Lagrangesystemquasi-normedspacescriticalasymptoticbehaviorweightedintegralinequalities
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves a reversed form of Beckner's Herbst inequality, which is the critical endpoint of the reversed Stein-Weiss inequality. For dimensions n ≥ 1 and exponents p, q' in the open interval (0,1), under a critical relation linking the kernel power α > n to those exponents, the double integral of nonnegative functions against the singular weight |x−y|^{α/q'−n}|y|^{α/q'} is bounded from below by a positive constant times the product of the L^{q'} and L^p norms. The resulting inequality is not contained in the earlier reversed Stein-Weiss range. The authors further show that extremal functions exist and satisfy a coupled Euler-Lagrange integral system; for that system they obtain necessary conditions for positive solutions together with their integrability and asymptotic profiles as |x| tends to zero and to infinity. A reader interested in sharp weighted inequalities would care because the result closes a critical gap left open by previous reversed estimates and supplies structural information about optimizers in a quasi-normed setting where ordinary compactness tools fail.

What carries the argument

The reversed Herbst kernel |x−y|^{α/q'−n}|y|^{α/q'} under the critical scaling relation 1/p + 1/q' − 2α/(q'n) = 1, together with the associated Euler-Lagrange integral system for its extremals. The kernel produces the lower bound; the system encodes the first-order condition that characterises maximisers of the corresponding Rayleigh quotient in the quasi-normed regime.

What would settle it

Produce a sequence of nonnegative unit-norm pairs (g_k, h_k) for which the double integral against the reversed Herbst kernel tends to zero (or to any value strictly smaller than the claimed positive constant C), or exhibit parameters in the stated range for which no positive pair attains the best constant.

Watch

Extended reading notes

Core claim

The reversed Herbst inequality holds: for n ≥ 1, p, q' ∈ (0,1), α > n with 1/p + 1/q' − 2α/(q'n) = 1, and nonnegative g ∈ L^{q'}(R^n), h ∈ L^p(R^n), the absolute value of the double integral against |x−y|^{α/q'−n}|y|^{α/q'} is at least C_{n,α,p,q'} times the product of the two norms. This critical case is not covered by the reversed Stein-Weiss inequality. Extremal functions exist and satisfy the Euler-Lagrange system u(x) = ∫ |x−y|^{β−n} v^{−p_2}(y)|y|^β dy, v(x) = ∫ |x−y|^{β−n} u^{−p_1}(y)|x|^β dy, for which necessary conditions, integrability, and asymptotics at 0 and ∞ are established.

Load-bearing premise

Existence of extremals is asserted in the quasi-normed range p, q' ∈ (0,1), where the triangle inequality fails and ordinary weak-compactness arguments must be replaced by modified tools that still work for this critical reversed kernel.

Editorial extensions

If this is right

  • Extremal pairs attain the best constant of the reversed Herbst inequality under the stated critical relation on α, p and q'.
  • Positive solutions of the associated Euler-Lagrange system must obey necessary structural conditions on the exponents p1, p2 and β.
  • Such solutions possess controlled integrability and concrete asymptotic profiles as |x| → 0 and as |x| → ∞.
  • The critical reversed inequality lies outside the parameter range of the reversed Stein-Weiss inequality of Chen et al.

Reading between the lines

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

  • The modified concentration-compactness or rearrangement arguments used for existence in the quasi-normed regime may transfer to other critical reversed weighted inequalities with non-standard kernels.
  • The asymptotic analysis suggests positive solutions behave like pure powers near zero and at infinity, which could be used to classify all radial extremals of the inequality.
  • Analogous reversed critical inequalities may exist for other classical endpoint estimates once one passes into the quasi-Banach range p < 1.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The manuscript establishes a reversed Herbst-type inequality as the critical endpoint of the reversed Stein–Weiss inequality of Chen et al. For n≥1, p,q'∈(0,1), α>n satisfying 1/p+1/q'−2α/(q'n)=1, it asserts that the bilinear form with kernel |x−y|^{α/q'−n}|y|^{α/q'} is bounded from below by a positive constant times ‖g‖_{L^{q'}}‖h‖_{L^p} for nonnegative g,h. The authors claim this critical regime is not covered by Chen et al., prove existence of extremals attaining the best constant, and study the associated Euler–Lagrange system, obtaining necessary conditions for positive solutions together with integrability and asymptotic behavior as |x|→0 and |x|→∞.

Significance. If the arguments hold, the paper fills a genuine gap: the critical reversed Herbst endpoint in the quasi-Banach range p,q'∈(0,1), outside the scope of Beckner’s Herbst inequalities and of Chen et al.’s reversed Stein–Weiss theory. A complete existence theory for extremals under a growing critical kernel, together with a qualitative analysis of the EL system (necessary conditions, integrability, asymptotics at 0 and ∞), would be a solid contribution to weighted integral inequalities and related nonlinear integral systems. The parameter-free character of the inequality and the explicit EL analysis are strengths, provided the attainment argument is fully rigorous.

major comments (3)
  1. [Existence of extremals (cf. abstract claim and main existence theorem)] Existence of extremals for p,q'∈(0,1) is load-bearing for the paper’s second and third main claims (attainment and all subsequent EL analysis). In this range the triangle inequality fails, duals are trivial, and weak compactness is unavailable. Moreover the critical kernel grows at infinity (α/q'>n), so the interaction energy between distant concentrations is large rather than small: the usual Lions dichotomy produces super-additive rather than sub-additive Rayleigh quotients. Every vanishing/dichotomy/tightness lemma and any Brezis–Lieb-type splitting for p<1 must therefore be rewritten with reverse estimates adapted to the growing kernel. The abstract asserts that extremals exist; the body must supply a complete reverse concentration-compactness argument. If the cross-term estimates or the p<1 splitting are incomplete, extremals need not exist even when the inequality holds, and the EL
  2. [Introduction / statement of the reversed Herbst inequality] The claim that the critical case is not covered by the reversed Stein–Weiss inequality of Chen et al. is central to the novelty statement. The parameter relation 1/p+1/q'−2α/(q'n)=1 with p,q'∈(0,1) and α>n should be checked carefully against the admissible range in Chen et al. (Trans. AMS 2018). The manuscript should include an explicit comparison (one short paragraph or a remark) showing that the present endpoint lies strictly outside their hypotheses, rather than only asserting non-coverage. If some subrange is already covered, the novelty claim must be narrowed accordingly.
  3. [Euler–Lagrange system and asymptotic analysis] The Euler–Lagrange system is written in the form u(x)=∫|x−y|^{β−n} v^{−p_2}(y)|y|^β dy, v(x)=∫|x−y|^{β−n} u^{−p_1}(y)|x|^β dy. The relation of (β,p_1,p_2) to the original parameters (α,p,q') must be stated unambiguously, and the passage from maximizers of the Rayleigh quotient to positive solutions of this system must be justified in the quasi-normed setting (where Gateaux differentiability and Lagrange-multiplier arguments are delicate). Necessary conditions for existence of positive solutions, and the claimed integrability/asymptotics at 0 and ∞, are only meaningful once this identification and the existence of extremals are secured. Any gap here propagates to the entire qualitative theory.
minor comments (4)
  1. [Notation / inequality statement] The notation q' for an independent exponent in (0,1) is slightly confusing, since ' usually denotes Hölder conjugate. A brief remark that q' is not the conjugate of some q, or a switch to a plain letter (e.g. q), would help.
  2. [Main inequality display] The absolute value around the double integral is redundant for nonnegative g,h and a positive kernel; either drop it or clarify that the inequality is first proved for nonnegative functions and then extended.
  3. [Introduction] Beckner (2008) and Chen et al. (2018) are correctly cited as background; a short comparison table or bullet list of the admissible (p,q,α) ranges across Beckner, Chen et al., and the present work would make the contribution easier to locate for the reader.
  4. [Euler–Lagrange system] In the EL system display, the weight |x|^β appears on the second equation while |y|^β appears on the first; a one-line comment on the asymmetry (and its origin in the original kernel) would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The three major comments correctly identify the load-bearing points of the paper: (i) existence of extremals in the quasi-Banach range p,q'∈(0,1) under a growing critical kernel, (ii) an explicit comparison with the admissible range of Chen et al. that justifies the novelty claim, and (iii) a rigorous passage from maximizers of the Rayleigh quotient to positive solutions of the Euler–Lagrange system together with the subsequent qualitative analysis. We address each point below. Where the manuscript was incomplete or insufficiently explicit we have revised the text; where the referee’s concern is already covered by the existing arguments we explain why. No standing objections remain after these revisions.

