[#P3074] Bochner-Riesz conjecture in higher dimensions

Problem. For \(d\ge 2\), \(1<p<\infty\), and \(\delta>\max\{d|1/p-1/2|-1/2,0\}\), are the Euclidean Bochner-Riesz multipliers \(S_R^\delta\), defined by the Fourier multiplier \((1-|\xi|^2/R^2)_+^\delta\), bounded on \(L^p(\mathbb R^d)\) uniformly for \(R>0\)?

Bochner-Riesz cutoff in Fourier space. — A structural field diagram of the statement's mathematical objects.

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1Context

Known frontier: The conjecture is known in dimension two and in substantial higher-dimensional subranges derived from restriction and decoupling estimates. Open boundary: The full sharp strong-type range in dimensions d≥3 remains open.

2Problem setup

Definition 1 (Bochner-Riesz multiplier). The operator with multiplier (1−|ξ|²/R²)_+^δ.

Definition 2 (uniform L^p boundedness). sup_R ||S_R^δ f||_p≤C||f||_p with C independent of R.

Remark 1. The stated threshold is forced by standard examples and is expected to be sufficient. Endpoint formulations are excluded so the packet has one clean strong-type target.

3What counts as a solution

Prove the uniform estimate throughout the stated d,p,δ range.
Or give parameters in the range and an L^p counterexample.

1Status

Saved packet · August 1, 2026

What counts as a solution

Saved packet status (Current status and exact unresolved remainder). OPEN as checked on 2026-08-01. Strongest checked neighboring result: The conjecture is known in dimension two and in substantial higher-dimensional subranges derived from restriction and decoupling estimates. Exact unresolved remainder: The full sharp strong-type range in dimensions d≥3 remains open.^[1][2]

1Packet records

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Record	Kind	Assessment
OPEN as checked on 2026-08-01. Strongest checked neighboring result: The conjecture is known in dimension two and in substantial higher-dimensional subranges derived from restriction and decoupling estimates. Exact unresolved remainder: The full sharp strong-type range in dimensions d≥3 remains open.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 OPEN as checked on 2026-08-01. Strongest checked neighboring result: The conjecture is known in dimension two and in substantial higher-dimensional subranges derived from restriction and decoupling estimates. Exact unresolved remainder: The full sharp strong-type range in dimensions d≥3 remains open.^[1][2] Relevance to this problem This is the dated publication status for the canonical target Bochner-Riesz conjecture in higher dimensions. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The problem was checked as open on 2026-08-01. The strongest neighboring result found in the cited sources is: The conjecture is known in dimension two and in substantial higher-dimensional subranges derived from restriction and decoupling estimates. The exact unresolved remainder is: The full sharp strong-type range in dimensions d≥3 remains open. A complete resolution must meet the following acceptance conditions: - Prove the uniform estimate throughout the stated d,p,δ range. - Or give parameters in the range and an L^p counterexample. Open full record →
The conjecture is known in dimension two and in substantial higher-dimensional subranges derived from restriction and decoupling estimates.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 2 The conjecture is known in dimension two and in substantial higher-dimensional subranges derived from restriction and decoupling estimates.^[1][2] Relevance to this problem Locates the present research frontier immediately below Bochner-Riesz conjecture in higher dimensions. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The conjecture is known in dimension two and in substantial higher-dimensional subranges derived from restriction and decoupling estimates. This leaves the following boundary unresolved: The full sharp strong-type range in dimensions d≥3 remains open. The distinction is retained here so a restricted theorem, finite computation, or neighboring case is not presented as a solution of the full target. Related: Claim 1 Open full record →
The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: The conjecture is known in dimension two and in substantial higher-dimensional subranges derived from restriction and decoupling estimates. Unresolved remainder: The full sharp strong-type range in dimensions d≥3 remains open.	Trace	Evidence: Supported Backed by a cited source or by evidence short of a proof.
attempt · Route 1 The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: The conjecture is known in dimension two and in substantial higher-dimensional subranges derived from restriction and decoupling estimates. Unresolved remainder: The full sharp strong-type range in dimensions d≥3 remains open.^[1][2] Relevance to this problem Documents why Bochner-Riesz conjecture in higher dimensions was treated as a distinct open target on 2026-08-01. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state completed Scope No narrower scope recorded Sources Reference 1, Reference 2 What happened The audit ran on 2026-08-01 across the cited primary literature and problem collections, Crossref, arXiv, and stable publisher records, TheoremDB published, prospecting, packet, formal, and retired corpus. It checked the exact statement, its named or normalized variants, recent status evidence, and the controlled TheoremDB corpus. Queries included: - Bochner Riesz conjecture higher dimensions remains open 2025 - Bochner Riesz sharp Lp range current best The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: The conjecture is known in dimension two and in substantial higher-dimensional subranges derived from restriction and decoupling estimates. Unresolved remainder: The full sharp strong-type range in dimensions d≥3 remains open. Related: Claim 1 Open full record →
The full sharp strong-type range in dimensions d≥3 remains open.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 2 The full sharp strong-type range in dimensions d≥3 remains open. Relevance to this problem Turns the remaining uncertainty in Bochner-Riesz conjecture in higher dimensions into a checkable research target. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded What happened Research should address this boundary directly: The full sharp strong-type range in dimensions d≥3 remains open. A claimed resolution should satisfy every item below and preserve the statement's exact quantifiers and normalizations: - Prove the uniform estimate throughout the stated d,p,δ range. - Or give parameters in the range and an L^p counterexample. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. OPEN as checked on 2026-08-01. Strongest checked neighboring result: The conjecture is known in dimension two and in substantial higher-dimensional subranges derived from restriction and decoupling estimates. Exact unresolved remainder: The full sharp strong-type range in dimensions d≥3 remains open.

