Uniform resolvent and Strichartz estimates for Schrödinger equations with critical singularities

Bouclet, Jean-Marc; Mizutani, Haruya

doi:10.1090/tran/7243

Uniform resolvent and Strichartz estimates for Schrödinger equations with critical singularities
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by Jean-Marc Bouclet and Haruya Mizutani PDF

Trans. Amer. Math. Soc. 370 (2018), 7293-7333 Request permission

Abstract:

This paper deals with global dispersive properties of Schrödinger equations with real-valued potentials exhibiting critical singularities, where our class of potentials is more general than inverse-square type potentials and includes several anisotropic potentials. We first prove weighted resolvent estimates, which are uniform with respect to the energy, with a large class of weight functions in Morrey–Campanato spaces. Uniform Sobolev inequalities in Lorentz spaces are also studied. The proof employs the iterated resolvent identity and a classical multiplier technique. As an application, the full set of global-in-time Strichartz estimates including the endpoint case, is derived. In the proof of Strichartz estimates, we develop a general criterion on perturbations ensuring that both homogeneous and inhomogeneous endpoint estimates can be recovered from resolvent estimates. Finally, we also investigate uniform resolvent estimates for long range repulsive potentials with critical singularities by using an elementary version of the Mourre theory.

References

Wolfgang Arendt, Charles J. K. Batty, Matthias Hieber, and Frank Neubrander, Vector-valued Laplace transforms and Cauchy problems, 2nd ed., Monographs in Mathematics, vol. 96, Birkhäuser/Springer Basel AG, Basel, 2011. MR 2798103, DOI 10.1007/978-3-0348-0087-7
Valeria Banica and Thomas Duyckaerts, Weighted Strichartz estimates for radial Schrödinger equation on noncompact manifolds, Dyn. Partial Differ. Equ. 4 (2007), no. 4, 335–359. MR 2376801, DOI 10.4310/DPDE.2007.v4.n4.a3
Juan Antonio Barceló, Luis Vega, and Miren Zubeldia, The forward problem for the electromagnetic Helmholtz equation with critical singularities, Adv. Math. 240 (2013), 636–671. MR 3046321, DOI 10.1016/j.aim.2013.03.012
Marius Beceanu, New estimates for a time-dependent Schrödinger equation, Duke Math. J. 159 (2011), no. 3, 417–477. MR 2831875, DOI 10.1215/00127094-1433394
Marius Beceanu and Michael Goldberg, Schrödinger dispersive estimates for a scaling-critical class of potentials, Comm. Math. Phys. 314 (2012), no. 2, 471–481. MR 2958960, DOI 10.1007/s00220-012-1435-x
Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275, DOI 10.1007/978-3-642-66451-9
Jean-Marc Bouclet and Nikolay Tzvetkov, Strichartz estimates for long range perturbations, Amer. J. Math. 129 (2007), no. 6, 1565–1609. MR 2369889, DOI 10.1353/ajm.2007.0039
Anne Boutet de Monvel, Galina Kazantseva, and Marius Mantoiu, Some anisotropic Schrödinger operators without singular spectrum, Helv. Phys. Acta 69 (1996), no. 1, 13–25. MR 1398558
Anne Boutet de Monvel and Marius Mantoiu, The method of the weakly conjugate operator, