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REVIEW 4 major objections 5 minor 28 references

Radial symmetry of positive solutions of an integral system associated with the reversed Stein-Weiss inequality

T0 review · 4 major / 5 minor · reviewed 2026-07-08 · glm-5.2

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Pith's one-line read Symmetry proven for reversed Stein-Weiss extremals

desk verdict Solid incremental symmetry result for reversed Stein-Weiss; one real gap in parameter scope needs checking read the letter →

arxiv 2607.05922 v1 pith:3SEPRV67 submitted 2026-07-07 math.AP

Tiantian Zhou , Yutian Lei This is my paper · ORCID
classification math.AP MSC 45G1545E1026D1545M05
keywords radialsymmetrymethodofmovingplanesreversedStein-WeissinequalityintegralsystemnegativeexponentsweightedEuler-Lagrange
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

The paper proves that any pair of positive functions (u, v) solving the Euler-Lagrange integral system associated with the reversed Stein-Weiss inequality must be radially symmetric and monotonically increasing about the origin. This system involves a kernel |x-y|^Λ with positive exponent Λ (so the kernel grows rather than decays with distance), double power-law weights |x|^α and |y|^β, and negative powers of the unknowns (v^{-p2}, u^{-p1}). The authors adapt the method of moving planes in integral form to this setting by introducing auxiliary rescaled functions that absorb the weights, then proving symmetry for those auxiliaries. The key difficulty is that the double weights create singularities near the origin and the negative exponents create non-standard growth, requiring careful gradient estimates and a multi-region narrow-band argument to push the moving plane from negative infinity to the origin.

What carries the argument

Method of moving planes in integral form, applied to weight-rescaled auxiliary functions w and s rather than to u and v directly, with gradient estimates split into near-origin, bounded-annulus, and far-field regimes.

What would settle it

A positive solution pair of system (1.12) satisfying all parameter conditions in Theorem 1.1 that is not radially symmetric would refute the claim.

Watch

Extended reading notes

Core claim

The central result is Theorem 1.1: under the parameter conditions min{p1,p2} > 1, 0 ≤ α < n/(p1-1), 0 ≤ β < n/(p2-1), and 1/(p1-1) + 1/(p2-1) ≤ (α+β+Λ)/n, every positive locally bounded solution pair of the weighted integral system (1.12) is radially symmetric and increasing about the origin. The proof works by defining w(x) = |x|^{-α/p1} u(x) and s(x) = |x|^{-β} v(x), establishing gradient bounds for these rescaled functions via asymptotic estimates, and then running the moving-plane argument on w and s. The auxiliary functions remove the weight singularities enough to apply the standard comparison framework, and a three-region decomposition (far field, compact interior, and narrow band) in

Load-bearing premise

The proof depends on asymptotic estimates and integrability properties of the solutions (Lemma 2.1) that are cited from prior work. If those estimates do not hold under the stated parameter conditions, the gradient bounds and the narrow-band argument that pushes the moving plane to the origin would fail.

Editorial extensions

If this is right

  • If the radial symmetry result holds, the extremal functions of the reversed Stein-Weiss inequality are fully classified up to scaling, which would pin down the sharp best constant in that inequality.
  • The technique of absorbing weights into auxiliary functions before running the moving-plane argument could extend to other weighted integral systems with negative exponents or reversed inequalities in settings such as the Heisenberg group.
  • Radial symmetry of extremals reduces the variational problem for the reversed Stein-Weiss best constant to a one-dimensional (radial) optimization, making numerical and analytical computation of the constant tractable.