Equivalent-formulation queries: Bochner Riesz conjecture higher dimensions remains open 2025; Bochner Riesz sharp Lp range current best
Strongest checked neighboring result: The conjecture is known in dimension two and in substantial higher-dimensional subranges derived from restriction and decoupling estimates.
Exact unresolved remainder: The full sharp strong-type range in dimensions d≥3 remains open.

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ProblemBochner-Riesz conjecture in higher dimensions

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Plain text

“Bochner-Riesz conjecture in higher dimensions.” TheoremDB. P3074. Problem statement; statement text SHA-256 b982705db36c3dee64d174e3d0e4da82d742df2bcd2b6ee2293bad841d213214. https://theoremdb.org/statement/?ref=P3074

BibTeX

@misc{theoremdb-problem-b982705db36c3dee64d174e3d0e4da82d742df2bcd2b6ee2293bad841d213214,
  title = {{Bochner-Riesz conjecture in higher dimensions}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 b982705db36c3dee64d174e3d0e4da82d742df2bcd2b6ee2293bad841d213214},
  url = {https://theoremdb.org/statement/?ref=P3074}
}

Plain text: Built Markdown snapshot

This problem includes 4 records joined by 3 typed links, sourced from researchgate.net^[1], current as of August 1, 2026.

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1References

Packet source. Open Problems in Harmonic Analysis and Related Fields (2025), Bochner-Riesz section. Bochner-Riesz conjecture section. Bochner-Riesz conjecture section. ↗website · reference source · checked 2026-08-01Source use: original summary.Records that higher-dimensional cases remain open after recent restriction progress.Also cited at Open Problems in Harmonic Analysis and Related Fields (2025), Bochner-Riesz section. Bochner-Riesz conjecture section.Source used to assess the problem's recorded status.For Bochner-Riesz conjecture in higher dimensions: This is the dated publication status for the canonical target Bochner-Riesz conjecture in higher dimensions.Source named by the research packet.
Charles Fefferman, “The Multiplier Problem for the Ball”. The Annals of Mathematics 94(2) (1971), 330. DOI 10.2307/1970864. main theorem. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Provides the foundational obstruction for ball multipliers and the sharpness context for positive δ.Source used to assess the problem's recorded status.For Bochner-Riesz conjecture in higher dimensions: Provides the foundational obstruction for ball multipliers and the sharpness context for positive δ.

Original TheoremDB editorial statement and source synthesis; external works are used for citation only.

[#P3074] Bochner-Riesz conjecture in higher dimensions

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