REVIEW 1 major objections 5 minor 19 references
A minimization problem involving a fractional Hardy-Sobolev type inequality
T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read A threshold parameter controls when minimizers exist for the fractional Hardy-Sobolev inequality.
desk verdict A solid, squarely-posed fractional analogue of Hashizume's threshold result, but the attainment proof in Proposition 3.2(i) has a repairable gap that as written leaves Theorem 1.1 incomplete. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the comparison between the local constant $\mu_{\alpha,\lambda}(\Omega)$ and the whole-space constant $\mu_\alpha$. Three pieces carry the argument. First, Lemma 2.2 shows $\mu_{\alpha,\lambda}(\Omega)$ is monotone, continuous, and bounded above by $\mu_\alpha$, using a scaled and cut-off whole-space extremal whose decay makes the test functions admissible. Second, Lemma 3.1 establishes a fractional Hardy–Sobolev inequality with an $\varepsilon$-loss, $\frac{\mu_\alpha}{1+\varepsilon} \|u\|_{s,\alpha,\Omega}^2 \le [u]_{s,\Omega}^2 + C(\varepsilon)\|u\|_{L^2(\Omega)}^2$, which prevents a minimizing sequence from vanishing at the singularity. Third, a decomposition identity for weakly convergent sequences upgrades weak convergence to strong convergence once the strict inequality below $\mu_\alpha$ holds. The sign of $\mu_{\alpha,\lambda}(\Omega)-\mu_\alpha$ is the switch: negative means attainment, zero means non-attainment.
What would settle it
Check the decay of the whole-space extremal used in Lemma 2.2: if the true bound is weaker than $C/|x|^{n-2s}$ at infinity, or if $n\le 4s$, the integral $\int_{|y|\ge 1}|y|^{-2(n-2s)}\,dy$ diverges and the test function $v_\varepsilon$ is not in $H^s(\Omega)$, so the upper-bound comparison collapses. Alternatively, compute $\mu_{\alpha,\lambda}(\Omega)$ for an explicit domain such as a ball and test numerically whether it ever equals $\mu_\alpha$; equality at some $\lambda$ confirms the nonexistence branch, while strict inequality for all $\lambda$ would mean $\lambda_*=\infty$.
Extended reading notes
Core claim
The paper's central claim is Theorem 1.1: for $0<s<1$, $n>4s$, $0<\alpha<2s$ and a bounded domain $\Omega$ with $0\in\Omega$, there exists $\lambda_*\in(0,\infty]$ such that $\mu_{\alpha,\lambda}(\Omega)$ is attained for every $0<\lambda<\lambda_*$, and if $\lambda_*<\infty$ it is not attained for every $\lambda>\lambda_*$. The mechanism is Proposition 3.2: strict inequality $\mu_{\alpha,\lambda}(\Omega)<\mu_\alpha$ forces the minimizing sequence to converge strongly in the weighted space, so a minimizer exists; equality $\mu_{\alpha,\lambda}(\Omega)=\mu_\alpha$ makes attainment impossible, because any would-be minimizer would have to beat the whole-space constant while living in a bounded domain. The threshold $\lambda_*$ is defined as the first value of $\lambda$ where the local constant meets the whole-space constant $\mu_\alpha$.
Load-bearing premise
The load-bearing premise is that the known whole-space extremal decays as $|u_0(x)|\le C/|x|^{n-2s}$ at infinity and that $n>4s$; if that decay were slower, the scaled cut-off functions used to prove $\mu_{\alpha,\lambda}(\Omega)\le\mu_\alpha$ would not be admissible, and the threshold comparison could fail.
Editorial extensions
If this is right
- For every bounded domain containing $0$, the set of parameters for which $\mu_{\alpha,\lambda}(\Omega)$ is attained is an interval $(0,\lambda_*)$, possibly the whole half-line.
- Whenever $\mu_{\alpha,\lambda}(\Omega)<\mu_\alpha$, every minimizing sequence converges strongly in the weighted Hardy–Sobolev space, so the minimizer is a genuine function in $H^s(\Omega)$.
- If equality $\mu_{\alpha,\lambda}(\Omega)=\mu_\alpha$ holds at some $\lambda$, no minimizer exists for any larger $\lambda$; the infimum can only be approached by sequences concentrating near the singularity.