Reading between the lines

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

  • The result suggests that the reversed Stein-Weiss inequality shares the same extremal structure as the classical Stein-Weiss and Hardy-Littlewood-Sobolev inequalities — namely, radial profiles — despite the reversal of the inequality direction and the sign change in the kernel exponent.
  • The restriction to increasing (rather than decreasing) symmetry about the origin is consistent with the reversed nature of the inequality: extremals grow at infinity rather than decay, mirroring the sign flip in the kernel.
  • The parameter condition 1/(p1-1) + 1/(p2-1) ≤ (α+β+Λ)/n appears to play the role of a subcriticality condition; the equality case might correspond to a critical threshold where symmetry could fail or additional analysis would be needed.
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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

4 major / 5 minor

Summary. The paper proves radial symmetry and monotonicity of positive solutions to the Euler-Lagrange system (1.12) associated with the reversed Stein-Weiss inequality, using the method of moving planes in integral form. The authors introduce auxiliary functions t, w, s to handle the double-weight structure and adapt the Dou-Guo-Zhu scheme previously used for the unweighted reversed HLS system. The proof proceeds in four steps: starting the moving plane from negative infinity, narrowing the band, showing the limiting plane is at the origin, and concluding radial symmetry.

Significance. The result extends the radial symmetry theory for reversed HLS-type systems to the weighted (Stein-Weiss) setting, which is a natural and non-trivial generalization. The auxiliary function technique to handle the double weights is a reasonable adaptation. The proof structure follows established methods (Chen-Li-Ou, Dou-Guo-Zhu, Liu) and the technical lemmas on differentiability and gradient estimates are carefully done with case splits based on Lambda. The result is a solid contribution to the symmetry classification literature for integral systems.

major comments (4)
  1. Lemma 2.1 is stated with 'alpha, beta, p1, p2, Lambda be positive,' requiring alpha > 0 and beta > 0, but Theorem 1.1 permits 0 <= alpha and 0 <= beta. The asymptotic estimates (2.1)-(2.3) are cited from Lemma 14 and Theorem 3 of [5]. The authors should verify and explicitly state that the estimates in [5] cover the boundary cases alpha = 0 or beta = 0, or restrict the theorem to alpha, beta > 0. If [5] does not cover these cases, the proof breaks down because Lemma 2.2, Lemma 2.3, and Steps 1-3 all depend on these estimates.
  2. Condition (1.14) allows the strict inequality 1/(p1-1) + 1/(p2-1) < (alpha+beta+Lambda)/n (subcritical case), whereas [5] establishes the reversed Stein-Weiss inequality and existence of extremal functions in the critical case (equality). The authors should clarify whether the asymptotic estimates in [5] were proved only for extremal functions at critical parameters, or for arbitrary positive solutions across both subcritical and critical regimes. If the estimates hold only at criticality, the theorem's scope should be adjusted accordingly.
  3. Step 3 (the contradiction argument showing lambda_0 = 0) uses the identity t(x) - t_{lambda_0}(x) = w(x)[|x|^{(1/p1-1)alpha} - |x_{lambda_0}|^{(1/p1-1)alpha}], which relies on the relation t(x) = |x|^{(1/p1-1)alpha} w(x) from (1.15). When alpha = 0, this factor becomes 1 and the left side vanishes by definition, while the right side of the integral identity (2.22) may still be strictly positive. The authors should verify that the contradiction argument in Step 3 remains valid for this boundary case.
  4. In Step 1, equation (3.3) computes the limit of |partial t / partial x1| / |x1|^{Lambda-1} by sending |x1| -> infinity while fixing x2,...,xn. The dominated convergence argument uses the bound |F(x,y) v^{-p2}(y) |y|^beta| <= 2 v^{-p2}(y) |y|^beta, which requires |F| <= 2 for large |x1|. The function F involves |x-y| in both numerator and denominator, and the uniform bound for a.e. y should be verified more carefully, particularly when |y| is also large. The authors should confirm this estimate or provide additional justification.
minor comments (5)
  1. The keyword 'Stein-Wiess' in the abstract should be 'Stein-Weiss'.
  2. In the proof of Lemma 2.2, Case 1, the gradient formula for t_3 states the domain of integration as R^n setminus B_R(0), but it should be B_delta(0) to match the definition of t_3.
  3. In equation (2.15), the derivation uses condition (1.14) and 0 <= beta < n/(p2-1) to deduce 1/(p1-1) < (Lambda+alpha)/(n-1). The appearance of n-1 in the denominator should be n; please verify this inequality.
  4. The notation in Case 3 of Step 2 is dense, with many sub-regions Omega_i and intermediate quantities. Some clarifying remarks or a figure showing the geometric setup of the regions would improve readability.
  5. Reference [13] is cited as 'Gilbrag, N. Trudinger'; the first author's name should be 'Gilbarg'.

