Strichartz estimates for convex co-compact hyperbolic surfaces

Wang, Jian

doi:10.1090/proc/14156

Strichartz estimates for convex co-compact hyperbolic surfaces
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by Jian Wang PDF

Proc. Amer. Math. Soc. 147 (2019), 873-883 Request permission

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