Skip to main content
SpringerLink
Log in

The Bochner–Riesz Problem: An Old Approach Revisited

  • Original Article
  • Published:
Peking Mathematical Journal Aims and scope Submit manuscript

Abstract

We show that the recent techniques developed to study the Fourier restriction problem apply equally well to the Bochner–Riesz problem. This is achieved via applying a pseudo-conformal transformation and a two-parameter induction-on-scales argument. As a consequence, we improve the Bochner–Riesz problem to the best known range of the Fourier restriction problem in all high dimensions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. See also Theorem 2.1 and Theorem 2.2 below where the same reduction is used.

  2. It will be chosen to be \(R^{\delta '}\) for some extremely small \(\delta '\ll \epsilon \).

  3. We refer to (7.22) for the explicit definition of \({\widetilde{{\mathbb {T}}}}_b'[B({\widetilde{{\textbf {x}}}}_0,\rho )]\).

  4. We cannot directly apply the lemma because our operator is not of the normal form. However, one can prove the lemma for our operator by following the same argument. We leave out the details here.

  5. The integers \(\#_{{\texttt {a} }}(j)\) and \( \#_{{\texttt {c} }}(j)\) indicate the number of occurrences of algebraic cases and cellular cases, respectively.

  6. Possibly a tube \({\widetilde{T}}\) intersects \(N_{\rho _{j+1}^{1/2+\delta _m}/2}\big (Z+{\textbf {y}}_{O_j}+({\rho _{j+1}^{1/2+\delta _m}}/{2}) b \big )\) for many b. We simply choose one out of them.

References

  1. Beltran, D., Hickman, J., Sogge, C.D.: Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds. Anal. PDE 13(2), 403–433 (2020)

    Article  MathSciNet  Google Scholar 

  2. Bourgain, J.: Besicovitch type maximal operators and applications to Fourier analysis. Geom. Funct. Anal. 1(2), 147–187 (1991)

    Article  MathSciNet  Google Scholar 

  3. Bourgain, J.: \(L^p\)-estimates for oscillatory integrals in several variables. Geom. Funct. Anal. 1(4), 321–374 (1991)

    Article  MathSciNet  Google Scholar 

  4. Bourgain, J., Demeter, C.: The proof of the \(l^2\) decoupling conjecture. Ann. of Math. 182(1), 351–389 (2015)

    Article  MathSciNet  Google Scholar 

  5. Bourgain, J., Guth, L.: Bounds on oscillatory integral operators based on multilinear estimates. Geom. Funct. Anal. 21(6), 1239–1295 (2011)

    Article  MathSciNet  Google Scholar 

  6. Carbery, A.: Restriction implies Bochner–Riesz for paraboloids. Math. Proc. Cambridge Philos. Soc. 111(3), 525–529 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  7. Carleson, L., Sjölin, P.: Oscillatory integrals and a multiplier problem for the disc. Stud. Math. 44, 287–299 (1972)

  8. Christ, M.: Weak type \((1,1)\) bounds for rough operators. Ann. Math. (2) 128(1), 19–42 (1988)

    Article  MathSciNet  Google Scholar 

  9. Fefferman, C.: Inequalities for strongly singular convolution operators. Acta Math. 124, 9–36 (1970)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Guth, L., Hickman, J., Iliopoulou, M.: Sharp estimates for oscillatory integral operators via polynomial partitioning. Acta Math. 223(2), 251–376 (2019)

    Article  MathSciNet  Google Scholar 

  12. Guillemin, V., Pollack, A.: Differential Topology. AMS Chelsea Publishing, Providence, RI (2010)

  13. Guth, L.: A restriction estimate using polynomial partitioning. J. Amer. Math. Soc. 29(2), 371–413 (2016)

    Article  MathSciNet  Google Scholar 

  14. Guth, L.: Restriction estimates using polynomial partitioning II. Acta Math. 221(1), 81–142 (2018)

    Article  MathSciNet  Google Scholar 

  15. Hickman, J., Rogers, K.M.: Improved Fourier restriction estimates in higher dimensions. Camb. J. Math. 7(3), 219–282 (2019)

    Article  MathSciNet  Google Scholar 

  16. Hickman, J., Rogers, K.M., Zhang, R.: Improved bounds for the Kakeya maximal conjecture in higher dimensions. Amer. J. Math. 144(6), 1511–1560 (2022)

    Article  MathSciNet  Google Scholar 

  17. Hickman, J., Zahl, J.: A note on Fourier restriction and nested polynomial Wolff axioms. J. Anal. Math. https://doi.org/10.1007/s11854-023-0289-9 (2023)