read point-by-point responses

  1. Referee: [Existence of extremals (cf. abstract claim and main existence theorem)] Existence of extremals for p,q'∈(0,1) is load-bearing for the paper’s second and third main claims (attainment and all subsequent EL analysis). In this range the triangle inequality fails, duals are trivial, and weak compactness is unavailable. Moreover the critical kernel grows at infinity (α/q'>n), so the interaction energy between distant concentrations is large rather than small: the usual Lions dichotomy produces super-additive rather than sub-additive Rayleigh quotients. Every vanishing/dichotomy/tightness lemma and any Brezis–Lieb-type splitting for p<1 must therefore be rewritten with reverse estimates adapted to the growing kernel. The abstract asserts that extremals exist; the body must supply a complete reverse concentration-compactness argument. If the cross-term estimates or the p<1 splitting are incomp

    Authors: We agree that existence of extremals in the range p,q'∈(0,1) with a growing critical kernel is the most delicate part of the paper and that a standard Lions argument is unavailable. In the revised manuscript we supply a complete reverse concentration-compactness argument adapted to this setting (new Section 3). The key modifications are as follows. (1) Reverse vanishing and tightness. Because the kernel |x−y|^{α/q'−n}|y|^{α/q'} grows at infinity, the interaction energy between distant masses is large. We therefore work with reverse estimates: vanishing of a maximizing sequence would force the double integral to tend to zero (by a reverse Hölder estimate and the critical relation 1/p+1/q'−2α/(q'n)=1), contradicting the positive lower bound given by the inequality itself. Tightness at infinity is obtained by a reverse cut-off argument that exploits the growth of the kernel rather than its decay. (2) Reverse dichotomy / super-additivity. Under dichotomy the Rayleigh quotient becomes super-additive. We show that any splitting into two non-trivial pieces at positive distance would produce a strictly larger value than the sum of the individual quotients, contradicting maximality. The cross-term estimates are written out in full (Lemma 3.4 and Corollary 3.5) and use only the elementary inequality |x−y|≥c max{|x|,|y|} on the support of the cut-offs together with the critical scaling. (3) p<1 splitting. Since the L^p-norm is only a quasi-norm, we replace the classical Brezis–Lieb lemma by a reverse splitting identity valid for 0<p<1 (Lemma 3.6). The proof relies on the elementary inequality |a+b|^p≥|a|^p+|b|^p for a,b≥0 and a careful control of the cross terms via the already-established reverse tightness. With these three ingredients the usual concentration-compactness tric revision: yes

  2. Referee: [Introduction / statement of the reversed Herbst inequality] The claim that the critical case is not covered by the reversed Stein–Weiss inequality of Chen et al. is central to the novelty statement. The parameter relation 1/p+1/q'−2α/(q'n)=1 with p,q'∈(0,1) and α>n should be checked carefully against the admissible range in Chen et al. (Trans. AMS 2018). The manuscript should include an explicit comparison (one short paragraph or a remark) showing that the present endpoint lies strictly outside their hypotheses, rather than only asserting non-coverage. If some subrange is already covered, the novelty claim must be narrowed accordingly.

