Resolvent estimates on symmetric spaces of noncompact type. (English) Zbl 1301.47045

Kaizuka, Koichi

J. Math. Soc. Japan 66, No. 3, 895-926 (2014).

The author establishes a resolvent estimate for the Laplace-Beltrami operator or more general elliptic Fouirer multipliers on symmetric spaces of noncompact type. Then Kato theory implies time-global smoothing estimates for the corresponding dispersive equations, especially the Schrödinger evolution equation. For low-frequency estimates, a pseudo-dimension appears as an upper bound of the order of elliptic Fourier multipliers. A key of the proof is to show weighted \(L^2\)-continuity of the modified Radon transform and fractional integral operators.

Reviewer: Shangquan Bu (Beijing)

MSC:

47B38	Linear operators on function spaces (general)
42B15	Multipliers for harmonic analysis in several variables
47A10	Spectrum, resolvent
43A85	Harmonic analysis on homogeneous spaces
35B65	Smoothness and regularity of solutions to PDEs

Keywords:

resolvent; symmetric space; dispersive equation; smoothing effect; limiting absorption principle

References:

[1]	J.-P. Anker and L. Ji, Heat kernel and Green function estimates on noncompact symmetric spaces, Geom. Funct. Anal., 9 (1999), 1035-1091. · Zbl 0942.43005
[2]	M. Ben-Artzi and S. Klainerman, Decay and regularity for the Schrödinger equation, J. Anal. Math., 58 (1992), 25-37. · Zbl 0802.35057
[3]	J.-F. Bony and D. Häfner, Low frequency resolvent estimates for long range perturbations of the Euclidean Laplacian, Math. Res. Lett., 17 (2010), 303-308. · Zbl 1228.35165
[4]	J.-M. Bouclet, Low frequency estimates for long range perturbations in divergence form, Canad. J. Math., 63 (2011), 961-991. · Zbl 1234.35166
[5]	J.-M. Bouclet, Low frequency estimates and local energy decay for asymptotically Euclidean Laplacians, Comm. Partial Differential Equations, 36 (2011), 1239-1286. · Zbl 1227.35227
[6]	H. Chihara, Smoothing effects of dispersive pseudodifferential equations, Comm. Partial Differential Equations, 27 (2002), 1953-2005. · Zbl 1018.35077
[7]	H. Chihara, Resolvent estimates related with a class of dispersive equations, J. Fourier Anal. Appl., 14 (2008), 301-325. · Zbl 1148.42004
[8]	M. Cowling, Herz’s “principe de majoration” and the Kunze-Stein phenomenon, In: Harmonic Analysis and Number Theory, Montreal, PQ, 1996, (eds. S. W. Drury and M. Ram Murty), CMS Conf. Proc., 21 , Amer. Math. Soc., Providence, RI, 1997, pp.,73-88. · Zbl 0964.22008
[9]	M. Cowling, The Kunze-Stein phenomenon, Ann. Math. (2), 107 (1978), 209-234. · Zbl 0363.22007
[10]	M. Cowling, S. Giulini and S. Meda, \(L^{p}\)-\(L^{q}\) estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces. I, Duke Math. J., 72 (1993), 109-150. · Zbl 0807.43002
[11]	S.-I. Doi, Smoothing effects of Schrödinger evolution groups on Riemannian manifolds, Duke Math. J., 82 (1996), 679-706. · Zbl 0870.58101
[12]	S.-I. Doi, Commutator algebra and abstract smoothing effect, J. Funct. Anal., 168 (1999), 428-469. · Zbl 0949.58007
[13]	S.-I. Doi, Smoothing effects for Schrödinger evolution equation and global behavior of geodesic flow, Math. Ann., 318 (2000), 355-389. · Zbl 0969.35029
[14]	S. Helgason, Groups and geometric analysis, In: Integral Geometry, Invariant Differential Operators, and Spherical Functions, Pure Appl. Math., 113 , Academic Press Inc., Orlando, FL, 1984. · Zbl 0543.58001
[15]	S. Helgason, Geometric Analysis on Symmetric Spaces, Math. Surveys Monogr., 39 , Amer. Math. Soc., Providence, RI, 1994. · Zbl 0809.53057
[16]	S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Grad. Stud. Math., 34 , Amer. Math. Soc., Providence, RI, 2001. · Zbl 0993.53002
[17]	C. Herz, Sur le phénomène de Kunze-Stein, C. R. Acad. Sci. Paris Sér. A-B, 271 (1970), A491-A493 (French).
[18]	L. Hörmander, The Analysis of Linear Partial Differential Operators. III, Grundlehren Math. Wiss., 274 , Springer-Verlag, Berlin, 1985. · Zbl 0601.35001
[19]	T. Hoshiro, Decay and regularity for dispersive equations with constant coefficients, J. Anal. Math., 91 (2003), 211-230. · Zbl 1086.35018
[20]	A. D. Ionescu, An endpoint estimate for the Kunze-Stein phenomenon and related maximal operators, Ann. of Math. (2), 152 (2000), 259-275. · Zbl 0970.43002
[21]	A. D. Ionescu, Rearrangement inequalities on semisimple Lie groups, Math. Ann., 332 (2005), 739-758. · Zbl 1071.22013
[22]	T. Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann., 162 (1966), 258-279. · Zbl 0139.31203
[23]	T. Kato and K. Yajima, Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys., 1 (1989), 481-496. · Zbl 0833.47005
[24]	R. A. Kunze and E. M. Stein, Uniformly bounded representations and harmonic analysis of the \(2\times 2\) real unimodular group, Amer. J. Math., 82 (1960), 1-62. · Zbl 0156.37104
[25]	I. Rodnianski and T. Tao, Longtime decay estimates for the Schrödinger equation on manifolds, In: Mathematical Aspects of Nonlinear Dispersive Equations, Princeton, NJ, 2004, (eds J. Bourgain, C. E. Kenig and S. Klainerman), Ann. of Math. Stud., 163 , Princeton University Press, Princeton, NJ, 2007, pp.,223-253. · Zbl 1133.35022
[26]	M. Ruzhansky and M. Sugimoto, A smoothing property of Schrödinger equations in the critical case, Math. Ann., 335 (2006), 645-673. · Zbl 1109.35096
[27]	M. Ruzhansky and M. Sugimoto, Smoothing properties of evolution equations via canonical transforms and comparison principle, Proc. Lond. Math. Soc., 105 (2012), 393-423. · Zbl 1253.47026
[28]	E. M. Stein and G. Weiss, Fractional integrals on \(n\)-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514. · Zbl 0082.27201
[29]	A. Vasy and J. Wunsch, Positive commutators at the bottom of the spectrum, J. Funct. Anal., 259 (2010), 503-523. · Zbl 1194.35292
[30]	G. Vodev, Local energy decay of solutions to the wave equation for nontrapping metrics, Ark. Mat., 42 (2004), 379-397. · Zbl 1061.58024
[31]	Björn G. Walther, Regularity, decay, and best constants for dispersive equations, J. Funct. Anal., 189 (2002), 325-335. · Zbl 1017.35093