$W^{1,p}$ estimates for Schr\"odinger equation in the region above a convex graph

Ziyi Xu

REVIEW 4 major objections 4 minor 25 references

$W^{1,p}$ estimates for Schr\"odinger equation in the region above a convex graph

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves that the Neumann Schrödinger problem above a convex graph is W^{1,p}-solvable for every 1<p<∞ under B∞ potentials.

desk verdict A credible C^2 convex-graph result that does not yet prove the stated Lipschitz theorem, and whose novelty is undermined by the paper's own citation of a forthcoming Lipschitz-graph paper. read the letter →

arxiv 2411.18852 v1 pith:MWCTGBYU submitted 2024-11-28 math.AP

Ziyi Xu This is my paper · ORCID

classification math.AP MSC 35J1035J2535B45

keywords SchrödingeroperatorNeumannproblemW^{1p}estimatesconvexgraphdomainsB∞weightsFefferman-Phong-ShenmaximalfunctionreverseHölderinequalityBesovboundarydata

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 an a priori $W^{1,p}$ estimate for the Neumann problem of $-\Delta + V$ in the unbounded region above a convex graph, for potentials $V$ in the $B_\infty$ class (a reverse Hölder condition). For every $1

What carries the argument

The central machinery is a two-step reduction. A real-variable perturbation theorem (Theorem 3.1) says the global $W^{1,p}$ bound follows once every local solution of the homogeneous Neumann problem satisfies a weak reverse Hölder inequality of the form (1.5)/(3.4). The paper proves that local inequality by combining three ingredients: the Fefferman–Phong–Shen maximal function $m(x,V)$ (the scale below which $V$ is small on average), the improved Fefferman–Phong inequality, and the nonnegativity of the second fundamental form on a convex boundary, which makes an integration-by-parts identity coercive. For $1<p<2$, the argument switches to duality, using estimates for the Neumann function $N(x,y)$ and dyadic annuli in the metric of $m(x,V)$.

What would settle it

Take the cone domain $\Omega=\{t>|x'|\}$ with $V\equiv1$ and run the asserted estimate for smooth compactly supported $f$ whose support touches the ridge; alternatively, smooth the cone through convex $C^2$ graphs and watch the constants in (3.17). If the constant blows up as the approximations converge, or if (1.3) fails for some $1<p<\infty$ on the cone, the theorem as stated is false. The minimal test is whether the uniform bound survives the $C^2$-to-Lipschitz passage.

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Extended reading notes

Core claim

On its own terms, the paper establishes the following. Let $\Omega = \{t>\varphi(x')\}$ with $\varphi$ convex and $\|\nabla\varphi\|_\infty\le M$, and let $0<V\in B_\infty$ satisfy (1.2). Then the Neumann problem (1.1) is uniquely solvable in $W^{1,p}(\Omega)$ for all $1<p<\infty$, and the solution satisfies $\|\nabla u\|_{L^p(\Omega)}+\|V^{1/2}u\|_{L^p(\Omega)}\le C(\|f\|_{L^p(\Omega)}+\|g\|_{B^{-1/p,p}(\partial\Omega)})$, with $C$ depending only on $d$, $p$ and the Lipschitz character of $\Omega$, not on $V$ or the diameter. The exponent range is asserted to be sharp even for the Laplacian. The route is: a refined real-variable perturbation criterion (Theorem 3.1) turns a weak reverse Hölder inequality for local null solutions into the global $W^{1,p}$ estimate; the local inequality is proved using the improved Fefferman–Phong inequality, the Fefferman–Phong–Shen maximal function $m(x,V)$, and convexity of the boundary; and the range $1<p<2$ is completed by duality through estimates for the Neumann function.

Load-bearing premise

The proof of the key local estimate uses a $C^2$ boundary so that the curvature term in an integration-by-parts identity is nonnegative, but the theorem only assumes a Lipschitz convex graph; the paper does not provide the approximation step that would carry the estimate across that gap with uniform constants.

Editorial extensions

If this is right

The full range $1<p<\infty$ is obtained for unbounded convex graph domains, matching the sharp range for the Laplacian in bounded convex domains.
The constant in the estimate is uniform in the potential $V$ and in the domain's diameter, so the result is available for limit and homogenization arguments.
Potentials of the form $|x|^a$ with $a\ge0$ are covered, so the estimate holds for unbounded potentials, not just bounded ones.
With zero Neumann data the gradient and $V^{1/2}u$ are controlled by the $L^p$ norm of $f$ alone; with $f=0$ the gradient is controlled by the Besov norm of $g$.
Uniqueness and existence follow for every $1<p<\infty$, so the Neumann problem is well posed throughout the full range.

