Intermittent symmetry breaking and stability of the sharp Agmon–Hörmander estimate on the sphere

Negro, Giuseppe; Oliveira e Silva, Diogo

doi:10.1090/proc/16072

Intermittent symmetry breaking and stability of the sharp Agmon–Hörmander estimate on the sphere
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by Giuseppe Negro and Diogo Oliveira e Silva

Proc. Amer. Math. Soc. 151 (2023), 87-99

DOI: https://doi.org/10.1090/proc/16072

Published electronically: October 19, 2022

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Abstract:

We compute the optimal constant and characterise the maximisers at all spatial scales for the Agmon–Hörmander $L^2$-Fourier adjoint restriction estimate on the sphere. The maximisers switch back and forth from being constants to being non-symmetric at the zeros of two Bessel functions. We also study the stability of this estimate and establish a sharpened version in the spirit of Bianchi–Egnell. The corresponding stability constant and maximisers again exhibit a curious intermittent behaviour.

References

S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics, J. Analyse Math. 30 (1976), 1–38. MR 466902, DOI 10.1007/BF02786703
J. A. Barcelo, A. Ruiz, and L. Vega, Weighted estimates for the Helmholtz equation and some applications, J. Funct. Anal. 150 (1997), no. 2, 356–382. MR 1479544, DOI 10.1006/jfan.1997.3131
Jonathan Bennett, Neal Bez, Taryn C. Flock, and Sanghyuk Lee, Stability of the Brascamp-Lieb constant and applications, Amer. J. Math. 140 (2018), no. 2, 543–569. MR 3783217, DOI 10.1353/ajm.2018.0013
Jonathan Bennett, Neal Bez, Michael G. Cowling, and Taryn C. Flock, Behaviour of the Brascamp-Lieb constant, Bull. Lond. Math. Soc. 49 (2017), no. 3, 512–518. MR 3723636, DOI 10.1112/blms.12049
Neal Bez, Chris Jeavons, Tohru Ozawa, and Mitsuru Sugimoto, Stability of trace theorems on the sphere, J. Geom. Anal. 28 (2018), no. 2, 1456–1476. MR 3790507, DOI 10.1007/s12220-017-9870-8
Neal Bez, Hiroki Saito, and Mitsuru Sugimoto, Applications of the Funk-Hecke theorem to smoothing and trace estimates, Adv. Math. 285 (2015), 1767–1795. MR 3406541, DOI 10.1016/j.aim.2015.08.025
Gabriele Bianchi and Henrik Egnell, A note on the Sobolev inequality, J. Funct. Anal. 100 (1991), no. 1, 18–24. MR 1124290, DOI 10.1016/0022-1236(91)90099-Q
Emanuel Carneiro and Diogo Oliveira e Silva, Some sharp restriction inequalities on the sphere, Int. Math. Res. Not. IMRN 17 (2015), 8233–8267. MR 3404013, DOI 10.1093/imrn/rnu194
Jean Dolbeault, Functional inequalities: nonlinear flows and entropy methods as a tool for obtaining sharp and constructive results, Milan J. Math. 89 (2021), no. 2, 355–386. MR 4356736, DOI 10.1007/s00032-021-00341-y
Damiano Foschi, Global maximizers for the sphere adjoint Fourier restriction inequality, J. Funct. Anal. 268 (2015), no. 3, 690–702. MR 3292351, DOI 10.1016/j.jfa.2014.10.015
D. Foschi and D. Oliveira e Silva, Some recent progress on sharp Fourier restriction theory, Anal. Math. 43 (2017), no. 2, 241–265. MR 3685152, DOI 10.1007/s10476-017-0306-2
F. Gonçalves, Orthogonal polynomials and sharp estimates for the Schrödinger equation., Int. Math. Res. Not. IMRN 2019, no. 8, 2356–2383.
Felipe Gonçalves, A sharpened Strichartz inequality for radial functions, J. Funct. Anal. 276 (2019), no. 6, 1925–1947. MR 3912796, DOI 10.1016/j.jfa.2018.07.016
Felipe Gonçalves and Giuseppe Negro, Local maximizers of adjoint Fourier restriction estimates for the cone, paraboloid and sphere, Anal. PDE 15 (2022), no. 4, 1097–1130. MR 4478298, DOI 10.2140/apde.2022.15.1097
F. Gonçalves and D. Zagier, Strichartz estimates with broken symmetries, arXiv:2011.02187, 2020.
G. Negro, A sharpened Strichartz inequality for the wave equation, arXiv:1802.04114, 2018.
Diogo Oliveira e Silva and René Quilodrán, Global maximizers for adjoint Fourier restriction inequalities on low dimensional spheres, J. Funct. Anal. 280 (2021), no. 7, Paper No. 108825, 73. MR 4211023, DOI 10.1016/j.jfa.2020.108825
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, NJ, 1971. MR 304972
Terence Tao, Some recent progress on the restriction conjecture, Fourier analysis and convexity, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2004, pp. 217–243. MR 2087245, DOI 10.1198/106186003321335099
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge; The Macmillan Company, New York, 1944. MR 10746