• [1] Jean-Philippe Anker and Vittoria Pierfelice, Wave and Klein-Gordon equations on hyperbolic spaces, Anal. PDE 7 (2014), no. 4, 953–995. MR 3254350, https://doi.org/10.2140/apde.2014.7.953
• [2] Jean-Philippe Anker, Vittoria Pierfelice, and Maria Vallarino, The wave equation on hyperbolic spaces, J. Differential Equations 252 (2012), no. 10, 5613–5661. MR 2902129, https://doi.org/10.1016/j.jde.2012.01.031
• [3] Jean-Philippe Anker, Vittoria Pierfelice, and Maria Vallarino, The wave equation on Damek-Ricci spaces, Ann. Mat. Pura Appl. (4) 194 (2015), no. 3, 731–758. MR 3345662, https://doi.org/10.1007/s10231-013-0395-x
• [4] M. Blair, Y. Sire, C. Sogge, Quasimode, eigenfunction and spectral projection bounds for Schrödinger operators on manifolds with critically singular potentials, J. Geom. Anal. (2019), doi.org/10.1007/s12220-019-00287-z.
• [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, https://doi.org/10.24033/bsmf.2584
• [6] Xi Chen, Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds III: Global-in-time Strichartz estimates without loss, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 3, 803–829 (English, with English and French summaries). MR 3778653, https://doi.org/10.1016/j.anihpc.2017.08.003
• [7] Xi Chen and Andrew Hassell, Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds I: Resolvent construction at high energy, Comm. Partial Differential Equations 41 (2016), no. 3, 515–578. MR 3473907, https://doi.org/10.1080/03605302.2015.1116561
• [8] Xi Chen and Andrew Hassell, Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds II: Spectral measure, restriction theorem, spectral multipliers, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 3, 1011–1075 (English, with English and French summaries). MR 3805767
• [9] Michael Christ and Alexander Kiselev, Maximal functions associated to filtrations, J. Funct. Anal. 179 (2001), no. 2, 409–425. MR 1809116, https://doi.org/10.1006/jfan.2000.3687
• [10] Jean Fontaine, A semilinear wave equation on hyperbolic spaces, Comm. Partial Differential Equations 22 (1997), no. 3-4, 633–659. MR 1443052, https://doi.org/10.1080/03605309708821277
• [11] Colin Guillarmou and Andrew Hassell, Uniform Sobolev estimates for non-trapping metrics, J. Inst. Math. Jussieu 13 (2014), no. 3, 599–632. MR 3211800, https://doi.org/10.1017/S1474748013000273
• [12] J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal. 133 (1995), no. 1, 50–68. MR 1351643, https://doi.org/10.1006/jfan.1995.1119
• [13] Andrew Hassell and Junyong Zhang, Global-in-time Strichartz estimates on nontrapping, asymptotically conic manifolds, Anal. PDE 9 (2016), no. 1, 151–192. MR 3461304, https://doi.org/10.2140/apde.2016.9.151
• [14] Fritz John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math. 28 (1979), no. 1-3, 235–268. MR 535704, https://doi.org/10.1007/BF01647974
• [15] R. A. Kunze and E. M. Stein, Uniformly bounded representations and harmonic analysis of the 2×2 real unimodular group, Amer. J. Math. 82 (1960), 1–62. MR 163988, https://doi.org/10.2307/2372876
• [16] Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955–980. MR 1646048
• [17] Andrew Lawrie, Sung-Jin Oh, and Sohrab Shahshahani, Profile decompositions for wave equations on hyperbolic space with applications, Math. Ann. 365 (2016), no. 1-2, 707–803. MR 3498926, https://doi.org/10.1007/s00208-015-1305-x
• [18] A. Lawrie, J. Lührmann, S.-J. Oh, and S. Shahshahani, Local smoothing estimates for Schrödinger equations on hyperbolic space, 1808.04777, 2018.
• [19] A. Lawrie, J. Lührmann, S.-J. Oh, and S. Shahshahani, Asymptotic stability of harmonic maps on the hyperbolic plane under the Schrödinger maps evolution, 1909.06899, 2019.
• [20] Noël Lohoué, Estimation des fonctions de Littlewood-Paley-Stein sur les variétés riemanniennes à courbure non positive, Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 4, 505–544 (French). MR 932796
• [21] Rafe Mazzeo, The Hodge cohomology of a conformally compact metric, J. Differential Geom. 28 (1988), no. 2, 309–339. MR 961517
• [22] Rafe R. Mazzeo and Richard B. Melrose, Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal. 75 (1987), no. 2, 260–310. MR 916753, https://doi.org/10.1016/0022-1236(87)90097-8
• [23] Jason Metcalfe and Michael Taylor, Nonlinear waves on 3D hyperbolic space, Trans. Amer. Math. Soc. 363 (2011), no. 7, 3489–3529. MR 2775816, https://doi.org/10.1090/S0002-9947-2011-05122-6
• [24] Jason Metcalfe and Michael Taylor, Dispersive wave estimates on 3D hyperbolic space, Proc. Amer. Math. Soc. 140 (2012), no. 11, 3861–3866. MR 2944727, https://doi.org/10.1090/S0002-9939-2012-11534-5
• [25] 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 512086
• [26] Yannick Sire, Christopher D. Sogge, and Chengbo Wang, The Strauss conjecture on negatively curved backgrounds, Discrete Contin. Dyn. Syst. 39 (2019), no. 12, 7081–7099. MR 4026182, https://doi.org/10.3934/dcds.2019296
• [27] Elias M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory., Annals of Mathematics Studies, No. 63, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970. MR 0252961
• [28] Walter A. Strauss, Nonlinear scattering theory at low energy, J. Functional Analysis 41 (1981), no. 1, 110–133. MR 614228, https://doi.org/10.1016/0022-1236(81)90063-X
• [29] Daniel Tataru, Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation, Trans. Amer. Math. Soc. 353 (2001), no. 2, 795–807. MR 1804518, https://doi.org/10.1090/S0002-9947-00-02750-1
• [30] Michael Taylor, Hardy spaces and BMO on manifolds with bounded geometry, J. Geom. Anal. 19 (2009), no. 1, 137–190. MR 2465300, https://doi.org/10.1007/s12220-008-9054-7
• [31] Terence Tao, Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics, vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. Local and global analysis. MR 2233925
• [32] C. Wang, Recent progress on the Strauss conjecture and related problems, Mathematica 48 (2018), no. 1, 111-130.
• [33] Junyong Zhang, Strichartz estimates and nonlinear wave equation on nontrapping asymptotically conic manifolds, Adv. Math. 271 (2015), 91–111. MR 3291858, https://doi.org/10.1016/j.aim.2014.11.013

