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arxiv: 2512.24990 · v7 · pith:5QXNWBAEnew · submitted 2025-12-31 · 🧮 math.CA

The Fourier extension conjecture for the paraboloid

Cristian Rios , Eric T. Sawyer This is my paper

Pith reviewed 2026-05-21 17:04 UTC · model grok-4.3

classification 🧮 math.CA
keywords fourierconjectureextensionthenalpertamplitudeaveragingbilinear
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The pith

Proof of the Fourier extension conjecture on the paraboloid in d>2 by decomposing smooth Alpert projections, applying a bilinear reduction, and bounding the resulting oscillatory integral with periodic amplitude via lattice averaging and stationary phase.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The Fourier extension conjecture asks for sharp bounds on how the Fourier transform of a function supported on a curved surface like the paraboloid extends to the whole space. The authors begin by breaking a smooth projection into localized pieces whose Fourier extensions can be handled separately. They then invoke a bilinear form of the conjecture to turn the main estimate into an average over mollified functions on grids. This converts a hard exponential sum into an oscillatory integral whose amplitude is periodic. Further averaging over lattices extracts Dirichlet kernels, which are then bounded using a stationary phase argument with periodic amplitude. The net effect is the desired localization on the Fourier side.

Core claim

We give a proof of Fourier extension conjecture on the paraboloid in all dimensions bigger than 2 that begins with a decomposition suggested in Sawyer [Saw8] of writing a smooth Alpert projection as a sum of pieces whose Fourier extensions are localized.

Load-bearing premise

The bilinear inequality, when taken over smooth Alpert projections, only requires an averaging over grids of functions mollified by discrete multipliers, which converts a difficult exponential sum into an oscillatory integral with periodic amplitude that can be controlled by stationary phase after extracting Dirichlet kernels.

read the original abstract

We give a proof of Fourier extension conjecture on the paraboloid in all dimensions bigger than 2 that begins with a decomposition suggested in Sawyer [Saw8] of writing a smooth Alpert projection as a sum of pieces whose Fourier extensions are localized. This is then used to establish a local inequality that is well known to be equivalent to the Fourier extension conjecture, and is accomplished by using a variant of the bilinear equivalence of the Fourier extension conjecture given by Tao, Vargas and Vega in [TaVaVe]. A key aspect of our proof is that the bilinear inequality, when taken over smooth Alpert projections, only requires an averaging over grids of functions mollified by discrete multipliers, which converts a difficult exponential sum into an oscillatory integral with periodic amplitude. After extracting Dirichlet kernels in yet another averaging over lattices, this is then controlled using a stationary phase estimate with periodic amplitude, and altogether we then obtain the desired localization on the Fourier side.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Circularity Check

1 steps flagged

Proof begins with decomposition suggested in co-author's prior work [Saw8]

specific steps
  1. self citation load bearing [Abstract]
    "We give a proof of Fourier extension conjecture on the paraboloid in all dimensions bigger than 2 that begins with a decomposition suggested in Sawyer [Saw8] of writing a smooth Alpert projection as a sum of pieces whose Fourier extensions are localized."

    The central proof chain is stated to begin with a decomposition suggested in the co-author's prior work [Saw8]. This makes the initial localization step dependent on self-citation rather than an independent derivation, so the overall argument reduces in its starting point to the authors' own earlier results.

full rationale

The paper's derivation explicitly begins with a decomposition from Sawyer [Saw8], a self-citation load-bearing step since Sawyer is a co-author. This anchors the localization of Fourier extensions for smooth Alpert projections. The argument then invokes an external bilinear equivalence from Tao-Vargas-Vega and proceeds via averaging over grids, Dirichlet kernels, and stationary phase estimates. While the subsequent oscillatory integral control adds independent content, the foundational decomposition reduces the claim's independence from the authors' earlier results, producing partial circularity. No equations are shown reducing by construction to fitted inputs, and the bilinear step is externally sourced, so the score is not higher.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on two domain assumptions: the known equivalence between the local inequality and the full conjecture, and the applicability of the Tao-Vargas-Vega bilinear form to the chosen projections. No free parameters or new postulated entities appear in the abstract.

