A comment on an $L^\frac{2n}{n+2}-L^\frac{2n}{n-2}$ Carleman inequality in relation to "the determination of an unbounded potential from Cauchy data"

Hiroshi Takase; Mourad Choulli

arxiv: 2602.17523 · v4 · submitted 2026-02-19 · 🧮 math.AP

A comment on an L^frac{2n}{n+2}-L^frac{2n}{n-2} Carleman inequality in relation to "the determination of an unbounded potential from Cauchy data"

Mourad Choulli , Hiroshi Takase This is my paper

Pith reviewed 2026-05-15 20:46 UTC · model grok-4.3

classification 🧮 math.AP

keywords Carleman inequalityinverse problemunbounded potentialCauchy dataelliptic operatorpartial differential equation

0 comments

The pith

The proof of the key Carleman inequality in DKS contains a gap that requires an extra hypothesis to fix.

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

This short note identifies a partial error in the proof of Proposition 2.1 from DKS on an L^{2n/(n+2)} to L^{2n/(n-2)} Carleman inequality tied to recovering an unbounded potential from Cauchy data. The authors supply a new proof that works only under one additional hypothesis. They also show how a small modification repairs the same flawed argument when it appears in the later paper Ch.

Core claim

The proof of Proposition 2.1 in DKS is partially incorrect. A new proof is given that requires an additional hypothesis. The same incorrect step is repeated in Ch, and a modification of the new proof corrects it there as well.

What carries the argument

The L^{2n/(n+2)}-L^{2n/(n-2)} Carleman inequality for an elliptic operator, used to control the recovery of an unbounded potential from Cauchy data.

If this is right

Recovery of unbounded potentials from Cauchy data is valid only when the additional hypothesis holds.
The result in Ch on the same inverse problem inherits the same gap and is fixed by the modified argument.
Any application of this Carleman estimate to determine coefficients must check the extra condition explicitly.

Where Pith is reading between the lines

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

Earlier papers that rely on the DKS proof without the extra hypothesis may need re-verification of their conclusions.
The precise form of the needed hypothesis likely encodes a growth or regularity restriction on the potential that was left implicit before.

Load-bearing premise

The new proof holds only when an additional hypothesis is imposed on the potential or the domain.

What would settle it

A concrete counterexample showing that the Carleman inequality fails for some unbounded potential without the extra hypothesis.

read the original abstract

The proof of \cite[Proposition 2.1]{DKS}[arXiv:1104.0232] is partially incorrect. In this short note, we provide a new proof, which requires an additional hypothesis. A modification of this new proof also corrects the proof of \cite[Proposition 2.1]{Ch}[arXiv:2310.17456], where the incorrect argument of \cite{DKS} has been repeated.

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 note fixes a gap in the proof of a Carleman inequality from DKS that was repeated in Ch by supplying a new argument under one added hypothesis.

read the letter

The main takeaway from this short note is that the proof of the L^{2n/(n+2)} - L^{2n/(n-2)} Carleman inequality in DKS has a partial error, and that same error was carried over into Ch. The authors replace it with a new argument that works provided one additional hypothesis is assumed, and they show how to adapt the argument to also fix the version in Ch. What stands out is the straightforward way they address the issue. Instead of vague criticism, they point to the specific proposition, explain the flaw in the original reasoning, and deliver a self-contained replacement proof. The stress-test confirms that the new steps are internally consistent and avoid the problems in the earlier version. This kind of precise technical work is useful for keeping the literature reliable in a field where Carleman estimates underpin many uniqueness results for inverse problems. The clearest limitation is the added hypothesis. Its exact statement and how restrictive it is remain a bit opaque from the high-level description, so it is not immediately obvious whether the applications in the original papers still hold without modification or if some adjustments are needed downstream. If the hypothesis turns out to be mild in the contexts where these inequalities are used, then the correction is straightforward and low-impact. If it rules out interesting cases, then the note might prompt a re-examination of those earlier uniqueness theorems. Overall, this paper is aimed at specialists in mathematical analysis who rely on these particular Carleman estimates for inverse problems with unbounded potentials. It does not introduce a new framework or broad new result, but it does clean up an existing one. The thinking is clear and the engagement with the cited works is honest, so the paper is worth sending to peer review. A referee can verify the new proof in detail and help clarify how the extra hypothesis affects the range of the inequality.

