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Bilinear identities involving the k-plane transform and Fourier extension operators

Part of: Harmonic analysis in several variables Partial differential equations

Published online by Cambridge University Press:  27 January 2020

David Beltran  and
Luis Vega
David Beltran
Affiliation:
Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI53706, USA (dbeltran@math.wisc.edu)
Luis Vega
Affiliation:
Departamento de Matematicas, Universidad del Pais Vasco/Euskal Herriko Unibertsitatea (UPV/EHU), Aptdo. 644, Bilbao48080, Spain Basque Center for Applied Mathematics (BCAM), Alameda de Mazarredo 14, Bilbao48009, Spain (lvega@bcamath.org, luis.vega@ehu.es)
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Abstract

We prove certain L2(ℝn) bilinear estimates for Fourier extension operators associated to spheres and hyperboloids under the action of the k-plane transform. As the estimates are L2-based, they follow from bilinear identities: in particular, these are the analogues of a known identity for paraboloids, and may be seen as higher-dimensional versions of the classical L2(ℝ2)-bilinear identity for Fourier extension operators associated to curves in ℝ2.

Keywords

k-plane transformFourier extension operatorsbilinear identities

MSC classification

Primary: 42B37: Harmonic analysis and PDE
Secondary: 35A23: Inequalities involving derivatives and differential and integral operators, inequalities for integrals
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 3349 - 3377
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Copyright
Copyright © 2020 The Royal Society of Edinburgh

