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On extremizers for Strichartz estimates for higher order Schrödinger equations


Authors: Diogo Oliveira e Silva and René Quilodrán
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 42B10
DOI: https://doi.org/10.1090/tran/7223
Published electronically: February 26, 2018
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Abstract: For an appropriate class of convex functions $ \phi $, we study the Fourier extension operator on the surface $ \{(y,\vert y\vert ^2+\phi (y)):y\in \mathbb{R}^2\}$ equipped with projection measure. For the corresponding extension inequality, we compute optimal constants and prove that extremizers do not exist. The main tool is a new comparison principle for convolutions of certain singular measures that holds in all dimensions. Using tools of concentration-compactness flavor, we further investigate the behavior of general extremizing sequences. Our work is directly related to the study of extremizers and optimal constants for Strichartz estimates of certain higher order Schrödinger equations. In particular, we resolve a dichotomy from the recent literature concerning the existence of extremizers for a family of fourth order Schrödinger equations and compute the corresponding operator norms exactly where only lower bounds were previously known.


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  • [1] Jong-Guk Bak and David McMichael, Convolution of a measure with itself and a restriction theorem, Proc. Amer. Math. Soc. 125 (1997), no. 2, 463-470. MR 1350932, https://doi.org/10.1090/S0002-9939-97-03569-7
  • [2] Matania Ben-Artzi, Herbert Koch, and Jean-Claude Saut, Dispersion estimates for fourth order Schrödinger equations, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 2, 87-92 (English, with English and French summaries). MR 1745182, https://doi.org/10.1016/S0764-4442(00)00120-8
  • [3] Jonathan Bennett, Neal Bez, Anthony Carbery, and Dirk Hundertmark, Heat-flow monotonicity of Strichartz norms, Anal. PDE 2 (2009), no. 2, 147-158. MR 2547132, https://doi.org/10.2140/apde.2009.2.147
  • [4] Neal Bez and Mitsuru Sugimoto, Optimal constants and extremisers for some smoothing estimates, J. Anal. Math. 131 (2017), 159-187. MR 3631453, https://doi.org/10.1007/s11854-017-0005-8
  • [5] Emanuel Carneiro, A sharp inequality for the Strichartz norm, Int. Math. Res. Not. IMRN 16 (2009), 3127-3145. MR 2533799, https://doi.org/10.1093/imrn/rnp045
  • [6] 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, https://doi.org/10.1093/imrn/rnu194
  • [7] Michael Christ and Shuanglin Shao, Existence of extremals for a Fourier restriction inequality, Anal. PDE 5 (2012), no. 2, 261-312. MR 2970708, https://doi.org/10.2140/apde.2012.5.261
  • [8] Michael Christ and Shuanglin Shao, On the extremizers of an adjoint Fourier restriction inequality, Adv. Math. 230 (2012), no. 3, 957-977. MR 2921167, https://doi.org/10.1016/j.aim.2012.03.020
  • [9] Luca Fanelli, Luis Vega, and Nicola Visciglia, On the existence of maximizers for a family of restriction theorems, Bull. Lond. Math. Soc. 43 (2011), no. 4, 811-817. MR 2820166, https://doi.org/10.1112/blms/bdr014
  • [10] Luca Fanelli, Luis Vega, and Nicola Visciglia, Existence of maximizers for Sobolev-Strichartz inequalities, Adv. Math. 229 (2012), no. 3, 1912-1923. MR 2871161, https://doi.org/10.1016/j.aim.2011.12.012
  • [11] Damiano Foschi, Maximizers for the Strichartz inequality, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 4, 739-774. MR 2341830, https://doi.org/10.4171/JEMS/95
  • [12] Damiano Foschi, Global maximizers for the sphere adjoint Fourier restriction inequality, J. Funct. Anal. 268 (2015), no. 3, 690-702. MR 3292351, https://doi.org/10.1016/j.jfa.2014.10.015
  • [13] 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, https://doi.org/10.1007/s10476-017-0306-2
  • [14] Rupert L. Frank, Elliott H. Lieb, and Julien Sabin, Maximizers for the Stein-Tomas inequality, Geom. Funct. Anal. 26 (2016), no. 4, 1095-1134. MR 3558306, https://doi.org/10.1007/s00039-016-0380-9