References

[1]

Jean-Philippe Anker and Vittoria Pierfelice, Wave and Klein-Gordon equations on hyperbolic spaces, Anal. PDE 7 (2014), no. 4, 953-995. MR 3254350, https://doi.org/10.2140/apde.2014.7.953

[2]

Jean-Philippe Anker, Vittoria Pierfelice, and Maria Vallarino, The wave equation on hyperbolic spaces, J. Differential Equations 252 (2012), no. 10, 5613-5661. MR 2902129, https://doi.org/10.1016/j.jde.2012.01.031

[3]

Jean-Philippe Anker, Vittoria Pierfelice, and Maria Vallarino, The wave equation on Damek-Ricci spaces, Ann. Mat. Pura Appl. (4) 194 (2015), no. 3, 731-758. MR 3345662, https://doi.org/10.1007/s10231-013-0395-x

[4]

M. Blair, Y. Sire, C. Sogge, Quasimode, eigenfunction and spectral projection bounds for Schrödinger operators on manifolds with critically singular potentials, J. Geom. Anal. (2019), doi.org/10.1007/s12220-019-00287-z.

[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, https://doi.org/10.24033/bsmf.2584

[6]

Xi Chen, Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds III: Global-in-time Strichartz estimates without loss, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 3, 803-829 (English, with English and French summaries). MR 3778653, https://doi.org/10.1016/j.anihpc.2017.08.003

[7]

Xi Chen and Andrew Hassell, Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds I: Resolvent construction at high energy, Comm. Partial Differential Equations 41 (2016), no. 3, 515-578. MR 3473907, https://doi.org/10.1080/03605302.2015.1116561

[8]

Xi Chen and Andrew Hassell, Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds II: Spectral measure, restriction theorem, spectral multipliers, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 3, 1011-1075 (English, with English and French summaries). MR 3805767

[9]

Michael Christ and Alexander Kiselev, Maximal functions associated to filtrations, J. Funct. Anal. 179 (2001), no. 2, 409-425. MR 1809116, https://doi.org/10.1006/jfan.2000.3687

[10]

Jean Fontaine, A semilinear wave equation on hyperbolic spaces, Comm. Partial Differential Equations 22 (1997), no. 3-4, 633-659. MR 1443052, https://doi.org/10.1080/03605309708821277

