Harmonic analysis and oscillatory integral estimates

Fourier restriction conjecture

Collaboration beta

The conjecture asks exactly when the Fourier transform of an ambient function can be meaningfully restricted to a curved surface. The source focuses on a paraboloid model and does not claim a proof.

| \hat{F} |_{S} |_{L^{q} (S)} ≲ | F |_{L^{p} (ℝ^{n})}, p' > \frac{2 n}{n - 1}, \frac{n - 1}{q} \geq \frac{n + 1}{p'} .

Known results and sources

Fourier restriction asks when ambient oscillations admit controlled traces on curved hypersurfaces.

Research problem

Exact mathematical statement

Let $S\subset\mathbb R^n$ be a compact smooth hypersurface with nonzero Gaussian curvature and surface measure $d\sigma$ . The sharp restriction conjecture predicts

\|\widehat F|_S\|_{L^q(S)}\lesssim\|F\|_{L^p(\mathbb R^n)}

in the Knapp-compatible range

p'>\frac{2n}{n-1},\qquad \frac{n-1}{q}\ge\frac{n+1}{p'}.

Equivalently, for $Eg(x)=\int_S e^{ix\cdot\omega}g(\omega)\,d\sigma(\omega)$ , the dual extension estimate is $\|Eg\|_{L^{p'}(\mathbb R^n)}\lesssim\|g\|_{L^{q'}(S)}$ . Endpoint conventions and surface hypotheses remain part of the statement. The source's active case is the paraboloid $P^2\subset\mathbb R^3$ , and it explicitly does not claim a proof.

Problem infographic

Problem at a glance

A landscape explainer maps an ambient function to its Fourier transform on a compact smooth hypersurface with nonzero Gaussian curvature, shows the sharp exponent inequalities, and lists the active paraboloid interfaces as open. — The exact sharp-range question is distinct from the source's incomplete paraboloid packet-assembly route.

Current mathematical picture

Where work on Fourier restriction conjecture stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureAll-visited-ball square summation

The source records a no-go theorem and replaces the assignment with bounded endpoint truncation. Endpoint-selected packet families at the second persistence scale remain the active nonterminal framework.

Route status · Narrowed route

Main reductionCurrent reduction

For the active P^2 case, an imported trilinear characterization is combined with polynomial partitioning, a second-persistence stopping rule, and separate nonterminal and terminal packet estimates.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeProve the nonresonant two-endpoint vector-valued TT-star estimate uniformly in persistence scale and determinant shell.Task status · Ready to work on

Work mapped so far

Fourier restriction conjecture in numbers

2.1kretained lines of mathematical investigation2,105 in the current working snapshot

Argument development: 1,849 · 88%
Explored or eliminated routes: 18 · 1%
Computational analysis: 4 · 0%
Open obligations: 104 · 5%
Definitions and setup: 130 · 6%

6selected mapped statements2routes investigated6open questions6contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byElias M. SteinStatus checkedAug 14, 2026

Source recordedAug 14, 2026

Recommended next task

Prove the nonresonant two-endpoint vector-valued TT-star estimate uniformly in persistence scale and determinant shell.

Suggested move: Define endpoint multipliers with both labels retained, factor normalized shells locally, and prove cross-term almost orthogonality before summing scales.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

14 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

2 recorded

Narrowed routeAll-visited-ball square summation

Route status · Narrowed route

Narrowed routePositive affine-line-space summation

The source reports an unavoidable R^(1/4) loss in that positive method. Oscillatory TT-star structure or additional geometric cancellation remains necessary and potentially viable.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

6 featured tasks

Prove the nonresonant two-endpoint vector-valued TT-star estimate uniformly in persistence scale and determinant shell.Suggested move: Define endpoint multipliers with both labels retained, factor normalized shells locally, and prove cross-term almost orthogonality before summing scales.

Ready to work on

Formalize the invariant global tangent-pair theorem for all compatible packet pairs in a complete regular semialgebraic class.Suggested move: Establish common-base-point jet transfer, the global h-halo estimate, bounded product-cap overlap, cutoff tensor Bessel, and independent bounded pair charging.

