# P3074: Bochner-Riesz conjecture in higher dimensions - ID: `P3074` - Reference: `bochner-riesz-conjecture` - Page: https://theoremdb.org/statements/P3074 - Record maturity: Reviewed problem with recorded work ## Problem For \(d\ge 2\), \(1\max\{d|1/p-1/2|-1/2,0\}\), 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 \(R>0\)? ### Context 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. ### Problem setup - **Definition (Bochner-Riesz multiplier).** The operator with multiplier (1−|ξ|²/R²)_+^δ. - **Definition (uniform L^p boundedness).** sup_R ||S_R^δ f||_p≤C||f||_p with C independent of R. - **Remark.** 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. ### What 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. ## 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. [1](#reference-1) [2](#reference-2) ## Work ### Evidence for the current status **Claim 1 (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. 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. ### Background and intake notes - 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. - The release review checked 2 structured sources on 2026-08-01. - 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. ### Other known results - **Claim 2** (supported): The conjecture is known in dimension two and in substantial higher-dimensional subranges derived from restriction and decoupling estimates. [1](#reference-1) [2](#reference-2) ### Prior approaches - **Route 1** (supported): 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](#reference-1) [2](#reference-2) ### Open directions - **Route 2** (reported): The full sharp strong-type range in dimensions d≥3 remains open. ### Working on this Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `bochner-riesz-conjecture`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome. ## References 1. Open Problems in Harmonic Analysis and Related Fields (2025), Bochner-Riesz section. Bochner-Riesz conjecture section. Bochner-Riesz conjecture section https://www.researchgate.net/publication/393644508_Open_Problems_in_Harmonic_Analysis_and_Related_Fields - Also cited at Open Problems in Harmonic Analysis and Related Fields (2025), Bochner-Riesz section. Bochner-Riesz conjecture section - website; reference source; checked 2026-08-01 - Source use: original_summary - Records that higher-dimensional cases remain open after recent restriction progress. - 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. 2. Charles Fefferman, “The Multiplier Problem for the Ball”. The Annals of Mathematics 94(2) (1971), 330. DOI 10.2307/1970864. main theorem https://doi.org/10.2307/1970864 - journal_article; primary source; checked 2026-08-01 - Source 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.