Reading between the lines

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

Testable extension: smooth the cone $\varphi(x')=|x'|$ by convex $C^2$ graphs and track the constant in (3.17); a uniform bound would confirm the full Lipschitz-level claim, while a blow-up would locate the missing approximation step.
The proof's ingredients are scale-invariant, so finite unions of convex graph domains or domains with bounded geometry should satisfy the same estimate by localization.
The same $m(x,V)$-based argument plausibly gives weighted versions with $V\,dx$ or $V^\alpha\,dx$ and Dirichlet analogues, since the Fefferman–Phong inequality and Neumann function bounds are the only potential-dependent inputs.
The Neumann function bounds used for the duality step suggest that related kernel estimates, for example for the heat semigroup or Riesz transforms of $-\Delta+V$ under $B_\infty$ potentials on convex epigraphs, may follow along the same lines.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The short version: this paper has a real theorem inside, but it is not exactly the theorem it states. The author proves the full-range W^{1,p} estimate for the Neumann problem with B∞ potential above C^2 convex graphs, and the proof is a credible adaptation of Geng's real-variable method and Shen's Fefferman-Phong-Shen machinery. The claimed passage from C^2 to Lipschitz convex graphs is missing, and the introduction cites a forthcoming paper [14] that appears to already cover the Lipschitz case. Those two issues are the whole story.

What is genuinely good: the C^2 argument is substantive. Lemma 3.3 is the heart, a reverse Hölder inequality for gradients of solutions, using convexity through the sign of the second fundamental form. The duality section for 1

The main gap is real: Theorem 1.1 assumes only convex Lipschitz φ, but Lemmas 3.3 and 3.4 assume C^2 boundary. The integration-by-parts identity (3.13) uses the second fundamental form, which is not defined for Lipschitz boundary. No approximation by smooth convex graphs is supplied, and it is not obvious that the constants are uniform under approximation, though they may be. The same gap carries into Section 4 via Theorem 3.4. Also, Lemma 3.3 uses the Sobolev exponent 2* = 2d/(d−2), which does not exist for d=2; the theorem claims d≥2, and no separate argument is given for d=2.

Novelty is a serious question. The introduction says [14] (submitted, by the same group) establishes W^{1,p} solvability above Lipschitz graphs. If so, every convex graph is a Lipschitz graph, and Theorem 1.1 is subsumed. The author does not state how this paper differs from [14]. The referee needs to see [14] and a clarification.

Minor issues: the constant in (1.3) is said to be independent of V, but the proof uses the B∞ constant; that should be tracked explicitly. There are typos, and the citation to [19] for Shen's Lipschitz result looks like a mis-reference, probably [20] or [21].

Who this is for: people working on Lp estimates for Schrödinger operators in rough domains. It deserves a serious referee because the C^2 result is substantial and the gap is likely repairable. I would not cite it as the Lipschitz result until the approximation is written down, but I would send it out with instructions to demand the approximation argument and a direct comparison with [14].

Referee Report

4 major / 4 minor

Summary. The paper studies the Neumann problem for the Schrödinger equation -Δu+Vu=div f in the unbounded domain Ω={t>φ(x')} above a convex graph, where V is a positive B∞ weight. Theorem 1.1 claims unique W^{1,p} solvability for all 1<p<∞ together with the estimate (1.3), including boundary data in B^{-1/p,p}(∂Ω), with constants independent of V and of the domain diameter. The proof combines a real-variable perturbation theorem, a reduction of W^{1,p} estimates to weak reverse Hölder inequalities for local solutions, proofs of those reverse Hölder inequalities using convexity and Fefferman-Phong-Shen estimates, and a duality argument for 1<p<2.

Significance. If the gaps identified below are repaired, the result would be a sharp full-range W^{1,p} estimate for Schrödinger operators with B∞ potentials in unbounded convex graph domains, extending existing results for bounded convex domains and for Lipschitz graphs. The overall strategy is sound and well matched to the problem: the reduction via the real-variable theorem in Section 3 is standard, the use of the Fefferman-Phong-Shen maximal function is appropriate, and the convexity of the boundary is exploited through the sign of the second fundamental form. The paper also gives explicit duality arguments for p<2 and for Besov boundary data. However, several load-bearing technical steps are not proved at the level of generality asserted in Theorem 1.1.

major comments (4)