Simulated Author's Rebuttal

4 responses · 0 unresolved

We thank the referee for a careful reading and for identifying several genuine gaps in the manuscript. The referee's comments are well-taken. In summary: (1) Lemma 2.1 is currently stated with alpha, beta > 0, which does not cover the boundary cases alpha = 0 or beta = 0 allowed in Theorem 1.1; we will verify whether the estimates in [5] extend to these cases and adjust the theorem or lemma accordingly. (2) The subcritical case in condition (1.14) needs clarification regarding whether the asymptotic estimates from [5] were proved only at criticality; we will verify and restrict the scope if necessary. (3) The Step 3 contradiction argument requires separate verification when alpha = 0, since the key identity degenerates; we will address this. (4) The dominated convergence bound in Step 1 (equation (3.3)) needs more careful justification, particularly for large |y|; we will provide the missing details. We are able to address all four comments, though some require verification against [5] that we commit to carrying out in the revision.

read point-by-point responses

  1. Referee: Lemma 2.1 is stated with alpha, beta, p1, p2, Lambda positive, requiring alpha > 0 and beta > 0, but Theorem 1.1 permits 0 <= alpha and 0 <= beta. The estimates (2.1)-(2.3) are cited from Lemma 14 and Theorem 3 of [5]. The authors should verify and explicitly state that the estimates in [5] cover the boundary cases alpha = 0 or beta = 0, or restrict the theorem to alpha, beta > 0.

    Authors: The referee is correct that there is a mismatch between the hypotheses of Lemma 2.1 (where alpha, beta are stated as positive) and Theorem 1.1 (where 0 <= alpha, 0 <= beta are permitted). We have examined the proofs of Lemma 14 and Theorem 3 in [5] (Chen-Liu-Lu-Tao). The asymptotic estimates there are established under the conditions 0 <= alpha < -n/q and 0 <= beta < -n/p', which include the boundary cases alpha = 0 and beta = 0. The proofs do not require alpha > 0 or beta > 0 strictly; the key integrability conditions and the existence argument go through with alpha = 0 or beta = 0. We will revise the statement of Lemma 2.1 to read 'Let 0 <= alpha, 0 <= beta, p1, p2 > 1, Lambda > 0' to match the scope of [5] and Theorem 1.1. We will also add a remark explicitly noting that the estimates in [5] cover the boundary cases. If upon further verification we find that [5] does not fully cover these cases, we will restrict Theorem 1.1 to alpha, beta > 0 as the referee suggests. revision: yes

  2. Referee: Condition (1.14) allows the strict inequality 1/(p1-1) + 1/(p2-1) < (alpha+beta+Lambda)/n (subcritical case), whereas [5] establishes the reversed Stein-Weiss inequality and existence of extremal functions in the critical case (equality). The authors should clarify whether the asymptotic estimates in [5] were proved only for extremal functions at critical parameters, or for arbitrary positive solutions across both subcritical and critical regimes.

    Authors: This is an important clarification. We have re-examined [5]. The reversed Stein-Weiss inequality (1.10) and the existence of extremal functions are established at the critical parameter, i.e., when equality holds in (1.14). The asymptotic estimates in Lemma 14 and Theorem 3 of [5] are derived for solutions of the Euler-Lagrange system (1.12) at criticality. For the subcritical case (strict inequality in (1.14)), the existence of solutions to (1.12) and their asymptotic behavior are not directly established in [5]. However, the asymptotic estimates (2.1)-(2.3) depend on the integral representation (1.12) and the integrability of the right-hand side, not on the criticality condition per se. For any positive solution of (1.12) satisfying the stated conditions, the estimates can be derived by the same arguments as in [5]. We will clarify in the revision that the estimates in Lemma 2.1 are derived for arbitrary positive solutions of (1.12) (not only extremal functions), and that the subcritical case is covered because the proof of the asymptotic estimates only uses the integral representation and the integrability conditions. If we find upon closer inspection that the subcritical case requires additional justification not available in [5], we will restrict Theorem 1.1 to the critical case. revision: partial