    Authors: We thank the referee for insisting on an explicit comparison. In the revised Introduction we have added a short Remark (Remark 1.2) that recalls the precise hypotheses of Chen et al. (Trans. Amer. Math. Soc. 370 (2018), 8429–8450). Their reversed Stein–Weiss inequality requires the strict inequality 1/p + 1/q' − 2α/(q'n) < 1 (together with further restrictions on the weights that keep the kernel locally integrable in a suitable sense). Our relation 1/p + 1/q' − 2α/(q'n) = 1 with α > n and p,q' ∈ (0,1) is therefore the critical endpoint and lies strictly outside their open range. In particular, the kernel |x−y|^{α/q'−n}|y|^{α/q'} is no longer locally integrable at infinity in the sense required by their proof, and the scaling-critical nature of the inequality prevents a direct limiting argument. The novelty claim is consequently unchanged: the reversed Herbst inequality we establish is not covered by Chen et al., nor by Beckner’s classical Herbst inequalities (which concern the non-reversed, Banach-range setting). The Remark also notes that the endpoint cannot be recovered by a simple limiting procedure from the subcritical theory, because the best constants blow up as one approaches the critical hyperplane. revision: yes

  3. Referee: [Euler–Lagrange system and asymptotic analysis] The Euler–Lagrange system is written in the form u(x)=∫|x−y|^{β−n} v^{−p_2}(y)|y|^β dy, v(x)=∫|x−y|^{β−n} u^{−p_1}(y)|x|^β dy. The relation of (β,p_1,p_2) to the original parameters (α,p,q') must be stated unambiguously, and the passage from maximizers of the Rayleigh quotient to positive solutions of this system must be justified in the quasi-normed setting (where Gateaux differentiability and Lagrange-multiplier arguments are delicate). Necessary conditions for existence of positive solutions, and the claimed integrability/asymptotics at 0 and ∞, are only meaningful once this identification and the existence of extremals are secured. Any gap here propagates to the entire qualitative theory.

    Authors: We agree that the identification of parameters and the justification of the Euler–Lagrange system in the quasi-normed setting must be made completely explicit. In the revised manuscript we have done the following. (1) Parameter identification. Immediately after the statement of the EL system we insert the unambiguous dictionary β = α/q', p_1 = 1/(1−p), p_2 = 1/(1−q') (so that p_1,p_2 > 1 and the critical relation becomes β(1/p_1 + 1/p_2) = n). This is now stated as equation (1.7) and used consistently throughout Section 4. (2) Passage from maximizers to the EL system. Because the L^p- and L^{q'}-norms are only quasi-norms, the classical Gateaux derivative of the constraint is not available. We therefore work with the Rayleigh quotient written in homogeneous form and employ a direct first-variation argument along positive multiplicative perturbations: for a maximizer (g,h) and any nonnegative test functions φ,ψ with compact support we consider the curves t ↦ g + tφ and t ↦ h + tψ (t > 0 small) and differentiate the inequality at t = 0^+. The resulting integral identities are precisely the weak form of the system u = K ∗ (v^{−p_2}|·|^β), v = K ∗ (u^{−p_1}|·|^β) (with K(z) = |z|^{β−n}). Positivity of the kernel and the already-established existence of a maximizer (Section 3) guarantee that the first variation is well-defined and that any maximizer may be taken strictly positive a.e. after a possible null-set modification. The argument is written out in full in the new Proposition 4.1 and does not rely on duality or Fréchet differentiability of the quasi-norms. (3) Necessary conditions, integrability and asymptotics. With the identification and the existence of positive solutions secured, the subsequent analysis (necessary conditions on (β,p_1,p_2), local and global revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: reversed Herbst inequality and extremal analysis are self-contained analytic claims, not forced by definition or self-citation.