[§3, Lemmas 3.3 and 3.4; Theorem 1.1] The reverse Hölder estimate is proved only for Ω with C^2 boundary: Lemma 3.3 assumes a C^2 boundary and uses the nonnegativity of the second fundamental form β in (3.13), and Theorem 3.4 repeats the C^2 assumption. Theorem 1.1, however, states the result for arbitrary convex φ with ‖∇φ‖∞≤M, which need not be C^2. No approximation by smooth convex functions is supplied, and the paper does not show that the constants in (3.12) and (3.4) are preserved under such an approximation. Since Theorem 3.4 is invoked in Lemma 4.1 and in the proof of Theorem 1.1, the estimate (1.3) is not established for the stated class of domains.
[§3, Eq. (3.15), Lemma 3.3, Theorem 1.1] The proof of Lemma 3.3 uses the Sobolev exponent 2*=2d/(d−2) in Eq. (3.15), which is defined only for d≥3. Theorem 1.1 claims the result for all d≥2, and the paper gives no separate argument for d=2. Consequently the reverse Hölder inequality (3.4) and the final estimate (1.3) are not proved in two dimensions.
[§3, Theorems 3.2 and 3.4; the approximation domains Ω_R] The hypothesis of Theorem 3.2 requires the reverse Hölder inequality (3.4) for every ball centered at x0∈∂Ω_R, which includes the artificial vertical sides of Ω_R. The proof of Lemma 3.3 is an integration-by-parts argument on ∂Ω using g·n=0 on ∂Ω and β≥0 on ∂Ω, so it does not apply to balls centered on the vertical portions of ∂Ω_R. No separate estimate for such balls is provided, and the reduction to Ω_R together with the limit R→∞ in Theorem 3.2 is therefore incomplete.
[§4, Lemma 4.2, Eq. (4.6)] The duality proof for boundary data introduces c=⨏_Ω v dx and uses the Poincaré inequality ‖v−c‖_{W^{1,q}(Ω)}≤C‖∇v‖_{L^q(Ω)}. Since Ω is unbounded, the average over Ω is not defined and this global Poincaré inequality is not available without additional decay assumptions or a bounded-domain approximation. The same issue recurs in the proof of (4.8) for the component u2. Hence the estimate (4.5) for g∈B^{-1/p,p}(∂Ω), and with it the boundary-data part of Theorem 1.1, is not proved as written.

minor comments (4)

[§3, Lemma 3.3] The statement of Lemma 3.3 is confusing: the hypothesis says u is a weak solution but the boundary condition is written for ∂v/∂n, and the cut-off ϕ is introduced only after the inequality. The proof also switches between Ω and Ω_R; the domain of the cut-off and the domain of integration should be stated consistently.
[§3, Theorem 3.1 proof] The text attributes Theorem 3.1 to [24, Theorem 3.2] but the proof line says 'See [12, Theorem 2.1]'; these references should be aligned.
[§1, paragraph on prior Schrödinger results] The sentence 'For Schrödinger equations, Z. Shen [19] obtained the W^{1,p} estimate...' cites reference [19], which is Righi and Shen (2024), for what appears to be an older result; the intended citation should be checked and corrected.
[§2, Lemma 2.5] Lemma 2.5 is stated for A∞ weights but is applied to B∞ weights; this is harmless because B∞ implies A∞, but the implication should be stated explicitly.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the reverse Hölder estimate is proved from potential-theoretic inputs, and the sole self-citation [14] is only contextual; the C^2-to-Lipschitz gap is a proof gap, not circularity.

full rationale

The target estimate (1.3) is obtained from Lemma 4.1 and Lemma 4.2, whose proofs reduce to Theorem 3.2 (real-variable sufficiency) and Theorem 3.4 (proof of the weak reverse Hölder inequality). The reverse Hölder condition (3.4) is not assumed as the output: Theorem 3.4 derives it via Lemma 3.3, the Fefferman-Phong inequality (Lemma 2.4), the boundary L-infinity estimate (Lemma 2.6), and the Neumann-function estimate (Lemma 2.7), all quoted from external work [1,12,20,21,24]. No parameter is fitted to data and no target quantity is renamed as an input. The only self-citation is [14], mentioned in the introduction as "The W^{1,p} solvability is formulated in forthcoming paper [14]"; it is never invoked in Sections 2–4 and carries no weight in the proof, so it does not make the derivation circular. The genuine weakness is a generality gap, not circularity: Lemma 3.3 and Theorem 3.4 assume Omega has C^2 boundary, whereas Theorem 1.1 states only convex phi with bounded gradient; no approximation or density argument is supplied to pass the uniform reverse Hölder estimate to merely Lipschitz convex graphs, and Lemma 3.3 uses the Sobolev exponent 2* = 2d/(d-2), leaving d=2 unaddressed. These are correctness risks that are likely repairable because the constants do not visibly depend on the second fundamental form except through its sign. Score 2 reflects only the minor non-load-bearing self-citation.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new physical entities. Its burden sits in the strong B∞ hypothesis on V, the imported Fefferman-Phong-Shen machinery, and the unstated C^2 regularity needed for the curvature argument.

