Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds III: Global-in-time Strichartz estimates without loss

Chen, Xi

Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 3, pp. 803-829.

Dans cet article, nous examinons les estimations de Strichartz sans perte et globales en temps, définies sur des variétés non-captives asymptotiquement hyperboliques. De par la nature hyperbolique de ces variétés, l'ensemble des paires admissibles pour les estimations de Strichartz est beaucoup plus grand que d'ordinaire. Ces résultats généralisent les travaux menés par Anker–Pierfelice et Ionescu–Staffilani sur les espaces hyperboliques. Toutefois, notre approche utilise ici les estimations de mesures spectrales obtenues par l'auteur en collaboration avec Hassell afin d'établir les estimations de dispersion pour des propagateurs de Schrödinger tronqués ou micro-localisés ainsi que les estimations d'énergie correspondantes. À la différence des espaces hyperboliques, l'élément crucial est ici de gérer les points conjugués de la variété. Enfin, ces estimations de Strichartz sont appliquées au caractère bien posé dans L2 et à la diffusion L2 pour les équations de Schrödinger avec des non-linearités de type puissance et des données initiales petites.

In the present paper, we investigate global-in-time Strichartz estimates without loss on non-trapping asymptotically hyperbolic manifolds. Due to the hyperbolic nature of such manifolds, the set of admissible pairs for Strichartz estimates is much larger than usual. These results generalize the works on hyperbolic space due to Anker–Pierfelice and Ionescu–Staffilani. However, our approach is to employ the spectral measure estimates, obtained in the author's joint work with Hassell, to establish the dispersive estimates for truncated/microlocalized Schrödinger propagators as well as the corresponding energy estimates. Compared with hyperbolic space, the crucial point here is to cope with the conjugate points on the manifold. Additionally, these Strichartz estimates are applied to the L2 well-posedness and L2 scattering for nonlinear Schrödinger equations with power-like nonlinearity and small Cauchy data.

Mots clés : Asymptotically hyperbolic manifolds, Spectral measure, Dispersive estimates, Strichartz estimates

[1] Anker, J.-P.; Pierfelice, V. Nonlinear Schrödinger equation on real hyperbolic spaces, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 26 (2009), pp. 1853–1869 | Numdam | MR | Zbl

[2] Anker, J.-P.; Pierfelice, V. Wave and Klein–Gordon equations on hyperbolic spaces, Anal. PDE, Volume 7 (2014), pp. 953–995 | MR | Zbl

[3] Banica, V.; Carles, R.; Staffilani, G. Scattering theory for radial nonlinear Schrödinger equations on hyperbolic spaces, Geom. Funct. Anal., Volume 18 (2008) no. 2, pp. 367–399 | DOI | MR | Zbl

[4] Bouclet, J.-M. Strichartz estimates on asymptotically hyperbolic manifolds, Anal. PDE, Volume 4 (2011) no. 1, pp. 1–84 | MR | Zbl

[5] Bourgain, J. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations: I. Schrödinger equations, Geom. Funct. Anal., Volume 3 (1993), pp. 107–156 | DOI | MR | Zbl

[6] Bourgain, J. Exponential sums and nonlinear Schrödinger equations, Geom. Funct. Anal., Volume 3 (1993), pp. 157–178 | DOI | MR | Zbl

[7] Burq, N.; Gérard, P.; Tzvetkov, N. Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Am. J. Math., Volume 126 (2004), pp. 569–605 | DOI | MR | Zbl

[8] Burq, N.; Gérard, P.; Tzvetkov, N. Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math., Volume 159 (2005), pp. 187–223 | DOI | MR | Zbl

[9] Burq, N.; Guillarmou, C.; Hassell, A. Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics, Geom. Funct. Anal., Volume 20 (2010), pp. 627–656 | DOI | MR | Zbl

[10] Cazenave, T. Semilinear Schrödinger Equations, Courant Lect. Notes Math., vol. 10, New York Univ., Courant Inst. in Math. Sci./Amer. Math. Soc., New York/Providence, 2003 | DOI | MR | Zbl