- The fractional Hardy–Sobolev inequality of Lemma 3.1 supplies the quantitative control needed to rule out vanishing, so the threshold result holds for all bounded domains with $0\in\Omega$, not just for special geometries.
- The theorem leaves open whether $\lambda_*$ is finite or infinite; both alternatives occur through Corollary 2.3.
Reading between the lines
- If the whole-space extremal's decay were weaker than $|x|^{-(n-2s)}$, the cut-off construction in Lemma 2.2(1) would break down; testing the theorem under a slower decay would show whether the threshold phenomenon depends on that specific estimate.
- For the local (non-fractional) analogue, the geometry of the boundary at $0$ can decide attainability; by analogy, whether $\lambda_*$ is finite for a given $\Omega$ may depend on the shape of $\Omega$ near $0$, which the paper does not address.
- One direct extension is to compute $\mu_{\alpha,\lambda}(\Omega)$ for an explicit domain such as a ball; if equality with $\mu_\alpha$ never occurs, then $\lambda_*=\infty$ and minimizers exist for every $\lambda$, whereas a finite $\lambda_*$ would exhibit the predicted nonexistence branch.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the minimization problem for the fractional Hardy-Sobolev type constant μ_{α,λ}(Ω), defined as the infimum over u∈H^s(Ω) of the fractional seminorm plus λ∫|u|², normalized by the weighted Hardy-Sobolev mass ∫|u|^{2_{s,α}}|x|^{-α}. The main result, Theorem 1.1, states the existence of a threshold λ*∈(0,∞] such that μ_{α,λ}(Ω) is attained for every λ<λ*, and, if λ*<∞, is not attained for every λ>λ*. The proof strategy is to compare μ_{α,λ}(Ω) with the global constant μ_α on R^n: Lemma 2.2 establishes μ_{α,λ}(Ω)≤μ_α, monotonicity, continuity, and the limit as λ→0; Lemma 3.1 proves a local fractional Hardy-Sobolev inequality with a lower-order L² term; Proposition 3.2 then claims attainment when μ_{α,λ}(Ω)<μ_α and non-attainment when equality holds; Theorem 1.1 follows from a corollary of Lemma 2.2. The central claim of the paper is thus a threshold phenomenon for the attainability of this optimal constant.
Significance. If the proof is repaired, the result is a natural and clean fractional analogue of the local Hardy-Sobolev minimization problem studied by Hashizume and Ghoussoub-Kang, and it gives a precise threshold in λ for the loss of compactness caused by the interior singularity. The paper is concise and its comparison lemmas are mostly sound. A notable strength is that the upper-bound construction in Lemma 2.2 is explicitly benchmarked against the external result of Marano-Mosconi, including the polynomial decay of the global extremal, rather than relying on an ad-hoc ansatz. The main weakness is concentrated in Proposition 3.2(i): as written, the proof of strong convergence of the minimizing sequence is incomplete, and since that step is the core of the attainment claim, the paper requires a substantive but local revision.
major comments (1)
- [Section 3, Proposition 3.2(i)] The proof that the minimizing sequence converges strongly is not complete. After Brezis-Lieb, the displayed chain establishes only that the two weighted masses, one for u and one for u_k-u, have exponents summing to 1, namely lim_k [(∫|u|^{2_{s,α}}/|x|^α)^{2/2_{s,α}} + (∫|u_k-u|^{2_{s,α}}/|x|^α)^{2/2_{s,α}}] = 1. This is compatible with a nontrivial split in which both terms have positive limits. The sentence 'Since u ≠ 0, we conclude u_k → u strongly' is therefore a non sequitur. The strict inequality μ_{α,λ}(Ω)<μ_α is not used at this point, and the displayed estimate with μ_{α,λ}(Ω) in the denominator cannot rule out a split. The standard repair is to apply the global Hardy-Sobolev inequality (1.3) with constant μ_α to u and to u_k-u separately, using [u]^2+[u_k-u]^2 ≤ [u_k]^2+o(1) and (a+b)^r ≤ a^r+b^r with r=2/2_{s,α}<1. This yields μ_α ≤ μ_{α,λ}(Ω), contradicting the strict inequality assumed in Part (1). Because this step is exactly what proves attainment for λ<λ*, the proof of Theorem 1.1 is incomplete as written, although the gap is local and repairable.
minor comments (5)
- [Section 3, Lemma 3.1] In the estimate of I2, the exponent in the factor d_1^{-2α/2_{s,α}} is misprinted once as '2*_{s,α}' instead of '2_{s,α}'; the surrounding lines show the intended exponent.