  3. Referee: Step 3 uses the identity t(x) - t_{lambda_0}(x) = w(x)[|x|^{(1/p1-1)alpha} - |x_{lambda_0}|^{(1/p1-1)alpha}], which relies on t(x) = |x|^{(1/p1-1)alpha} w(x) from (1.15). When alpha = 0, this factor becomes 1 and the left side vanishes by definition, while the right side of the integral identity (2.22) may still be strictly positive. The authors should verify that the contradiction argument in Step 3 remains valid for this boundary case.

    Authors: The referee has identified a genuine issue. When alpha = 0, the relation t(x) = |x|^{(1/p1-1)alpha} w(x) reduces to t(x) = w(x), and the factor |x|^{(1/p1-1)alpha} - |x_{lambda_0}|^{(1/p1-1)alpha} becomes 0, so the left side of the identity in Step 3 vanishes identically. However, the right side of (2.22) involves the kernel |x_{lambda_0} - y|^Lambda - |x - y|^Lambda and the weight difference |y_{lambda_0}|^{(1-p2)beta} - |y|^{(1-p2)beta}, which is still strictly positive when beta > 0. So the contradiction argument as written does not directly apply when alpha = 0. We will address this as follows. When alpha = 0 and beta > 0, we can work directly with the integral identity for w(x) - w_{lambda_0}(x) (which equals t(x) - t_{lambda_0}(x) since alpha = 0) and use the weight structure in the s-equation to derive the contradiction. Specifically, the symmetry w = w_{lambda_0} forces s = s_{lambda_0} via (2.23), and then the integral identity (2.22) for t (with the |y|^{(1-p2)beta} weight) yields a contradiction because the kernel and weight differences are both strictly positive. When both alpha = 0 and beta = 0, the system reduces to the unweighted reversed HLS system already handled by Liu [24], and our argument reduces to that case. We will add a separate paragraph in Step 3 covering the case alpha = 0 explicitly. revision: yes

  4. Referee: In Step 1, equation (3.3) computes the limit of |partial t / partial x1| / |x1|^{Lambda-1} by sending |x1| -> infinity while fixing x2,...,xn. The dominated convergence argument uses the bound |F(x,y) v^{-p2}(y) |y|^beta| <= 2 v^{-p2}(y) |y|^beta, which requires |F| <= 2 for large |x1|. The function F involves |x-y| in both numerator and denominator, and the uniform bound for a.e. y should be verified more carefully, particularly when |y| is also large. The authors should confirm this estimate or provide additional justification.

    Authors: The referee is right to ask for more detail here. The function F is defined as F(x,y) = |x-y|^{Lambda-1}(x_1 - y_1) / (|x_1|^{Lambda-1} |x-y|). We need to show that for sufficiently large |x_1|, |F(x,y)| <= 2 for a.e. y. Write x_1 - y_1 = x_1(1 - y_1/x_1). For |x_1| sufficiently large relative to |y| (say |x_1| > 2|y|), we have |y_1/x_1| <= |y|/|x_1| < 1/2, so |x_1 - y_1| >= |x_1|/2 and |x_1 - y_1| <= 3|x_1|/2. Also, |x - y| >= |x_1| - |y| >= |x_1|/2 and |x - y| <= |x| + |y|. For the region where |y| <= |x_1|/2, we get |F| = |x-y|^{Lambda-2} |x_1 - y_1| / |x_1|^{Lambda-1} <= (|x|+|y|)^{Lambda-2} . (3|x_1|/2) / |x_1|^{Lambda-1}. When Lambda >= 1, |x-y|^{Lambda-2} is bounded (or decays), and the ratio is bounded by a constant. When 0 < Lambda < 1, |x-y|^{Lambda-2} = 1/|x-y|^{2-Lambda}, and since |x-y| >= |x_1|/2, we get |F| <= C |x_1|^{Lambda-2} . |x_1| / |x_1|^{Lambda-1} = C. For the region where |y| > |x_1|/2, we note that v^{-p2}(y)|y|^beta is integrable by (2.1), and |F| can be bounded using |x-y| ~ |y| and |x_1 - y_1| <= |x_1| + |y| <= 3|y|, giving |F| <= C|y|^{Lambda-1}/|x_1|^{Lambda-1}, which tends to 0 as |x_1| -> infinity for fixed y, and the integrability of v^{-p2}(y)|y|^beta handles the tail. We will expand the justification in the revised manuscript to make the dominated convergence argument fully rigorous, including the case split for Lambda >= 1 and 0 < Lambda < 1, and the treatment of large |y|. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the moving-planes derivation is self-contained with standard external citations

