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A restriction estimate on Morrey spaces

  • Junren Pan and Wenchang Sun EMAIL logo
Published/Copyright: June 29, 2022
Forum Mathematicum
From the journal Forum Mathematicum Volume 34 Issue 5

Abstract

We study the Fourier extension operator E on the truncated paraboloid

ℙ 2 := { ( ω 1 , ω 2 , ω 1 2 + ω 2 2 ) : ω 1 2 + ω 2 2 ≤ 1 } ⊂ ℝ 3 .

For a function f in the Morrey space L 2 , λ ⁢ ( B 2 ) , where B 2 denotes the unit disk in ℝ 2 , we prove that for each R > 1 and ε > 0 ,

∥ E ⁢ f ∥ L p ⁢ ( B 3 ⁢ ( 0 , R ) ) ≲ ε , λ R ε ⁢ ∥ f ∥ L 2 , λ ⁢ ( B 2 )

holds for all p ≥ ( 6 ⁢ λ + 7 ) / ( 2 ⁢ λ + 2 ) if 5 6 < λ ≤ 1 and for all p ≥ 6 / ( 1 + λ ) if 1 2 < λ ≤ 5 6 . The main tool used in our proof is the polynomial partitioning technique.

Keywords: Fourier restriction estimate; Morrey spaces
MSC 2010: 42B20

Communicated by Christopher D. Sogge

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12171250

Award Identifier / Grant number: U21A20426

Funding statement: This work was partially supported by the National Natural Science Foundation of China (Grant numbers 12171250 and U21A20426).

Acknowledgements

The authors thank the referees very much for valuable suggestions.

References

[1] D. R. Adams and J. Xiao, Morrey spaces in harmonic analysis, Ark. Mat. 50 (2012), no. 2, 201–230. 10.1007/s11512-010-0134-0Search in Google Scholar

[2] J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991), no. 2, 147–187. 10.1007/BF01896376Search in Google Scholar

[3] J. Bourgain and L. Guth, Bounds on oscillatory integral operators based on multilinear estimates, Geom. Funct. Anal. 21 (2011), no. 6, 1239–1295. 10.1007/s00039-011-0140-9Search in Google Scholar

[4] C. Demeter, Fourier Restriction, Decoupling, and Applications, Cambridge Stud. Adv. Math. 184, Cambridge University, Cambridge, 2020. 10.1017/9781108584401Search in Google Scholar

[5] J. Duoandikoetxea and M. Rosenthal, Muckenhoupt-type conditions on weighted Morrey spaces, J. Fourier Anal. Appl. 27 (2021), no. 2, Paper No. 32. 10.1007/s00041-021-09839-wSearch in Google Scholar

[6] L. Guth, A restriction estimate using polynomial partitioning, J. Amer. Math. Soc. 29 (2016), no. 2, 371–413. 10.1090/jams827Search in Google Scholar

[7] L. Guth, Polynomial Methods in Combinatorics, Univ. Lecture Ser. 64, American Mathematical Society, Providence, 2016. 10.1090/ulect/064Search in Google Scholar

[8] L. Guth, Restriction estimates using polynomial partitioning II, Acta Math. 221 (2018), no. 1, 81–142. 10.4310/ACTA.2018.v221.n1.a3Search in Google Scholar

[9] L. Guth and N. H. Katz, On the Erdős distinct distances problem in the plane, Ann. of Math. (2) 181 (2015), no. 1, 155–190. 10.4007/annals.2015.181.1.2Search in Google Scholar

[10] Y. Lu, D. Yang and W. Yuan, Interpolation of Morrey spaces on metric measure spaces, Canad. Math. Bull. 57 (2014), no. 3, 598–608. 10.4153/CMB-2013-009-4Search in Google Scholar

[11] C. B. Morrey, Jr., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), no. 1, 126–166. 10.1090/S0002-9947-1938-1501936-8Search in Google Scholar

[12] E. M. Stein, Some problems in harmonic analysis, Harmonic Analysis in Euclidean Spaces (Williamstown 1978), Proc. Sympos. Pure Math. 35 Part 1, American Mathematical Society, Providence (1979), 3–20. 10.1090/pspum/035.1/545235Search in Google Scholar