axioms (2)
  • domain assumption The local inequality is well known to be equivalent to the Fourier extension conjecture
    Explicitly stated in the abstract as the target of the local inequality.
  • domain assumption The bilinear inequality variant from Tao, Vargas and Vega applies when the functions are smooth Alpert projections
    Invoked to reduce the problem to an averaged oscillatory integral.

pith-pipeline@v0.9.0 · 5682 in / 1318 out tokens · 71128 ms · 2026-05-21T17:04:46.035003+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    We give a proof of Fourier extension conjecture on the paraboloid in all dimensions bigger than 2 that begins with a decomposition ... averaging over grids ... oscillatory integral with periodic amplitude ... controlled using a new periodic stationary phase estimate

  • IndisputableMonolith/Foundation/AlexanderDuality.lean D3_admits_circle_linking echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    Theorem 1. (Fourier extension conjecture on the paraboloid) Suppose d≥3 ...

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A discussion of two new proofs of Fefferman's Fourier extension theorem in the plane

    math.CA 2026-05 unverdicted novelty 6.0

    Discusses two alternative proofs of Fefferman's Fourier extension theorem using decoupling and wavelet decompositions, with one method extended to higher dimensions.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · cited by 1 Pith paper · 4 internal anchors

  1. [1]

    Bennett, A

    J. Bennett, A. Carbery and T. Tao , On the multilinear restriction and Kakeya conjectures, Acta. Math. 196 (2006), 261-302

    work page 2006

  2. [2]

    Essays on Fourier analysis in honor of Elias M. Stein

    J. Bourgain, Some new estimates for oscillatory integrals, in "Essays on Fourier analysis in honor of Elias M. Stein", ed. C.\ Fefferman, R. Fefferman and S. Wainger, Princeton University Press, 1994

    work page 1994

  3. [3]

    Bounds on oscillatory integral operators based on multilinear estimates

    J. Bourgain and L. Guth ,\ Bounds on oscillatory integral operators based on multilinear estimates, arXiv:1012.3760v3

    work page internal anchor Pith review Pith/arXiv arXiv

  4. [4]

    Carbery, Restriction implies B\^ o chner-Riesz for paraboloids , Math

    A. Carbery, Restriction implies B\^ o chner-Riesz for paraboloids , Math. Proc. Camb. Phil. Soc., 111\ (1992), 525-529

    work page 1992

  5. [5]

    Carleson and P

    L. Carleson and P. Sj\" o lin, Oscillatory integrals and a multiplier problem for the disc, Studia Math. 44 (1972), 287--299

    work page 1972

  6. [6]

    Fefferman, A note on spherical summation multipliers, Israel J

    C. Fefferman, A note on spherical summation multipliers, Israel J. Math. (1973) 15, 44--52

    work page 1973

  7. [7]

    Fefferman and E.M

    C. Fefferman and E.M. Stein , Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115

    work page 1971

  8. [8]

    Tuomas Hyt\" o nen, The two weight inequality for the Hilbert transform with general measures, arXiv:1312.0843v2

    work page arXiv

  9. [9]

    Yitzhak Katznelson, An introduction to HARMONIC ANALYSIS, John Wiley & Sons, Inc. 1968

    work page 1968

  10. [10]

    Lacey, \ Two weight inequality for the Hilbert transform: A real variable characterization, II, Duke Math

    Michael T. Lacey, \ Two weight inequality for the Hilbert transform: A real variable characterization, II, Duke Math. J. Volume 163, Number 15 (2014), 2821-2840

    work page 2014

  11. [11]

    Lacey, Eric T

    Michael T. Lacey, Eric T. Sawyer, Chun-Yen Shen, and Ignacio Uriarte-Tuero, Two weight inequality for the Hilbert transform: A real variable characterization I, Duke Math. J, Volume 163, Number 15 (2014), 2795-2820

    work page 2014

  12. [12]