assumptions (9)

domain assumption V > 0 is a B∞ weight: ||V||_{L∞(B)} ≤ C ⨍_B V dx for all balls B (equation 1.2).
Class of potentials considered; feeds Proposition 2.1, Lemma 2.5, boundary L∞ estimates, and Neumann function bounds. Examples are |x|^a with a ≥ 0.
domain assumption Ω is the region above a convex graph, with φ convex and ||∇φ||∞ ≤ M.
Convexity supplies β ≥ 0 in Lemma 3.3 and the boundary estimates; bounded slope supplies the Lipschitz character.
ad hoc to paper Ω has C^2 boundary in Lemma 3.3 and Theorem 3.4.
The second fundamental form β(g-(g·n)n, g-(g·n)n) ≥ 0 requires C^2 boundary; the passage to general Lipschitz convex graphs in Theorem 1.1 is not written.
standard math Fefferman-Phong-Shen maximal function properties: Proposition 2.1 (V ≤ C m^2), Proposition 2.2 (ψ scaling), Lemma 2.3 (doubling and comparison).
Cited from [20,21]; used throughout Sections 3 and 4 without proof.
standard math Fefferman-Phong inequality (2.5) and refined version Lemma 2.5.
Cited from [20] and [1]; used to handle V-weighted integrals in Theorem 3.4.
standard math Boundary L∞ estimate Lemma 2.6 and Neumann function estimate (2.6).
Taken from [20]; used in Theorem 3.4 and in the duality estimates for u1 and u2.
standard math Real-variable theorem 3.1 (Calderón-Zygmund type) from [12] and [24].
Used as a black box to convert local reverse Hölder into global Lp estimates.
standard math Caccioppoli inequality and Poincaré/Sobolev inequalities in Lipschitz domains.
Used in Lemma 3.3 and in (4.10); not stated or proved in the paper.
standard math Duality and trace theory: Besov space B^{-1/p,p}(∂Ω) dual pairing and trace embedding.
Used in Lemma 4.2 and in the final proof of Theorem 1.1; no details are given.

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Cite this review

Pith. "Pith review of $W^{1,p}$ estimates for Schr\"odinger equation in the region above a convex graph." pith.science (2026). https://pith.science/paper/MWCTGBYU

@misc{pith2026241118852,
  author       = {Pith},
  title        = {Pith review of: $W^1,p$ estimates for Schr\"odinger equation in the region above a convex graph},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MWCTGBYU}},
  note         = {Machine review of arXiv:2411.18852}
}

abstract

We investigate the $W^{1,p}$ estimates of the Neumann problem for the Schr\"odinger equation $-\Delta u+ V u={\rm div}(f)$ in the region above a convex graph. For any $p>2$, we obtain a sufficient condition for the $W^{1,p}$ solvability. As a result, we obtain sharp $W^{1,p}$ estimate $$\|\nabla u\|_{L^p(\Omega)}+\|V^\frac{1}{2}u\|_{L^p(\Omega)}\leq C\|f\|_{L^p(\Omega)}$$ for $1 <p<\infty$ with $d\geq2$ under the assumption that $V$ is a $B_\infty$ weight.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [14]

Geng and Z

J. Geng and Z. Xu, On the Schr¨ odinger equations withB∞ potentials in the region above a Lipschitz graph , submitted

work page

[2] [19]

Dirichlet Problems in Perforated Domains

R. Righi and Z. Shen, Dirichlet problems in perforated domains , arXiv:2402.13021, 2024

work page Pith review arXiv 2024

[3] [20]

Shen, On the Neumann problem for Schr ¨odinger operators in Lipschitz domains , Indiana Univ

Z. Shen, On the Neumann problem for Schr ¨odinger operators in Lipschitz domains , Indiana Univ. Math. J. 43 (1994), no. 1, 143–176

work page 1994

[4] [21]

, Lp estimates for Schr ¨odinger operators with certain potentials , Ann. Inst. Fourier (Grenoble) 45 (1995), no. 2, 224–251

work page 1995

[5] [1]

Auscher, On necessary and suﬃcient conditions for Lp-estimates of Riesz transforms associated to elliptic operators on Rn and related estimates , Mem

P. Auscher, On necessary and suﬃcient conditions for Lp-estimates of Riesz transforms associated to elliptic operators on Rn and related estimates , Mem. Amer. Math. Soc. 186 (2007), no. 871, xviii+75 pp

work page 2007

[6] [2]

Auscher and M

P. Auscher and M. Qafsaoui, Observations on W 1,p estimates for divergence elliptic equations with VMO coeﬃcients , Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 5 (2002), no. 2, 487–509

2002

[7] [3]

Byun, Elliptic equations with BMO coeﬃcients in Lipschitz domain s, Trans

S. Byun, Elliptic equations with BMO coeﬃcients in Lipschitz domain s, Trans. Amer. Math. Soc. 357 (2005), no. 3, 1025–1046

work page 2005

[8] [4]

Byun and L

S. Byun and L. Wang, Elliptic equations with BMO coeﬃcients in Reifenberg domai ns, Comm. Pure Appl. Math. 57 (2004), no. 10, 1283–1310

work page 2004