[13] R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705–714. 10.1215/S0012-7094-77-04430-1Search in Google Scholar

[14] J. Tao, D. Yang and W. Yuan, A bridge connecting Lebesgue and Morrey spaces via Riesz norms, Banach J. Math. Anal. 15 (2021), no. 1, Paper No. 20. 10.1007/s43037-020-00106-6Search in Google Scholar

[15] T. Tao, The Bochner–Riesz conjecture implies the restriction conjecture, Duke Math. J. 96 (1999), no. 2, 363–375. 10.1215/S0012-7094-99-09610-2Search in Google Scholar

[16] T. Tao, A sharp bilinear restrictions estimate for paraboloids, Geom. Funct. Anal. 13 (2003), no. 6, 1359–1384. 10.1007/s00039-003-0449-0Search in Google Scholar

[17] H. Wang, A restriction estimate in ℝ 3 using brooms, preprint (2018), https://arxiv.org/abs/1802.04312. 10.29007/584lSearch in Google Scholar

[18] W. Yuan, W. Sickel and D. Yang, Interpolation of Morrey–Campanato and related smoothness spaces, Sci. China Math. 58 (2015), no. 9, 1835–1908. 10.1007/s11425-015-5047-8Search in Google Scholar
Received: 2022-02-17
Revised: 2022-04-06
Published Online: 2022-06-29
Published in Print: 2022-09-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Recognizing pro-𝖱 closures of regular languages
  3. Reverse Stein–Weiss, Hardy–Littlewood–Sobolev, Hardy, Sobolev and Caffarelli–Kohn–Nirenberg inequalities on homogeneous groups
  4. Non-perturbative graph languages, halting problem and complexity
  5. Spectral synthesis via moment functions on hypergroups
  6. A function field analogue of the Stark conjecture
  7. Continuity properties and Bargmann mappings of quasi-Banach Orlicz modulation spaces
  8. Krull dimension for C*-algebras
  9. The sharp bound of the third Hankel determinant for starlike functions
  10. The role of gentle algebras in higher homological algebra
  11. Classifying spaces for the family of virtually abelian subgroups of orientable 3-manifold groups
  12. L p boundedness for a maximal singular integral operator
  13. Common reducing subspaces and decompositions of contractions
  14. A restriction estimate on Morrey spaces
  15. On ideals in group algebras: An uncertainty principle and the Schur product
  16. Spectrality of self-similar measures with product-form digits
  17. Borderline Lipschitz regularity for bounded minimizers of functionals with (p,q)-growth
  18. On the L q spectra of in-homogeneous self-similar measures
https://doi.org/10.1515/forum-2022-0056
Keywords for this article
Fourier restriction estimate; Morrey spaces

Articles in the same Issue

  1. Frontmatter
  2. Recognizing pro-𝖱 closures of regular languages
  3. Reverse Stein–Weiss, Hardy–Littlewood–Sobolev, Hardy, Sobolev and Caffarelli–Kohn–Nirenberg inequalities on homogeneous groups
  4. Non-perturbative graph languages, halting problem and complexity
  5. Spectral synthesis via moment functions on hypergroups
  6. A function field analogue of the Stark conjecture
  7. Continuity properties and Bargmann mappings of quasi-Banach Orlicz modulation spaces
  8. Krull dimension for C*-algebras
  9. The sharp bound of the third Hankel determinant for starlike functions
  10. The role of gentle algebras in higher homological algebra
  11. Classifying spaces for the family of virtually abelian subgroups of orientable 3-manifold groups
  12. L p boundedness for a maximal singular integral operator
  13. Common reducing subspaces and decompositions of contractions
  14. A restriction estimate on Morrey spaces
  15. On ideals in group algebras: An uncertainty principle and the Schur product
  16. Spectrality of self-similar measures with product-form digits
  17. Borderline Lipschitz regularity for bounded minimizers of functionals with (p,q)-growth
  18. On the L q spectra of in-homogeneous self-similar measures
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