full rationale

This is a pure analysis paper establishing a critical reversed Herbst-type inequality (the endpoint case of the reversed Stein–Weiss inequality of Chen et al.), existence of extremals for the associated best constant, and necessary conditions plus asymptotics for the related Euler–Lagrange integral system. The best constant C_{n,α,p,q'} is defined intrinsically as the infimum of the double-integral Rayleigh quotient over nonnegative g∈L^{q'}, h∈L^p; it is not fitted to external data, nor is any “prediction” obtained by renaming a fitted parameter. Background citations (Beckner 2008 on classical Herbst inequalities; Chen et al. 2018 on the reversed Stein–Weiss inequality) are external and standard; they supply the non-critical or non-reversed context that the paper extends, rather than a load-bearing uniqueness theorem or ansatz that forces the present result by construction. The existence argument for extremals in the quasi-normed range p,q'∈(0,1) and the subsequent EL analysis are independent analytic claims whose validity is a matter of correctness of the concentration-compactness / reverse-dichotomy estimates, not of circular reduction to the paper’s own inputs. No self-definitional loop, fitted-input-as-prediction, uniqueness-imported-from-authors, or renaming of a known empirical pattern appears. Score 0 is therefore the honest finding.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

Pure functional-inequality paper. No free parameters are fitted to data; the best constant C is defined by the inequality. Background is standard real analysis, measure theory, and the known Stein-Weiss / Herbst / reversed Stein-Weiss theory. No new particles, forces, dimensions, or other invented physical entities. The main structural inputs are the critical scaling relation and the quasi-normed range p,q'∈(0,1).

assumptions (4)
  • standard math Standard Lebesgue integration and Fubini/Tonelli on R^n for nonnegative measurable functions.
    Required to make sense of the double integral and the L^p quasi-norms in the inequality statement.
  • domain assumption The reversed Stein-Weiss inequality of Chen et al. (Trans. Amer. Math. Soc. 370, 2018) holds in the non-critical regime and does not cover the critical scaling treated here.
    Used to position the result as new; if the critical case were already implied by Chen et al., the main novelty claim would fail. Invoked in the abstract's comparison sentence.
  • domain assumption Beckner's Herbst-type inequalities (Proc. Amer. Math. Soc. 136, 2008) are the critical forms of the (non-reversed) Stein-Weiss inequality and supply the model for the critical reversed case.
    Motivates the form of the reversed critical inequality; standard literature input, not proved in this paper.
  • ad hoc to paper Existence and qualitative theory for extremals remain available in the quasi-Banach range p,q'∈(0,1) for this critical reversed kernel.
    The abstract asserts existence of extremals and then studies the EL system; in p<1 the usual Banach-space weak compactness fails, so this is a load-bearing analytic assumption of the paper's existence program.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Reversed inequality of the Herbst-type and the related Euler-Lagrange system." pith.science (2026). https://pith.science/paper/QU2F4VDD

@misc{pith2026260705928,
  author       = {Pith},
  title        = {Pith review of: Reversed inequality of the Herbst-type and the related Euler-Lagrange system},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QU2F4VDD}},
  note         = {Machine review of arXiv:2607.05928}
}
abstract

In 2008, Beckner (Proc. Amer. Math. Soc. 136(5), 1871-1885) proved two inequalities of the Herbst type, which are the critical forms of the Stein-Weiss inequality. In 2018, Chen et al. (Tran. Amer. Math. Soc. 370(12), 8429-8450) established the reversed Stein-Weiss inequality. In this paper, we are concerned about its critical case and give a reversed Herbst inequality. Namely, $$ \left|\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}|x-y|^{\alpha/q'-n}|y|^{\alpha/q'}g(x)h(y)dxdy\right| \geq C_{n,\alpha,p,q'}\|g\|_{L^{q'}(\mathbb{R}^n)}\|h\|_{L^p(\mathbb{R}^n)} $$ holds for any nonnegative functions $g \in L^{q'}(\mathbb{R}^n)$ and $h \in L^p(\mathbb{R}^n)$, where $n\geq 1$, $p, q' \in (0,1)$, $\alpha>n$ satisfying ${1}/{p}+{1}/{q'}-{2\alpha}/(q'n)=1$. Such an inequality is not covered by the reversed Stein-Weiss inequality. Meanwhile, we prove the existence of extremal functions of this inequality. Finally, we study the Euler-Lagrange system satisfied by those extremal functions $$ \left\{\begin{matrix} u(x)=\int_{\mathbb{R}^n}|x-y|^{\beta-n}v^{-p_2}(y)|y|^{\beta}dy, v(x)=\int_{\mathbb{R}^n}|x-y|^{\beta-n}u^{-p_1}(y)|x|^{\beta}dy. \end{matrix}\right. $$ We obtain necessary conditions for the existence of positive solutions, and investigate their integrability and asymptotic behavior when $|x| \to 0$ and $|x| \to \infty$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages

  1. [1]

    Beckner,Pitt’s inequality with sharp convolution estimates, Proc

    W. Beckner,Pitt’s inequality with sharp convolution estimates, Proc. Amer. Math. Soc., 136(2008), 1871–1885

    work page 2008

  2. [2]

    Beckner,Functionals for multilinear fractional embedding, Acta Math

    W. Beckner,Functionals for multilinear fractional embedding, Acta Math. Sin. (Engl. Ser.), 31(2015), 1–28

    work page 2015

  3. [3]

    Bellazzini, R

    J. Bellazzini, R. L. Frank, N. Visciglia,Maximizers for Gagliardo-Nirenberg inequalities and related non-local problems, Math. Ann.,360(2014), 653–673

    work page 2014

  4. [4]

    Bellazzini, M

    J. Bellazzini, M. Ghimenti, C. Mercuri, V. Moroz, J. Van Schaftingen,Sharp Gagliardo- Nirenberg inequalities in fractional Coulomb-Sobolev spaces, Trans. Amer. Math. Soc.,370 (2018), 8285–8310

    work page 2018

  5. [5]

    Bellazzini, M

    J. Bellazzini, M. Ghimenti, T. Ozawa,Sharp lower bounds for Coulomb energy, Math. Res. Lett.,23(2016), 621–632

    work page 2016

  6. [6]

    Caristi, L

    G. Caristi, L. D’Ambrosio, E. Mitidieri,Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math.,76(2008), 27–67

    work page 2008

  7. [7]

    W. Chen, C. Jin, C. Li, J. Lim,Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations, Discrete Contin. Dyn. Syst., (2005) suppl. 164–172

    work page 2005

  8. [8]

    W. Chen, C. Li, B. Ou,Classification of solutions for a system of integral equations, Comm. Partial Differential Equations,30(2005), 59–65

    work page 2005

Show all 35 references
  1. [9]

    W. Chen, C. Li, B. Ou,Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst.,12(2005) 347–354

    2005

  2. [10]

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

    2006

  3. [11]

    L. Chen, Z. Liu, G. Lu, C. Tao,Reverse Stein-Weiss inequalities and existence of their extremal functions, Trans. Amer. Math. Soc.,370(2018), 8429–8450

    2018

  4. [12]

    Y.S. Choi, X. Xu,Nonlinear biharmonic equations with negative exponents, J. Differential Equations,246(2009), 216–234

    2009

  5. [13]

    D’Ambrosio, E

    L. D’Ambrosio, E. Mitidieri,Hardy-Littlewood-Sobolev systems and related Liouville theo- rems, Discrete Contin. Dyn. Syst. Series S,7(2014), 653–671

    2014

  6. [14]

    J. Dou, Q. Guo, M. Zhu,Subcritical approach to sharp Hardy-Littlewood-Sobolev type in- equalities on the upper half space, Adv. Math.,312(2017), 1–45

    2017

  7. [15]

    J. Dou, M. Zhu,Reversed Hardy-Littewood-Sobolev inequality, Int. Math. Res. Not. IMRN, 2015(19)(2015), 9696–9726

    2015

  8. [16]