[11]

Colin Guillarmou and Andrew Hassell, Uniform Sobolev estimates for non-trapping metrics, J. Inst. Math. Jussieu 13 (2014), no. 3, 599-632. MR 3211800, https://doi.org/10.1017/S1474748013000273

[12]

J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal. 133 (1995), no. 1, 50-68. MR 1351643, https://doi.org/10.1006/jfan.1995.1119

[13]

Andrew Hassell and Junyong Zhang, Global-in-time Strichartz estimates on nontrapping, asymptotically conic manifolds, Anal. PDE 9 (2016), no. 1, 151-192. MR 3461304, https://doi.org/10.2140/apde.2016.9.151

[14]

Fritz John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math. 28 (1979), no. 1-3, 235-268. MR 535704, https://doi.org/10.1007/BF01647974

[15]

R. A. Kunze and E. M. Stein, Uniformly bounded representations and harmonic analysis of the real unimodular group, Amer. J. Math. 82 (1960), 1-62. MR 163988, https://doi.org/10.2307/2372876

[16]

Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955-980. MR 1646048

[17]

Andrew Lawrie, Sung-Jin Oh, and Sohrab Shahshahani, Profile decompositions for wave equations on hyperbolic space with applications, Math. Ann. 365 (2016), no. 1-2, 707-803. MR 3498926, https://doi.org/10.1007/s00208-015-1305-x

[18]

A. Lawrie, J. Lührmann, S.-J. Oh, and S. Shahshahani, Local smoothing estimates for Schrödinger equations on hyperbolic space, 1808.04777, 2018.

[19]

A. Lawrie, J. Lührmann, S.-J. Oh, and S. Shahshahani, Asymptotic stability of harmonic maps on the hyperbolic plane under the Schrödinger maps evolution, 1909.06899, 2019.

[20]

Noël Lohoué, Estimation des fonctions de Littlewood-Paley-Stein sur les variétés riemanniennes à courbure non positive, Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 4, 505-544 (French). MR 932796

[21]

Rafe Mazzeo, The Hodge cohomology of a conformally compact metric, J. Differential Geom. 28 (1988), no. 2, 309-339. MR 961517

[22]

Rafe R. Mazzeo and Richard B. Melrose, Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal. 75 (1987), no. 2, 260-310. MR 916753, https://doi.org/10.1016/0022-1236(87)90097-8

[23]

Jason Metcalfe and Michael Taylor, Nonlinear waves on 3D hyperbolic space, Trans. Amer. Math. Soc. 363 (2011), no. 7, 3489-3529. MR 2775816, https://doi.org/10.1090/S0002-9947-2011-05122-6

[24]

Jason Metcalfe and Michael Taylor, Dispersive wave estimates on 3D hyperbolic space, Proc. Amer. Math. Soc. 140 (2012), no. 11, 3861-3866. MR 2944727, https://doi.org/10.1090/S0002-9939-2012-11534-5

[25]

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 512086

[26]

Yannick Sire, Christopher D. Sogge, and Chengbo Wang, The Strauss conjecture on negatively curved backgrounds, Discrete Contin. Dyn. Syst. 39 (2019), no. 12, 7081-7099. MR 4026182, https://doi.org/10.3934/dcds.2019296

[27]

Elias M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory., Annals of Mathematics Studies, No. 63, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970. MR 0252961

[28]

Walter A. Strauss, Nonlinear scattering theory at low energy, J. Functional Analysis 41 (1981), no. 1, 110-133. MR 614228, https://doi.org/10.1016/0022-1236(81)90063-X

[29]

Daniel Tataru, Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation, Trans. Amer. Math. Soc. 353 (2001), no. 2, 795-807. MR 1804518, https://doi.org/10.1090/S0002-9947-00-02750-1

[30]

Michael Taylor, Hardy spaces and BMO on manifolds with bounded geometry, J. Geom. Anal. 19 (2009), no. 1, 137-190. MR 2465300, https://doi.org/10.1007/s12220-008-9054-7

[31]

Terence Tao, Nonlinear dispersive equations: Local and global analysis, CBMS Regional Conference Series in Mathematics, vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. MR 2233925

[32]

C. Wang, Recent progress on the Strauss conjecture and related problems, Mathematica 48 (2018), no. 1, 111-130.

[33]

Junyong Zhang, Strichartz estimates and nonlinear wave equation on nontrapping asymptotically conic manifolds, Adv. Math. 271 (2015), 91-111. MR 3291858, https://doi.org/10.1016/j.aim.2014.11.013