full rationale

The paper proves radial symmetry of positive solutions to the Euler-Lagrange system (1.12) via the method of moving planes. The main external dependency is Lemma 2.1, cited from Chen et al. [5] (2018), which provides asymptotic estimates (2.1)-(2.3). These estimates are used to establish differentiability (Lemma 2.2), gradient bounds (Lemma 2.3), and to control behavior at infinity and near the origin throughout Steps 1-3. The citation [5] is to a paper by different authors (L. Chen, Z. Liu, G. Lu, C. Tao) and provides independent input — the asymptotic behavior of solutions — that is not itself the conclusion of the present paper. The moving-planes argument (Lemmas 2.2-2.4, Steps 1-4) is then carried out self-containedly: the integral representations (2.22)-(2.23) are derived directly from the system (1.12), the narrow-band estimates use the mean value theorem and dominated convergence, and the contradiction in Step 3 follows from the sign structure of the kernel. No step in the proof chain reduces to its own inputs by construction. The auxiliary functions t, w, s defined in (1.15) are transformations of u and v, not definitions that presuppose the radial symmetry conclusion. The result is derived from the structure of the integral system and standard analytical tools, not from a self-citation chain or a fitted parameter renamed as prediction.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. It relies on standard mathematical axioms and domain-specific results from prior literature.

assumptions (3)
  • domain assumption Asymptotic estimates and integrability of solutions to (1.12) as stated in Lemma 2.1, originally from Chen et al. (2018) [5].
    The entire proof of the main theorem relies on these estimates to control the behavior of solutions at infinity and near the origin.
  • standard math Differentiability theorem for integrals with parameter variables (Theorem 3.16 in [3]).
    Used to justify differentiation under the integral sign in Lemma 2.2.
  • standard math Method of moving planes in integral form.
    The primary proof technique used to establish radial symmetry.

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Cite this review

Pith. "Pith review of Radial symmetry of positive solutions of an integral system associated with the reversed Stein-Weiss inequality." pith.science (2026). https://pith.science/paper/3SEPRV67

@misc{pith2026260705922,
  author       = {Pith},
  title        = {Pith review of: Radial symmetry of positive solutions of an integral system associated with the reversed Stein-Weiss inequality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3SEPRV67}},
  note         = {Machine review of arXiv:2607.05922}
}
read the original abstract

Whether the solutions of conformal equations in the whole space are radially symmetric is an interesting topic. Chen-Li-Ou proved the radial symmetry for integral systems of the Hardy-Littlewood-Sobolev type and the Stein-Weiss type by the method of moving planes in integral form. In 2015, Dou-Zhu obtained the radial symmetry of extremal functions of the reversed Hardy-Littlewood-Sobolev inequality by the method of moving spheres, and Liu proved the radial symmetry of solutions of the Euler-Lagrange system by the method of moving planes developed by Dou-Guo-Zhu. In this paper, we also use the method of moving planes to prove the radial symmetry of positive solutions of the Euler-Lagrange system satisfied by the extremal functions of the reversed Stein-Weiss inequality established by Chen-Liu-Lu-Tao in 2018.

Discussion (0). Continue with ORCID to comment.

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