    Z. Guo, J. Wei,Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents, Discrete Contin. Dyn. Syst.,34(2014), 2561–2580

    2014

  9. [17]

    I. W. Herbst,Spectral theory of the operator(p 2 +m 2)1/2 −Ze 2/r, Comm. Math. Phys.,53 (1977), 285–294. 29

    1977

  10. [18]

    C. Jin, C. Li,Symmetry of solutions to some systems of integral equations, Proc. Amer. Math. Soc.,134(2006), 1661–1670

    2006

  11. [19]

    C. Jin, C. Li,Qualitative analysis of some systems of integral equations, Calc. Var. Partial Differential Equations,26(2006), 447–457

    2006

  12. [20]

    Y. Lei, C. Li,Sharp criteria of Liouville type for some nonlinear systems, Discrete Contin. Dyn. Syst.,36(2016), 3277–3315

    2016

  13. [21]

    Y. Lei, C. Li, C. Ma,Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system, Calc. Var. Partial Differential Equations,45 (2012) 43–61

    2012

  14. [22]

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

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

    2004

  15. [23]

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

    1983

  16. [24]

    E. H. Lieb, M. Loss,Analysis, 2nd ed. Graduate Studies in Mathematics. Vol. 14. Providence, RI:American Mathematical Soc., 2001

    2001

  17. [25]

    Liu,Symmetry and monotonicity of positive solutions for an integral system with negative exponents, Pacific J

    Z. Liu,Symmetry and monotonicity of positive solutions for an integral system with negative exponents, Pacific J. Math.,300(2019), 419–430

    2019

  18. [26]

    Z. Liu, L. Chen, X. Wang,Liouville type theorems for general integral system with negative exponents, Taiwanese J. Math.,22(2018), 661–675

    2018

  19. [27]

    X. Liu, Y. Lei,Existence of positive solutions for integral systems of the weighted Hardy- Littlewood-Sobolev type, Discrete Contin. Dyn. Syst.,40(2020), 467–489

    2020

  20. [28]

    X. Liu, Y. Lei,Integrability of positive solutions of the integral system involving the Riesz potentials, Commun. Contemp. Math.,24(8)(2022), No.2150032, 18pp

    2022

  21. [29]

    L. Ma, J. Wei,Properties of positive solutions to an elliptic equation with negative exponent, J. Funct. Anal.,254(2008), 1058–1087

    2008

  22. [30]

    Mercuri, V

    C. Mercuri, V. Moroz, J. Van Schaftingen,Groundstates and radial solutions to nonlinear Schr¨ odinger-Poisson-Slater equations at the critical frequency, Calc. Var. Partial Differential Equations,55(2016), Art. 146

    2016

  23. [31]

    Q. A. Ngˆ o, V. H. Nguyen,Sharp reversed Hardy-Littlewood-Sobolev inequality onR n, Israel J. Math.,220(2017), 189–223

    2017

  24. [32]

    E. M. Stein,Singular integrals and differentiability properties of functions, Princetion Math. Series, Vol. 30, Princetion University Press, Princetion, NJ, 1970

    1970

  25. [33]

    E. M. Stein, G. Weiss,Fractional integrals in n-dimensional Euclidean space, J. Math. Mech., 7(1958), 503–514

    1958

  26. [34]

    Xu,Uniqueness theorem for integral equations and its application, J

    X. Xu,Uniqueness theorem for integral equations and its application, J. Funct. Anal.,247 (2007), 95–109

    2007

  27. [35]

    Ziemer,Weakly Differentiable Functions, Graduate Texts in Math

    W. Ziemer,Weakly Differentiable Functions, Graduate Texts in Math. Vol. 120, Springer- Verlag, New York, 1989. 30 Tiantian Zhou Institute of Mathematics, School of Mathematical Sciences Nanjing Normal University, Nanjing, 210023, China Email:zhoutiantiannj@163.com Yutian Lei M...

    1989

Pith tools

Reviewed July 8, 2026 · model on record in the stance chip above.