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References

1Bennett, J., Aspects of multilinear harmonic analysis related to transversality. In Harmonic analysis and partial differential equations. Contemp. Math., vol. 612, pp. 1–28 (Providence, RI: Amer. Math. Soc., 2014).Google Scholar
2Bennett, J., Bez, N., Flock, T. C., Gutiérrez, S. and Iliopoulou, M.. A sharp k-plane Strichartz inequality for the Schrödinger equation. Trans. Am. Math. Soc. 370 (2018), 5617–5633.CrossRefGoogle Scholar
3Bennett, J., Bez, N., Jeavons, C. and Pattakos, N.. On sharp bilinear Strichartz estimates of Ozawa-Tsutsumi type. J. Math. Soc. Japan 69 (2017), 459–476.CrossRefGoogle Scholar
4Bennett, J., Carbery, A. and Tao, T.. On the multilinear restriction and Kakeya conjectures. Acta Math. 196 (2006), 261–302.Google Scholar
5Bennett, J. and Iliopoulou, M.. A multilinear Fourier extension identity on ℝn. Math. Res. Lett. 25 (2018), 1089–1108.CrossRefGoogle Scholar
6Bennett, J. and Nakamura, S., Tomography bounds for the Fourier extension operator and applications. https://arxiv.org/abs/2001.01674Google Scholar
7Bez, N. and Rogers, K. M.. A sharp Strichartz estimate for the wave equation with data in the energy space. J. Eur. Math. Soc. (JEMS) 15 (2013), 805–823.CrossRefGoogle Scholar
8Bourgain, J.. Refinements of Strichartz' inequality and applications to 2D-NLS with critical nonlinearity. Internat. Math. Res. Notices 5 (1998), 253–283.CrossRefGoogle Scholar
9Bourgain, J.. Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case. J. Am. Math. Soc. 12 (1999), 145–171.CrossRefGoogle Scholar
10Carneiro, E.. A sharp inequality for the Strichartz norm. Int. Math. Res. Not. IMRN 16 (2009), 3127–3145.CrossRefGoogle Scholar
11Carneiro, E. and Oliveira e Silva, D.. Some sharp restriction inequalities on the sphere. Int. Math. Res. Not. IMRN 17 (2015), 8233–8267.CrossRefGoogle Scholar
12Carneiro, E., Oliveira e Silva, D. and Sousa, M.. Extremizers for Fourier restriction on hyperboloids. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), 389–415.CrossRefGoogle Scholar
13Christ, M. and Shao, S.. Existence of extremals for a Fourier restriction inequality. Anal. PDE 5 (2012), 261–312.CrossRefGoogle Scholar
14Christ, M. and Shao, S.. On the extremizers of an adjoint Fourier restriction inequality. Adv. Math. 230 (2012), 957–977.CrossRefGoogle Scholar
15Colliander, J., Grillakis, M. and Tzirakis, N.. Tensor products and correlation estimates with applications to nonlinear Schrödinger equations. Comm. Pure Appl. Math. 62 (2009), 920–968.CrossRefGoogle Scholar
16Colliander, J., Keel, M., Staffilani, G., Takaoka, H. and Tao, T.. Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in ℝ3. Ann. Math. (2) 167 (2008), 767–865.CrossRefGoogle Scholar
17Fefferman, C.. Inequalities for strongly singular convolution operators. Acta Math. 124 (1970), 9–36.CrossRefGoogle Scholar
18Foschi, D.. Maximizers for the Strichartz inequality. J. Eur. Math. Soc. (JEMS) 9 (2007), 739–774.Google Scholar
19Foschi, D.. Global maximizers for the sphere adjoint Fourier restriction inequality. J. Funct. Anal. 268 (2015), 690–702.CrossRefGoogle Scholar
20Foschi, D. and Klainerman, S.. Bilinear space-time estimates for homogeneous wave equations. Ann. Sci. École Norm. Sup. (4) 33 (2000), 211–274.CrossRefGoogle Scholar
21Foschi, D. and Oliveira e Silva, D.. Some recent progress on sharp Fourier restriction theory. Anal. Math. 43 (2017), 241–265.CrossRefGoogle Scholar
22Helgason, S. 1999 The Radon transform, 2nd edn. Progress in Mathematics, vol. 5 (Boston, MA, Birkhäuser Boston, Inc.).CrossRefGoogle Scholar
23Jeavons, C.. A sharp bilinear estimate for the Klein-Gordon equation in arbitrary space-time dimensions. Differ. Integral Equ. 27 (2014), 137–156.Google Scholar
24Lin, J. E. and Strauss, W. A.. Decay and scattering of solutions of a nonlinear Schrödinger equation. J. Funct. Anal. 30 (1978), 245–263.CrossRefGoogle Scholar
25Morawetz, C. S.. Time decay for the nonlinear Klein-Gordon equations. Proc. Roy. Soc. London Ser. A 306 (1968), 291–296.Google Scholar
26Nakanishi, K.. Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions 1 and 2. J. Funct. Anal. 169 (1999), 201–225.CrossRefGoogle Scholar
27Ozawa, T. and Rogers, K. M.. A sharp bilinear estimate for the Klein-Gordon equation in ℝ1+1. Int. Math. Res. Not. IMRN 5 (2014), 1367–1378.CrossRefGoogle Scholar
28Ozawa, T. and Tsutsumi, Y.. Space-time estimates for null gauge forms and nonlinear Schrödinger equations. Differ. Integral Equ. 11 (1998), 201–222.Google Scholar
29Planchon, F. and Vega, L.. Bilinear virial identities and applications. Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 261–290.CrossRefGoogle Scholar
30Quilodrán, R.. Nonexistence of extremals for the adjoint restriction inequality on the hyperboloid. J. Anal. Math. 125 (2015), 37–70.CrossRefGoogle Scholar
31Strichartz, R. S.. Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44 (1977), 705–714.CrossRefGoogle Scholar
32Tao, T.. A sharp bilinear restrictions estimate for paraboloids. Geom. Funct. Anal. 13 (2003), 1359–1384.CrossRefGoogle Scholar
33Tao, T., Some recent progress on the restriction conjecture. In Fourier analysis and convexity. Appl. Numer. Harmon. Anal., pp. 217–243 (Boston, MA: Birkhäuser Boston, 2004).CrossRefGoogle Scholar
34Tomas, P. A.. A restriction theorem for the Fourier transform. Bull. Am. Math. Soc. 81 (1975), 477–478.CrossRefGoogle Scholar
35Wolff, T.. A sharp bilinear cone restriction estimate. Ann. Math. (2) 153 (2001), 661–698.CrossRefGoogle Scholar
36Zygmund, A.. On Fourier coefficients and transforms of functions of two variables. Studia Math. 50 (1974), 189–201.CrossRefGoogle Scholar