arXiv:2508.17272v2

A variation norm Carleson theorem in higher dimensions

Himali Dabhi

math.CA42B0537A46

Paper metadata

Original submission: August 24, 2025
Latest revision: July 30, 2026 · version 2
Claimed MSC: 42B05, 37A46
arXiv categories: math.CA
Paper license: arXiv non-exclusive distribution license ↗
Comments: To appear in Colloquium Mathematicum, 18 pages

Abstract

In 1971, C. Fefferman established a higher dimensional extension of the celebrated Carleson--Hunt theorem which gives pointwise almost everywhere convergence of the partial Fourier sums of functions in $L^p(\mathbb T), 1 < p < \infty.$ More precisely, Fefferman proved a maximal function bound for polygonal Fourier partial sums of functions in $L^p(\mathbb T^d), p>1.$ In this note, we extend Fefferman's maximal function bound to strong $r$ -variation norm bounds whenever $r>2$ as well as uniform $2$ -oscillation bounds. Furthermore, for functions in $L^2(\mathbb T^d)$ , we establish $r$ -variational and $2$ -oscillation bounds for partial Fourier sums over nested rectangles whenever $r>2$ .

AI-generated audit

Audit summary

Audited against arXiv v2

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsCorrect

The full set of new estimates is supported. Theorems 1.10 and 1.12 establish convergence and $r$ -variation bounds for dilates of a fixed convex polytope. Theorem 1.16 establishes the analogous $r$ -variation bound uniformly over arbitrary nested axis-parallel rectangles, strengthening the recalled maximal theorem. The printed $p$ - and $r$ -ranges match the one-dimensional inputs used in each reduction.

Theorems 1.10 and 1.12Correct

Variational estimates for convex-polytope partial Fourier integrals

Pages 4–5 · Theorems 1.10 and 1.12 · arXiv:2508.17272v2

For a convex polytope containing the origin, Theorem 1.12 proves the $r$ -variation bound for its dilated Fourier truncations when $r>2$ and $r'<p<\infty$ . Theorem 1.10 is the corresponding almost-everywhere convergence consequence, obtained in the $p$ -range shown in the paper. The variation is taken over the full ordered dilation parameter, not only dyadic scales, and the operator norm depends on the fixed polytope but is uniform over every finite increasing sequence used to define variation.

Theorem 1.16Correct

Variation bound for nested axis-parallel rectangles

Pages 5–6 · Theorem 1.16 · arXiv:2508.17272v2

For every nested sequence of axis-parallel rectangles, Theorem 1.16 bounds the $r$ -variation of the associated partial Fourier integrals on $L^p$ for $r>2$ and $r'<p<\infty$ . The estimate is uniform in the number, eccentricities, and side-length jumps of the rectangles; only dimension and the displayed exponents enter the constant. The nestedness assumption is essential and is retained in every combinatorial decomposition used by the proof.

Transference consequencesCorrect

Discrete and ergodic convergence formulations

Pages 5 and 18–20 · theorem consequences and Appendix A · arXiv:2508.17272v2

The appendix transfers the finite-parameter variation inequality before any limit is taken. Translation invariance and truncation of the input permit averaging over large boxes, and the boundary-to-volume ratio tends to zero. Because the variation seminorm is computed on a finite parameter set during transference, the argument yields the maximal, variation, and almost-everywhere convergence formulations claimed in the introduction without a circular use of convergence.

02Proofs3 reported findingsCorrect

Proofs of the polytope and rectangle theorems. A triangulation of the polytope boundary decomposes each multiplier increment into finitely many simplicial cone contributions. After a linear change of variables, Fubini reduces a cone contribution to the one-dimensional variational Carleson–Hunt estimate along one coordinate, with maximal functions controlling the transverse coordinates. Finite overlap and the fixed Jacobians keep the constants independent of the dilation sequence. Interpolation between the $L^2$ oscillation input and the maximal estimate gives exactly $r'<p<\infty$ . Passing from finite sequences to the full variation by monotone exhaustion proves the asserted almost-everywhere convergence.

Geometric decomposition and one-dimensional inputCorrect and complete

Proofs of the polytope and rectangle theorems

Pages 8–18 · Sections 2–3 and Appendix A · arXiv:2508.17272v2

A triangulation of the polytope boundary decomposes each multiplier increment into finitely many simplicial cone contributions. After a linear change of variables, Fubini reduces a cone contribution to the one-dimensional variational Carleson–Hunt estimate along one coordinate, with maximal functions controlling the transverse coordinates. Finite overlap and the fixed Jacobians keep the constants independent of the dilation sequence. Interpolation between the $L^2$ oscillation input and the maximal estimate gives exactly $r'<p<\infty$ . Passing from finite sequences to the full variation by monotone exhaustion proves the asserted almost-everywhere convergence.

