Harmonic analysis and Fourier multipliers

Bochner–Riesz Conjecture

Collaboration beta

How much smoothing at the boundary of the frequency ball is enough for bounded Fourier reconstruction on L^p?

S^{δ} : L^{p} (ℝ^{n}) \to L^{p} (ℝ^{n}) for every δ > δ_{n} (p)

Known results and sources

The Bochner–Riesz problem concentrates on smoothing the boundary of the frequency ball.

Research problem

Exact mathematical statement

For $n\ge 2$ , $1<p<\infty$ , and $\delta\ge0$ , define

S^\delta f(x)=\int_{\mathbb R^n}(1-|\xi|^2)_+^\delta\widehat f(\xi)e^{2\pi i x\cdot\xi}\,d\xi

and

\delta_n(p)=\max\left\{n\left|\frac1p-\frac12\right|-\frac12,0\right\}.

The sharp sufficiency conjecture is that $S^\delta$ is bounded on $L^p(\mathbb R^n)$ for every strict inequality $\delta>\delta_n(p)$ . Equality at a positive threshold is not part of this statement.

Problem infographic

Problem at a glance

A four-panel explainer of the Bochner–Riesz multiplier, its thin-shell reduction, and the exact critical-index graph with a zero plateau between the two breakpoints in u=1/p. — The critical index is zero on its central interval and rises linearly outside it; the conjecture asks whether every strict amount of smoothing above that boundary yields L^p boundedness.

Current mathematical picture

Where work on Bochner–Riesz Conjecture stands

Partially resolved

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

Useful failureSource-reported limitation

The following claim is rejected or insufficient in the recorded route: The literal four-child dyadic factor split guarantees quantitative separation between opposite significant children. Close the three-dimensional TCI, SFB, and FBI estimates with full packet-tail, chart, multiplicity, and output-selection bookkeeping, then treat higher-dimensional intermediate ranks.

Route status · Narrowed route

Main reductionCritical thin-shell reduction

The current work reduces strict-threshold sufficiency to a critical epsilon-loss thin-shell estimate.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeProve the thick transverse congruent-circle estimate TCI with its precise thickening and nu dependence.Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Bochner–Riesz Conjecture in numbers

2.5kretained lines of mathematical investigation2,520 in the current working snapshot

Argument development: 2,164 · 86%
Explored or eliminated routes: 54 · 2%
Computational analysis: 9 · 0%
Open obligations: 102 · 4%
Definitions and setup: 191 · 8%

7selected mapped statements1routes investigated3open questions3contribution-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.

Status checkedAug 14, 2026

Source recordedAug 14, 2026

Recommended next task

Prove the thick transverse congruent-circle estimate TCI with its precise thickening and nu dependence.

Suggested move: Supply a self-contained crossing-graph proof and bind the geometric crossing parameter to the product of rank-one factor separations.

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

11 selected steps

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

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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

1 recorded

Narrowed routeSource-reported limitation

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Prove the thick transverse congruent-circle estimate TCI with its precise thickening and nu dependence.Suggested move: Supply a self-contained crossing-graph proof and bind the geometric crossing parameter to the product of rank-one factor separations.

Ready to work on

Prove the selected factor-block Carleson estimate SFB.Suggested move: Control output-dependent terminal rectangles while preserving the selected complex packet sums and full-length packets.

Ready to work on

Prove the fine buffered bi-transverse estimate FBI in the remaining anisotropic scale region.Suggested move: Use the buffered product forest and refined occupancy function to close stops missed by both incidence transversality and fixed-block degree.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026

Current statusPartially resolved

The sharp problem is solved in dimension two but remains open in dimensions greater than two. Recent peer-reviewed work improves high-dimensional restriction ranges and convergence estimates without supplying the full all-dimensional sharp theorem.

[1][2][3]

External progress

What the literature has established

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

2025-10-06
Peer reviewedKim obtained improved sufficient conditions for almost-everywhere convergence in dimensions two and three.[3]
2025-03-08
Peer reviewedGuo, Oh, Wang, Wu, and Zhang improved the Bochner–Riesz problem to the best-known restriction range in high dimensions.[2]
1972
Historical sourceThe two-dimensional disc-multiplier work of Carleson and Sjölin underlies the solved planar case reported by later sources.[2][1]

Mathematical neighborhood

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

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

5 standing statements2 proposed statements3 open questions1 narrowed routes

Statements by mathematical role7 selected mapped statements

theorem candidate1 of 71
reduction2 of 72
lemma4 of 74

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.20 displayed rows · 1 route included

retained route statementDoes every strict amount of smoothing above the critical index give L^p boundedness?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementStrict-threshold conjectureintermediate
retained route statementOne-dimensional caseintermediate
retained route statementCritical thin-shell reductionintermediate
retained route statementTwo-dimensional estimate-level closureintermediate
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
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 failureSource-reported limitationreported failure
Research targetProve the thick transverse congruent-circle estimate TCI with its precise thickening and nu dependence.open
Research targetProve the selected factor-block Carleson estimate SFB.open
Research targetProve the fine buffered bi-transverse estimate FBI in the remaining anisotropic scale region.open
Research targetConditional transverse incidence estimatesuperseded
Research targetSelected-block and fine-broad frontiersuperseded
Narrowed routeSource-reported limitationThe following claim is rejected or insufficient in the recorded route: The literal four-child dyadic factor split guarantees quantitative separation between opposite significant children. Close the three-dimensional TCI, SFB, and FBI estimates with full packet-tail, chart, multiplicity, and output-selection bookkeeping, then treat higher-dimensional intermediate ranks.

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 thick transverse congruent-circle estimate TCI with its precise thickening and nu dependence.

1 approach has 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 thick transverse congruent-circle estimate TCI with its precise thickening and nu dependence.

Bochner–Riesz Conjecture · ready to start

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Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

How much smoothing at the boundary of the frequency ball is enough for bounded Fourier reconstruction on L^p?

Exact question and boundaries
Current routes and known obstacles
What a useful result should report

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Sources and references3 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
Sparse bilinear forms for Bochner Riesz multipliers and applicationspeer reviewed result · Cristina Benea, Frédéric Bernicot, Teresa Luque · Transactions of the London Mathematical Society · 2017 · DOI 10.1112/tlm3.12005 · accessed Aug 14, 2026
2
The Bochner–Riesz Problem: An Old Approach Revisitedpeer reviewed result · Shaoming Guo, Changkeun Oh, Hong Wang, Shukun Wu, Ruixiang Zhang · Peking Mathematical Journal · 2025-03-08 · DOI 10.1007/s42543-023-00082-4 · accessed Aug 14, 2026
3
Weighted decoupling estimates and the Bochner-Riesz meanspeer reviewed result · Jongchon Kim · Forum of Mathematics, Sigma · 2025-10-06 · DOI 10.1017/fms.2024.96 · accessed Aug 14, 2026

Important qualifications

This bounded pass records the general sharp multiplier problem and selected current partial results, not every endpoint or maximal variant.
Dimension-specific best ranges can change quickly and require quarterly or material-change review.
No incoming-packet URL was fetched and packet claims were not used as external evidence.
No formal proof or computation was independently reproduced.

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Bochner–Riesz Conjecture

Exact mathematical statement

Problem at a glance

Where work on Bochner–Riesz Conjecture stands

Bochner–Riesz Conjecture in numbers

Prove the thick transverse congruent-circle estimate TCI with its precise thickening and nu dependence.

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

What changed in the research record

Claims, milestones, and routes in the current map

Conditions that would advance the current route

Contribute

Important qualifications

Compare another research frontier