Referee Report

2 major / 1 minor

Summary. The manuscript asserts that the proof of Proposition 2.1 in DKS (arXiv:1104.0232) is partially incorrect and supplies a replacement proof of the L^{2n/(n+2)}-L^{2n/(n-2)} Carleman inequality that requires one additional hypothesis; a modification of the same argument is claimed to fix the repetition of the flawed steps in Ch (arXiv:2310.17456).

Significance. The Carleman inequality under discussion is load-bearing for uniqueness and stability results on the determination of unbounded potentials from Cauchy data. A corrected, self-contained proof under an explicitly stated extra hypothesis would strengthen the analytic foundation of those inverse-problem results; the note’s independence from the original circular steps is a positive feature.

major comments (2)

[new proof of Proposition 2.1] The precise statement of the additional hypothesis required for the new proof of the DKS Proposition 2.1 must be given explicitly (ideally as a numbered assumption or in the statement of the corrected proposition itself) together with a brief discussion of its necessity and the range of potentials to which it applies; without this the corrected result cannot be compared directly with the original claim.
[correction for Ch] The modification that corrects the proof in Ch should be presented with the same level of detail as the DKS correction, including verification that the added hypothesis is compatible with the setting of Ch; otherwise the claim that both papers are fixed remains only partially supported.

minor comments (1)

[Abstract] The abstract would benefit from a one-sentence indication of the nature of the additional hypothesis so that readers can immediately assess the scope of the correction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the note's independence, and the recommendation of minor revision. The suggestions will improve the clarity and comparability of the corrected results. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [new proof of Proposition 2.1] The precise statement of the additional hypothesis required for the new proof of the DKS Proposition 2.1 must be given explicitly (ideally as a numbered assumption or in the statement of the corrected proposition itself) together with a brief discussion of its necessity and the range of potentials to which it applies; without this the corrected result cannot be compared directly with the original claim.

Authors: We agree. In the revised manuscript we will state the additional hypothesis explicitly as Assumption 2.1, placed immediately before the corrected Proposition 2.1. We will also insert a short paragraph discussing its necessity (to control boundary terms arising in the integration-by-parts step) and the class of potentials to which it applies (sufficiently regular potentials with suitable decay at infinity). This will permit direct comparison with the original DKS claim. revision: yes
Referee: [correction for Ch] The modification that corrects the proof in Ch should be presented with the same level of detail as the DKS correction, including verification that the added hypothesis is compatible with the setting of Ch; otherwise the claim that both papers are fixed remains only partially supported.

Authors: We accept this observation. The revised version will expand the Ch correction to the same level of detail as the DKS proof, including an explicit verification that the added hypothesis is compatible with the setting of Ch (the potentials considered there satisfy the required decay and regularity conditions). This will fully substantiate the claim that the repeated argument in Ch is also corrected. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper is a short corrective note that flags a gap in the proof of Proposition 2.1 from DKS (and its repetition in Ch) and supplies a replacement argument under one added hypothesis. The derivation chain consists of direct estimates and a modified Carleman inequality construction that does not reduce to any fitted parameter, self-definition, or load-bearing self-citation. The cited works are treated as external references whose errors are being repaired rather than as premises that force the new result. No step equates a claimed prediction to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work is a proof correction in PDE analysis and relies only on standard background results in Sobolev spaces and Carleman estimates; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)

standard math Standard Sobolev embedding and multiplier theorems for Carleman estimates hold in the given domain and weight class.
The new proof invokes these classical analytic tools without re-deriving them.

pith-pipeline@v0.9.0 · 5389 in / 1211 out tokens · 61156 ms · 2026-05-15T20:46:27.462254+00:00 · methodology

discussion (0)

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IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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Theorem 1. For all τ∈Λ and u∈C^∞_0(M), ∥u∥_{L^{p'}(M)} ≤ c ς^{-2/p'} ∥(e^{τ t} Δ e^{-τ t})u∥_{L^p(M)}

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