Abstract
We prove Lp → Lq Fourier restriction estimates for three dimensional quadratic surfaces in ℝ5. Our results are sharp, up to endpoints, for a few classes of surfaces.
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References
A. D. Banner, Restriction of the Fourier transform to quadratic submanifolds, Ph.D. Thesis, Princeton University, Princeton, NJ, 2002.
J. Bennett, N. Bez, T. C. Flock and S. Lee, Stability of the Brascamp-Lieb constant and applications, Amer. J. Math. 140 (2018), 543–569.
J. Bennett, A. Carbery, M. Christ and T. Tao, The Brascamp-Lieb inequalities: finiteness, structure and extremals, Geom. Funct. Anal. 17 (2008), 1343–1415.
J. Bourgain and C. Demeter, The proof of the l2decoupling conjecture, Ann. of Math. (2) 182 (2015), 351–389.
J. Bourgain and C. Demeter, Decouplings for surfaces in ℝ4, J. Funct. Anal. 270 (2016), 1299–1318.
J. Bourgain and C. Demeter, Mean value estimates for Weyl sums in two dimensions, J. Lond. Math. Soc. (2) 94 (2016), 814–838.
J. Bourgain and C. Demeter, Decouplings for curves and hypersurfaces with nonzero Gaussian curvature, J. Anal. Math. 133 (2017), 279–311.
J. Bourgain, C. Demeter and S. Guo, Sharp bounds for the cubic Parsell—Vinogradov system in two dimensions, Adv. Math. 320 (2017), 827–875.
J. Bourgain and L. Guth, Bounds on oscillatory integral operators based on multilinear estimates Geom. Funct. Anal. 21 (2011), 1239–1295.
J.-G. Bak and S. Lee, Restriction of the Fourier transform to a quadratic surface in ℝn, Math. Z. 247 (2004), 409–422.
J.-G. Bak, J. Lee and S. Lee, Bilinear restriction estimates for surfaces of codimension bigger than 1, Anal. PDE 10 (2017), 1961–1985.
M. Christ, Restriction of the Fourier transform to submanifolds of low codimension, Ph.D. Thesis, University of Chicago, Chicago, IL, 1982.
M. Christ, On the restriction of the Fourier transform to curves: endpoint results and the degenerate case, Trans. Amer. Math. Soc. 287 (1985), 223–238.
L. Carleson and P. Sjölin, Oscillatory integrals and a multiplier problem for the disc, Studia Math. 44 (1972), 287–299. (errata insert)
C. Demeter, S. Guo and F. Shi, Sharp decouplings for three dimensional manifolds in ℝ5, Rev. Mat. Iberoam. 35 (2019), 423–460.
C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9–36.
S. Guo, C. Oh, J. Roos, P.-L. Yung and P. Zorin-Kranich, Decoupling for two quadratic forms in three variables: a complete characterization, arXiv:1912.03995 [math.CA].
L. Guth, A short proof of the multilinear Kakeya inequality, Math. Proc. Cambridge Philos. Soc. 158 (2015), 147–153.
S. Guo and R. Zhang, On integer solutions of Parsell—Vinogradov systems, Invent. Math. 218 (2019), 1–81.
S. Guo and P. Zorin-Kranich, Decoupling for certain quadratic surfaces of low co-dimensions, J. Lond. Math. Soc. (2) 102 (2020), 319–344.
S. Guo and P. Zorin-Kranich, Decoupling for moment manifolds associated to Arkhipov—Chubarikov—Karatsuba systems, Adv. Math. 360 (2020), Article no. 106889.
J. Kim, Some remarks on Fourier restriction estimates, arXiv:1702.01231 [math.CA].
J. Lee and S. Lee, Restriction estimates to complex hypersurfaces, J. Math. Anal. Appl. 506 (2022), 125702.
G. Mockenhaupt, Bounds in Lebesgue Spaces of Oscillatory Integral Operators, Habilitationsschrift, Universität Siegen, Siegen, 1996.
D. M. Oberlin, A restriction theorem for a k-surface in ℝn, Canad. Math. Bull. 48 (2002), 260–266.
D. M. Oberlin, Convolution and restriction estimates for a 3-surface in ℝ5, J. Fourier Anal. Appl. 10 (2004), 377–382.
C. Oh, Decouplings for three-dimensional surfaces in ℝ6, Math. Z. 290 (2018), 389–419.
T. Tao, The Bochner—Riesz conjecture implies the restriction conjecture, Duke Math. J. 96 (1999), 363–375.
A. Zygmund, On Fourier coefficients and transforms of functions of two variables, Studia Math. 50 (1974), 189–201.
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The authors were supported in part by the NSF grant 1800274. The authors would like to thank the referee for valuable suggestions.
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Guo, S., Oh, C. Fourier restriction estimates for surfaces of co-dimension two in ℝ5. JAMA 148, 471–499 (2022). https://doi.org/10.1007/s11854-022-0235-2
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DOI: https://doi.org/10.1007/s11854-022-0235-2