  • [15] Wei Han, The sharp Strichartz and Sobolev-Strichartz inequalities for the fourth-order Schrödinger equation, Math. Methods Appl. Sci. 38 (2015), no. 8, 1506-1514. MR 3343568, https://doi.org/10.1002/mma.3164
  • [16] Jean-Baptiste Hiriart-Urruty and Claude Lemaréchal, Fundamentals of convex analysis, abridged version of Convex analysis and minimization algorithms. I [Springer, Berlin, 1993; MR1261420 (95m:90001)] and II [ibid.; MR1295240 (95m:90002)], Grundlehren Text Editions, Springer-Verlag, Berlin, 2001. MR 1865628
  • [17] Dirk Hundertmark and Vadim Zharnitsky, On sharp Strichartz inequalities in low dimensions, Int. Math. Res. Not. , posted on (2006), Art. ID 34080, 18. MR 2219206, https://doi.org/10.1155/IMRN/2006/34080
  • [18] Jin-Cheng Jiang, Benoit Pausader, and Shuanglin Shao, The linear profile decomposition for the fourth order Schrödinger equation, J. Differential Equations 249 (2010), no. 10, 2521-2547. MR 2718708, https://doi.org/10.1016/j.jde.2010.06.014
  • [19] Jin-Cheng Jiang, Shuanglin Shao and Betsy Stovall,
    Linear profile decompositions for a family of fourth order Schrödinger equations,
    preprint, 2014, arXiv:1410.7520.
  • [20] Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955-980. MR 1646048
  • [21] Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), no. 1, 33-69. MR 1101221, https://doi.org/10.1512/iumj.1991.40.40003
  • [22] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109-145 (English, with French summary). MR 778970
  • [23] Diogo Oliveira e Silva, Extremizers for Fourier restriction inequalities: convex arcs, J. Anal. Math. 124 (2014), 337-385. MR 3286057, https://doi.org/10.1007/s11854-014-0035-4
  • [24] Diogo Oliveira e Silva,
    Nonexistence of extremizers for certain convex curves,
    preprint, 2012. To appear in Mathematical Research Letters, arXiv:1210.0585.
  • [25] Benoit Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dyn. Partial Differ. Equ. 4 (2007), no. 3, 197-225. MR 2353631, https://doi.org/10.4310/DPDE.2007.v4.n3.a1
  • [26] René Quilodrán, Nonexistence of extremals for the adjoint restriction inequality on the hyperboloid, J. Anal. Math. 125 (2015), 37-70. MR 3317897, https://doi.org/10.1007/s11854-015-0002-8
  • [27] Javier Ramos, A refinement of the Strichartz inequality for the wave equation with applications, Adv. Math. 230 (2012), no. 2, 649-698. MR 2914962, https://doi.org/10.1016/j.aim.2012.02.020
  • [28] Michael Ruzhansky and Mitsuru Sugimoto, Smoothing properties of evolution equations via canonical transforms and comparison principle, Proc. Lond. Math. Soc. (3) 105 (2012), no. 2, 393-423. MR 2959931, https://doi.org/10.1112/plms/pds006
  • [29] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, with the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. MR 1232192
  • [30] Robert 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. MR 0512086
  • [31] Peter A. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 477-478. MR 0358216, https://doi.org/10.1090/S0002-9904-1975-13790-6

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Additional Information

Diogo Oliveira e Silva
Affiliation: Hausdorff Center for Mathematics 53115 Bonn, Germany
Email: dosilva@math.uni-bonn.de

René Quilodrán
Affiliation: Departamento de Ciencias Exactas Universidad de Los Lagos Avenida Fuchslocher 1305, Osorno, Chile
Email: rene.quilodran@ulagos.cl

DOI: https://doi.org/10.1090/tran/7223
Keywords: Fourier extension theory, extremizers, optimal constants, convolution of singular measures, concentration-compactness, Strichartz inequalities.
Received by editor(s): June 13, 2016
Received by editor(s) in revised form: December 26, 2016
Published electronically: February 26, 2018
Additional Notes: The first author was partially supported by the Hausdorff Center for Mathematics.
Article copyright: © Copyright 2018 American Mathematical Society

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