Section 3Correct and complete

Nested-rectangle partition and oscillation estimate

Pages 13–18 · Section 3 · arXiv:2508.17272v2

Each change from $R_k$ to $R_{k+1}$ is partitioned according to the coordinates in which a side length crosses the relevant scale. Nestedness makes the resulting pieces disjoint in the ordered parameter and permits successive use of one-dimensional Carleson variation estimates. The square-function summation controls all coordinate patterns, and interpolation yields the stated $p$ -range without a factor depending on the number of rectangles. The terminal unbounded-side cases are obtained by monotone limits of finite rectangles.

Appendix ACorrect and complete

Finite-family transference and limiting passage

Pages 18–20 · Appendix A · arXiv:2508.17272v2

The continuous estimate is applied to compactly supported lifts of a discrete or dynamical input and averaged over translations. On interior points the lifted operators agree with the target finite family; the error is confined to a boundary layer whose relative measure vanishes. Uniformity in the chosen finite parameter set allows exhaustion and Fatou's lemma to recover the full variation. This verifies the announced transference rather than merely citing a pointwise maximal principle.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

Detailed audit reportFull reasoning, manuscript locations, and references.

Open report PDF ↗

Author response

Challenge an audit finding

Local workflow preview

A listed author may submit formal evidence that an audit is inaccurate. The response would be considered in a fresh AI re-evaluation; it would not edit the audit automatically.

Paper: arXiv:2508.17272v2
Authors listed: Himali Dabhi
Audit date: August 18, 2026

01Establish identityMatch an authenticated scholarly identity to this paper.
02Submit evidenceIdentify the finding and give a formal mathematical response.
03Re-evaluateA separate agent checks the response and records a disposition.

Email fallback for papers without a linked ORCID

A production fallback could send a one-time link only when the submitted address matches an independently maintained author-contact allowlist for this paper. MathAudit must return the same message for every address so the form cannot reveal which contacts are on that list.

Structured response preview

This form remains unavailable until production identity verification succeeds. Nothing entered here is submitted.

Audit finding challengedWhy the evaluation is inaccurateFormal evidence or repairVerifiable sources, if anyI am submitting evidence, not instructions for the reviewing agent.

This panel never establishes authorship in the local prototype. A production result should be described narrowly as an authenticated ORCID match or control of a separately allowlisted author-contact mailbox.

Audit controls

Status: Report ready
Source: arXiv v2
Prompt: Standard prompt · locked

The report is available below and in the public summary.

Review prompt

You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. For every adverse or unverified outcome, restate the exact claim and hypotheses, identify its location in the reviewed version, and give the decisive reasoning. Do not infer that a theorem is false merely because its printed proof is defective. Assign the overall statements status by the following priority: 1. Contains wrong statements, if at least one central statement is Incorrect. 2. Contains unsupported statements, if no central statement is Incorrect but at least one is Not able to verify. 3. Correct, if every central statement is Correct, apart from any typographical or minor formal findings allowed by Part 4. PART 2 — CORRECTNESS AND COMPLETENESS OF THE PROOFS Audit the proofs of the central statements and every material internal or cited input on which they depend. For each proof or proof segment, use the narrowest supported outcome: - Correct and complete: every substantive step is verifiable and no case or crucial nontrivial step is missing. - Incorrect as written: a displayed calculation, inference, lemma, or cited application is formally false or inapplicable. - Incomplete as written: a necessary substantive step or case is missing, even though no explicit contradiction has been proved. - Not able to verify: the available paper and sources are insufficient to decide whether the step is valid. Do not flag omitted routine details, compressed arguments, or “analogously” steps when the omitted reasoning is recoverable and correct. For every defect, identify its smallest exact location, explain its downstream dependency, and classify any proposed repair as one of: - Verified repair: supply enough reasoning to verify that the repair works throughout every affected use. - Plausible repair only: describe the possible repair but do not treat it as established. - No repair supplied. Assign the overall proofs status by the following priority: 1. Contains incorrect or incomplete proofs, if at least one proof is Incorrect as written or Incomplete as written. 2. Contains unverified proofs, if there is no proved incorrectness or incompleteness but at least one material proof is Not able to verify. 3. Correct, if all audited proofs are Correct and complete, apart from any typographical or minor formal findings allowed by Part 4. PART 3 — NOVELTY OF THE MAIN STATEMENTS This is a one-sided, evidence-only check. Evaluate only central results. Report a result only when a verifiable primary source, publicly available before the earliest public version of the paper that contains the result, proves the same statement up to a change of notation, or when the result follows directly and immediately from such an earlier theorem. For each non-novelty finding, provide the earlier source, its public date, the exact theorem or proposition, and an explicit hypothesis-and-conclusion mapping. A later source cannot establish non-novelty. Do not flag a result merely because an easier proof exists or because it follows through a nontrivial argument. Do not flag a result that the paper itself explicitly presents as known, a survey item, or an alternative proof. Mention only statements that meet this non-novelty threshold. Say nothing about statements for which no such evidence was found: do not certify novelty, list unsuccessful searches, or add statement-level “no prior source found” entries. If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. Do not report a gap, unsupported statement, minor formal correction, or recommendation to add an explanatory sentence, and do not mention the suppressed item in the report. This suppression does not apply to a genuine typographical error as defined below, because a typo is a literal notation defect that still requires a textual correction. For example, if