[11] Chen, X.; Hassell, A. Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds I: resolvent construction at high energy | arXiv | DOI | Zbl

[12] Chen, X.; Hassell, A. Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds II: spectral measure, restriction theorem, spectral multiplier | arXiv | DOI | Zbl

[13] Christ, M.; Kiselev, A. Maximal functions associated to filtrations, J. Funct. Anal., Volume 179 (2001), pp. 409–425 | DOI | MR | Zbl

[14] Cowling, M. The Kunze–Stein phenomenon, Ann. Math., Volume 107 (1978), pp. 209–234 | DOI | MR | Zbl

[15] Ginibre, J.; Velo, G. Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., Volume 133 (1995) no. 1, pp. 50–68 | DOI | MR | Zbl

[16] Grafakos, L. Classical and Modern Fourier Analysis, Prentice Hall, New Jersey, 2004 | MR | Zbl

[17] Guillarmou, C. Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds, Duke Math. J., Volume 129 (2005) no. 1, pp. 1–37 | DOI | MR | Zbl

[18] Guillarmou, C.; Hassell, A. Uniform Sobolev estimates for non-trapping metrics, J. Inst. Math. Jussieu, Volume 13 (2014) no. 3, pp. 599–632 | DOI | MR | Zbl

[19] Guillarmou, C.; Hassell, A.; Sikora, A. Restriction and spectral multiplier theorems on asymptotically conic manifolds, Anal. PDE, Volume 6 (2013) no. 4, pp. 893–950 | DOI | MR | Zbl

[20] Guillarmou, C.; Qing, J. Spectral characterization of Poincaré–Einstein manifolds with infinity of positive Yamabe type, Int. Math. Res. Not., Volume 2010 (2010) no. 9, pp. 1720–1740 | MR | Zbl

[21] Hassell, A.; Zhang, J. Global-in-time Strichartz estimates on non-trapping asymptotically conic manifolds | arXiv | Zbl

[22] Ionescu, A.; Staffilani, G. Semilinear Schrödinger flows on hyperbolic spaces: scattering in H1 , Math. Ann., Volume 345 (2009), pp. 133–158 | DOI | MR | Zbl

[23] Kato, T. On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., Volume 46 (1987), pp. 113–129 | Numdam | MR | Zbl

[24] Keel, M.; Tao, T. Endpoint Strichartz estimates, Am. J. Math., Volume 120 (1998) no. 5, pp. 955–980 | DOI | MR | Zbl

[25] Kunze, R.A.; Stein, E.M. Uniformly bounded representations and harmonic analysis of the 2×2 unimodular group, Am. J. Math., Volume 82 (1960), pp. 1–62 | DOI | MR | Zbl

[26] Mazzeo, R. The Hodge cohomology of a conformally compact metric, J. Differ. Geom., Volume 28 (1988), pp. 309–339 | DOI | MR | Zbl

[27] Mazzeo, R. Unique continuation at infinity and embedded eigenvalues for asymptotically hyperbolic manifolds, Am. J. Math., Volume 113 (1991) no. 1, pp. 25–45 | DOI | MR | Zbl

[28] Mazzeo, R.; Melrose, R.B. Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal., Volume 75 (1987), pp. 260–310 | DOI | MR | Zbl

[29] Melrose, R.B.; Sá Barreto, A.; Vasy, A. Analytic continuation and semiclassical resolvent estimates on asymptotically hyperbolic spaces, Commun. Partial Differ. Equ., Volume 39 (2014), pp. 452–511 | DOI | MR | Zbl

[30] Strichartz, R.S. Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., Volume 44 (1977) no. 3, pp. 705–714 | DOI | MR | Zbl

[31] Tao, T. Nonlinear Dispersive Equations, Local and Global Analysis, CBMS Reg. Conf. Ser. Math., vol. 106, Amer. Math. Soc., Providence, RI, 2006 | DOI | MR | Zbl

[32] Wang, Y. Resolvent and radiation fields on asymptotically hyperbolic manifolds | arXiv