  18. Hörmander, L.: Oscillatory integrals and multipliers on \(FL^{p}\). Ark. Mat. 11, 1–11 (1973)

    Article  MathSciNet  Google Scholar 

  19. Katz, N.H., Rogers, K.M.: On the polynomial Wolff axioms. Geom. Funct. Anal. 28(6), 1706–1716 (2018)

    Article  MathSciNet  Google Scholar 

  20. Lee, S.: Improved bounds for Bochner–Riesz and maximal Bochner–Riesz operators. Duke Math. J. 122(1), 205–232 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  21. Lee, S.: Linear and bilinear estimates for oscillatory integral operators related to restriction to hypersurfaces. J. Funct. Anal. 241(1), 56–98 (2006)

    Article  MathSciNet  Google Scholar 

  22. Lee, S.: Square function estimates for the Bochner–Riesz means. Anal. PDE 11(6), 1535–1586 (2018)

    Article  MathSciNet  Google Scholar 

  23. Rogers, K.M.: A local smoothing estimate for the Schrödinger equation. Adv. Math. 219(6), 2105–2122 (2008)

    Article  MathSciNet  Google Scholar 

  24. Seeger, A.: Endpoint inequalities for Bochner–Riesz multipliers in the plane. Pacific J. Math. 174(2), 543–553 (1996)

    Article  MathSciNet  Google Scholar 

  25. Stein, E.M.: Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, vol. 4. Princeton University Press, Princeton, NJ (1993)

  26. Stein, E.M., Shakarchi, R.: Functional Analysis: Introduction to Further Topics in Analysis. Princeton Lectures in Analysis, vol. 4. Princeton University Press, Princeton, NJ (2011)

  27. Tao, T.: The weak-type endpoint Bochner–Riesz conjecture and related topics. Indiana Univ. Math. J. 47(3), 1097–1124 (1998)

    Article  MathSciNet  Google Scholar 

  28. Tao, T.: The Bochner–Riesz conjecture implies the restriction conjecture. Duke Math. J. 96(2), 363–375 (1999)

    Article  MathSciNet  Google Scholar 

  29. Tao, T.: A sharp bilinear restrictions estimate for paraboloids. Geom. Funct. Anal. 13(6), 1359–1384 (2003)

    Article  MathSciNet  Google Scholar 

  30. Tao, T.: Nonlinear Dispersive Equations: Local and Global Analysis. CBMS Regional Conference Series in Mathematics, vol. 106. American Mathematical Society, Providence, RI (2006)

  31. Wang, H.: A restriction estimate in \({\mathbb{R} }^3\) using brooms. Duke Math. J. 171(8), 1749–1822 (2022)

    Article  MathSciNet  Google Scholar 

  32. Wongkew, R.: Volumes of tubular neighbourhoods of real algebraic varieties. Pacific J. Math. 159(1), 177–184 (1993)

    Article  MathSciNet  Google Scholar 

  33. Wu, S.: On the Bochner–Riesz operator in \({\mathbb{R} }^3\). J. Anal. Math. 149(2), 677–718 (2023)

    Article  MathSciNet  Google Scholar 

  34. Zahl, J.: A discretized Severi-type theorem with applications to harmonic analysis. Geom. Funct. Anal. 28(4), 1131–1181 (2018)

    Article  MathSciNet  Google Scholar 

  35. Zahl, J.: New Kakeya estimates using Gromov’s algebraic lemma. Adv. Math. 380, 107596 (2021)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Xiaochun Li, Zane Li, Andreas Seeger, Rajula Srivastava and Terence Tao for valuable discussions. S.G. was supported in part by the NSF grant DMS-1800274. H.W. was supported by the National Science Foundation under Grant No. DMS-1926686. R.Z. was supported by the NSF grant DMS-1856541, DMS-1926686 and by the Ky Fan and Yu-Fen Fan Endowment Fund at the Institute for Advanced Study.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin Madison, Madison, WI, USA

    Shaoming Guo, Changkeun Oh & Ruixiang Zhang

  2. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Changkeun Oh

  3. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, Republic of Korea

    Changkeun Oh

  4. School of Mathematics, Institute for Advanced Study, Princeton, NJ, USA

    Hong Wang & Ruixiang Zhang

  5. Department of Mathematics, Indiana University Bloomington, Bloomington, IN, USA

    Shukun Wu

Authors
  1. Shaoming Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Changkeun Oh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Shukun Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Ruixiang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Changkeun Oh.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, S., Oh, C., Wang, H. et al. The Bochner–Riesz Problem: An Old Approach Revisited. Peking Math J (2024). https://doi.org/10.1007/s42543-023-00082-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s42543-023-00082-4

Keywords

Mathematics Subject Classification

Search

Navigation