P(delta)

is proved for every sufficiently small

delta>0

and

P

is immediately monotone in the required direction, then the claim that

P(delta)

holds for every

delta>0

requires no correction; the omitted monotonicity step is exposition rather than an audit finding. 6. Immediate-consequence closure permits reconstruction of an omitted inference; it does not permit silent alteration of the printed domain, hypotheses, or quantifiers of an intermediate statement later invoked outside its stated scope. If formal validity requires changing the text of that statement or its stated range, report the appropriate finding even when the repair is easy. Thus a result stated only for

t>0

and subsequently applied at

t=0

remains a formal scope mismatch unless the endpoint case was separately established before the application. 7. Apply this mandatory gate before recording any adverse, unverified, typographical, or minor formal finding: (a) identify the exact assertion or inference alleged to be defective; (b) reconstruct the strongest conclusion following from the preceding argument after all admissible immediate formal consequences are added; (c) identify the specific defect that remains — a formally demonstrated falsehood, a concrete nontrivial unresolved proof obligation, or a genuine domain, hypothesis, quantifier, or notation mismatch requiring a textual correction; and (d) state the evidence and minimal repair. If no specific defect remains, or if the only possible addition would be explanatory, suppress the finding entirely. Use Not able to verify only when a precise, nontrivial unresolved obligation can be identified. 8. A typographical error is an incorrect symbol, index, exponent, sign, variable name, or notation mismatch whose unique intended correction is mechanically determined from the surrounding text and does not alter the mathematical argument. Report it in yellow as Typo, with the printed text, correction, exact location, and reason it is harmless. It does not lower an overall status. 9. A minor formal correction is a boundary, endpoint, or displayed parameter-range defect that can be repaired consistently in every affected statement by a local change, without a new argument, without invalidating any related result or application, and without creating another substantive defect. Report it in yellow as Minor formal correction, with the same details. It does not lower an overall status. 10. Do not use either minor category if more than one materially different repair is possible; if the repair changes the substance or scope of a central result; if an excluded case is used later; if the correction fails to propagate consistently; or if a new mathematical idea is required. In such cases use the strongest substantive classification actually supported by evidence. 11. Write every mathematical expression as valid, compilable LaTeX. Use

...

for inline mathematics and

...

for displayed mathematics, with balanced delimiters and braces. Do not leave raw LaTeX commands outside math delimiters. 12. Separate theorem truth from the correctness of the printed proof. A theorem can be Correct in Part 1 because of a fully verified independent proof while its printed proof is adverse in Part 2. Conversely, a defective proof alone never licenses an Incorrect theorem classification. Organize the report under Parts 1–3 and apply Part 4 throughout. Do not add editorial commentary, scores, rankings, recommendations, or a list of suppressed observations.

This panel is visibly marked as owner-only for the concept prototype. Authentication is not enabled locally.

A variation norm Carleson theorem in higher dimensions

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Variational estimates for convex-polytope partial Fourier integrals

Variation bound for nested axis-parallel rectangles

Discrete and ergodic convergence formulations

Proofs of the polytope and rectangle theorems

Nested-rectangle partition and oscillation estimate

Finite-family transference and limiting passage

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match