Ready to work on

Sharp restriction range

The conjecture predicts the restriction estimate in the stated Knapp-compatible range for compact smooth hypersurfaces with nonzero Gaussian curvature.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

Global packet-halo gap

The local scalar geometry still requires a global coefficient-uniform packet-halo theorem, cutoff-compatible tensor Bessel, and bounded packet-pair charging.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

Principal open seams

Nonresonant vector-valued analysis, global pair packing, cutoff orthogonality, degenerate strata, and recurrence closure remain open in the current work.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

Control singular, gradient-degenerate, and lower-dimensional wall strata and then close the full three-dimensional polynomial-partitioning recurrence.Suggested move: Stratify repeated, planar, singular, and gradient-degenerate loci, assign them without coefficient loss, and feed curve or point strata into dimension induction.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026

Current statusOpen conjecture

The 2025 peer-reviewed source describes the higher-dimensional non-L2 range as incomplete. Current arXiv:2512.24990v9 presents a testing characterization and reports that an incorrect claim regarding simpler proofs was removed due to an error; it does not claim a full proof. The sphere formulation, the full paraboloid conjecture, and endpoint conventions remain open in this bounded collection.

[1][2]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2026-07-27
PreprintArXiv v9 presents a smooth Alpert testing characterization for the Fourier extension conjecture on paraboloids. Its comments say that an incorrect claim regarding simpler proofs was removed due to an error;…[2]

Mathematical neighborhood

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

4 standing statements2 proposed statements6 open questions2 narrowed routes

Statements by mathematical role6 selected mapped statements

theorem candidate1 of 61
reduction1 of 61
lemma2 of 62
negative result2 of 62

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.21 displayed rows · 2 routes included

retained route statementWhich Lp functions admit an Lq Fourier restriction to a curved hypersurface?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementNo source-level proofintermediate
retained route statementAll-ball assignment is falseintermediate
retained route statementInvariant pair-volume cancellationintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureAll-visited-ball square summationreported failure
Useful failurePositive affine-line-space summationreported failure
Research targetProve the nonresonant two-endpoint vector-valued TT-star estimate uniformly in persistence scale and determinant shell.open
Research targetFormalize the invariant global tangent-pair theorem for all compatible packet pairs in a complete regular semialgebraic class.open
Research targetControl singular, gradient-degenerate, and lower-dimensional wall strata and then close the full three-dimensional polynomial-partitioning recurrence.open
Research targetSharp restriction rangeopen
Research targetGlobal packet-halo gapopen
Research targetPrincipal open seamsopen
Narrowed routeAll-visited-ball square summationThe source records a no-go theorem and replaces the assignment with bounded endpoint truncation. Endpoint-selected packet families at the second persistence scale remain the active nonterminal framework.
Narrowed routePositive affine-line-space summationThe source reports an unavoidable R^(1/4) loss in that positive method. Oscillatory TT-star structure or additional geometric cancellation remains necessary and potentially viable.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeProve the nonresonant two-endpoint vector-valued TT-star estimate uniformly in persistence scale and determinant shell.

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Continue the mathematics

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Prepared starting pointProve the nonresonant two-endpoint vector-valued TT-star estimate uniformly in persistence scale and determinant shell.

Fourier restriction conjecture · ready to start

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Research contextPrepared context for any AI agent

The conjecture asks exactly when the Fourier transform of an ambient function can be meaningfully restricted to a curved surface. The source focuses on a paraboloid model and does not claim a proof.

Exact question and boundaries
Current routes and known obstacles
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Sources and references2 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

1
Fourier Restriction for Schatten Class Operators and Functions on Phase Spacepeer reviewed result · Franz Luef, Helge J. Samuelsen · Oxford University Press · 2025-01-17 · DOI 10.1093/imrn/rnae291 · accessed Aug 14, 2026
2
A smooth Alpert testing characterization of convolution type for the Fourier extension conjecture on paraboloidspreprint · Cristian Rios, Eric T. Sawyer · arXiv · 2026-07-27 · ARXIV 2512.24990 · accessed Aug 14, 2026

Important qualifications

Bounded primary and publisher search; not an exhaustive literature review.
The current arXiv v9 testing-characterization result was not independently checked and no peer-reviewed disposition was located.
The earlier full-proof presentation is not current: arXiv v9 reports that an incorrect claim regarding simpler proofs was removed due to an error.

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Fourier restriction conjecture

Exact mathematical statement

Problem at a glance

Where work on Fourier restriction conjecture stands

Fourier restriction conjecture in numbers

Prove the nonresonant two-endpoint vector-valued TT-star estimate uniformly in persistence scale and determinant shell.

How the current approaches connect

Research route map

Other routes

Open questions

The known mathematical landscape

What the literature has established

Related results and reusable starting points

Formalization opportunities

Claims, milestones, and routes in the current map

Conditions that would advance the current route

Contribute

Important qualifications

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