- [Section 2, Lemma 2.1] The statement 'It is clear that φu∈L^2(Ω)' appears twice in the proof; the second occurrence seems intended for the seminorm finiteness and should be rephrased to avoid repetition.
- [Abstract and Introduction] The abstract speaks of 'existence of nontrivial solutions' to a minimization problem, but the paper actually proves existence and non-existence of minimizers; the wording could be aligned with the theorem statement.
- [References] Several reference entries appear to contain typographical artifacts (for example, entries [7] and [8]); the final version should ensure the bibliographic data is clean.
- [Section 2, proof of Lemma 2.2(3)] The constant test function c is used without explicitly noting that for a bounded domain the constant function belongs to H^s(Ω); a one-line justification would improve clarity.
Circularity Check
No significant circularity: the paper's inputs are independent external results and the threshold result is derived, not assumed.
full rationale
The paper's central comparison mu_{alpha,lambda}(Omega) <= mu_alpha is not derived from the target theorem; it uses the extremal u_0 and the decay bound supplied by Marano-Mosconi [18, Theorem 1.1], an external source whose assumptions (global Hardy-Sobolev minimization on R^n) do not include the bounded-domain attainment claim. Lemma 3.1 is an independent local inequality proved by applying the global mu_alpha inequality to a localized function and estimating the remaining term away from the singularity; it does not assume the target inequality. Proposition 3.2 then combines the strict comparison mu_{alpha,lambda}(Omega) < mu_alpha with Lemma 3.1 through the Br\'ezis-Lieb theorem, and the threshold lambda* is defined from the monotone continuity of mu_{alpha,lambda} in lambda, not fitted to the attainment conclusion. No step equates a prediction with an input by construction, and there are no load-bearing self-citations: the author cites no previous work of Ritorto. The reviewer's noted gap in the proof of strong convergence in Proposition 3.2(i) is a completeness or correctness concern, not a circularity, so it does not raise the circularity score.
Assumptions & free parameters
assumptions (5)
- domain assumption Existence, positivity, and decay of a minimizer u_0 for the global constant μ_α on R^n, with |u_0(x)| ≤ C/|x|^{n-2s} for |x|≥1.
- domain assumption Global fractional Hardy-Sobolev inequality on R^n with best constant μ_α: μ_α (∫ |u|^{2_{s,α}}/|x|^α dx)^{2/2_{s,α}} ≤ [u]_s^2.
- standard math Compactness of the embedding H^s(Ω) into L^p(Ω) for 1 ≤ p < 2*_s = 2n/(n-2s).
- standard math BreZis-Lieb lemma on the convergence of integrals under a.e. convergence and boundedness in L^q.
- standard math Positivity of the best fractional Sobolev constant κ_{Ω_1} for functions vanishing in Ω_1.
Cite this review
Pith. "Pith review of A minimization problem involving a fractional Hardy-Sobolev type inequality." pith.science (2026). https://pith.science/paper/DSA6MSJT
@misc{pith2026190805095,
author = {Pith},
title = {Pith review of: A minimization problem involving a fractional Hardy-Sobolev type inequality},
year = {2026},
howpublished = {\url{https://pith.science/paper/DSA6MSJT}},
note = {Machine review of arXiv:1908.05095}
}
abstract
In this work, we obtain an existence of nontrivial solutions to a minimization problem involving a fractional Hardy-Sobolev type inequality in the case of inner singularity. Precisely, for $\lambda>0$ we analyze the attainability of the optimal constant $$ \mu_{\alpha, \lambda}(\Omega):=\inf\left\{ [u]^2_{s,\Omega}+\lambda\int_{\Omega}|u|^2 \, dx \colon u\in H^s(\Omega), \, \int_{\Omega} \frac{|u(x)|^{2_{s,\alpha}}}{|x|^{\alpha}} \, dx=1 \right\}, $$ where $0<s<1, n>4s, 0<\alpha<2s$, $2_{s,\alpha}=\frac{2(n-\alpha)}{n-2s}$, and $\Omega \subset \mathbb{R}^n$ be a bounded domain such that $0\in \Omega$.
Reference graph
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