Strichartz estimates for convex co-compact hyperbolic surfaces
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- by Jian Wang
- Proc. Amer. Math. Soc. 147 (2019), 873-883
- DOI: https://doi.org/10.1090/proc/14156
- Published electronically: October 31, 2018
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Abstract:
Using recent work of Bourgain–Dyatlov (2018) we show that for any convex co-compact hyperbolic surface Strichartz estimates for the Schrö- dinger equation hold with an arbitrarily small loss of regularity.References
- V. Banica, The nonlinear Schrödinger equation on hyperbolic space, Comm. Partial Differential Equations 32 (2007), no. 10-12, 1643–1677. MR 2372482, DOI 10.1080/03605300600854332
- Nicolas Burq, Colin Guillarmou, and Andrew Hassell, Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics, Geom. Funct. Anal. 20 (2010), no. 3, 627–656. MR 2720226, DOI 10.1007/s00039-010-0076-5
- N. Burq, P. Gérard, and N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math. 126 (2004), no. 3, 569–605. MR 2058384
- David Borthwick, Spectral theory of infinite-area hyperbolic surfaces, Progress in Mathematics, vol. 256, Birkhäuser Boston, Inc., Boston, MA, 2007. MR 2344504, DOI 10.1007/978-0-8176-4653-0
- Jean Bourgain and Semyon Dyatlov, Spectral gaps without the pressure condition, Ann. of Math. (2) 187 (2018), no. 3, 825–867. MR 3779959, DOI 10.4007/annals.2018.187.3.5
- Jean-Marc Bouclet, Littlewood-Paley decompositions on manifolds with ends, Bull. Soc. Math. France 138 (2010), no. 1, 1–37 (English, with English and French summaries). MR 2638890, DOI 10.24033/bsmf.2584
- Jean-Marc Bouclet, Strichartz estimates on asymptotically hyperbolic manifolds, Anal. PDE 4 (2011), no. 1, 1–84. MR 2783305, DOI 10.2140/apde.2011.4.1
- Jean-Marc Bouclet, Semi-classical functional calculus on manifolds with ends and weighted $L^p$ estimates, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 3, 1181–1223 (English, with English and French summaries). MR 2918727, DOI 10.5802/aif.2638
- Michael Christ and Alexander Kiselev, Maximal functions associated to filtrations, J. Funct. Anal. 179 (2001), no. 2, 409–425. MR 1809116, DOI 10.1006/jfan.2000.3687
- Kiril Datchev, Local smoothing for scattering manifolds with hyperbolic trapped sets, Comm. Math. Phys. 286 (2009), no. 3, 837–850. MR 2472019, DOI 10.1007/s00220-008-0684-1
- Semyon Dyatlov and Maciej Zworski, Mathematical theory of scattering resonances, book in progress, http://math.mit.edu/~dyatlov/res/res_20170323.pdf.
- Emmanuel Hebey, Nonlinear analysis on manifolds: Sobolev spaces and inequalities, Courant Lecture Notes in Mathematics, vol. 5, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. MR 1688256
- Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955–980. MR 1646048
- Herbert Koch, Daniel Tataru, and Maciej Zworski, Semiclassical $L^p$ estimates, Ann. Henri Poincaré 8 (2007), no. 5, 885–916. MR 2342881, DOI 10.1007/s00023-006-0324-2
- Gigliola Staffilani and Daniel Tataru, Strichartz estimates for a Schrödinger operator with nonsmooth coefficients, Comm. Partial Differential Equations 27 (2002), no. 7-8, 1337–1372. MR 1924470, DOI 10.1081/PDE-120005841
- Maciej Zworski, Semiclassical analysis, Graduate Studies in Mathematics, vol. 138, American Mathematical Society, Providence, RI, 2012. MR 2952218, DOI 10.1090/gsm/138
Bibliographic Information
- Jian Wang
- Affiliation: Department of Mathematics, University of California, Berkeley, California, 94720
- Email: wangjian@berkeley.edu
- Received by editor(s): August 24, 2017
- Received by editor(s) in revised form: February 28, 2018, and March 8, 2018
- Published electronically: October 31, 2018
- Additional Notes: Partial support by the National Science Foundation grant DMS-1500852 is gratefully acknowledged.
- Communicated by: Michael Hitrik
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 873-883
- MSC (2010): Primary 58JXX; Secondary 35Q41
- DOI: https://doi.org/10.1090/proc/14156
- MathSciNet review: 3894924