    Two weight estimate for the Hilbert transform and corona decomposition for non-doubling measures

    F. Nazarov, S. Treil and A. Volberg, Two weight estimate for the Hilbert transform and corona decomposition for non-doubling measures, preprint (2004) arXiv:1003.1596

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  13. [13]

    Rios and E

    C. Rios and E. Sawyer, Trilinear characterizations of the Fourier extension conjecture on the paraboloid in three dimensions, arXiv:2506.03992

    work page arXiv

  14. [14]

    Rios and E

    C. Rios and E. Sawyer, Equivalence of linear and trilinear Kakeya conjectures in three dimensions,\ arXiv:2507.21315

    work page arXiv

  15. [15]

    Rios and E

    C. Rios and E. Sawyer, A single scale smooth Alpert trilinear characterization of the Fourier extension conjecture on the paraboloid in three dimensions, arXiv:2506.23376v4

    work page arXiv

  16. [16]

    A probabilistic analogue of the Fourier extension conjecture

    E. Sawyer, A probabilistic analogue of the Fourier extension conjecture, arXiv:2311.03145v14

    work page internal anchor Pith review Pith/arXiv arXiv

  17. [17]

    Sawyer, A comparison of trilinear testing conditions for the paraboloid Fourier extension and Kakeya conjectures in three dimensions, arXiv:2411.18457v2

    E. Sawyer, A comparison of trilinear testing conditions for the paraboloid Fourier extension and Kakeya conjectures in three dimensions, arXiv:2411.18457v2

    work page arXiv

  18. [18]

    E. M. Stein, Some problems in harmonic analysis, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part 1, pp. 3-20, Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979

    work page 1978

  19. [19]

    E. M. Stein, Harmonic Analysis: real-variable methods, orthogonality, and oscillatory integrals ,\ Princeton University Press, Princeton, N. J., 1993

    work page 1993

  20. [20]

    Betsy Stovall, Waves, Spheres, and Tubes: A Selection of Fourier Restriction Problems, Methods, and Applications, Notices of the A.M.S. 66, No. 7 (2019), 1013-1022

    work page 2019

  21. [21]

    Some recent progress on the Restriction conjecture

    Terence Tao, Some recent progress on the restriction conjecture,\ arXiv:math/0303136 , In: Brandolini, L., Colzani, L., Travaglini, G., Iosevich, A. (eds) Fourier Analysis and Convexity. Applied and Numerical Harmonic Analysis. Birkh\" a user, Boston, MA. https://doi.org/10.1007/978-0-8176-8172-2\_10

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/978-0-8176-8172-2

  22. [22]

    Tao , The B\^ o chner-Riesz conjecture implies the restriction conjecture , Duke Math

    T. Tao , The B\^ o chner-Riesz conjecture implies the restriction conjecture , Duke Math. J. 96 (1999), no. 2, 363-375

    work page 1999

  23. [23]

    T. Tao, A. Vargas, and L. Vega , A bilinear approach to the restriction and Kakeya conjectures, JAMS 11 (1998), no. 4, 967--1000

    work page 1998

  24. [24]

    Tomas, A restriction theorem for the Fourier transform, Bull

    P.A. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 477--478

    work page 1975

  25. [25]

    Hong Wang and Joshua Zahl, Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions, arXiv:2502.17655v1

    work page arXiv

  26. [26]

    Thomas Wolff, Recent work connected with the Kakeya problem, DOI:10.1090/ulect/029/11Corpus ID: 14402276, (2007)

    work page doi:10.1090/ulect/029/11corpus 2007

  27. [27]

    Zygmund, On Fourier coefficients and transforms of functions of two variables, Studia Mathematica 50 (1974), no

    A. Zygmund, On Fourier coefficients and transforms of functions of two variables, Studia Mathematica 50 (1974), no. 2, 189-201

    work page 1974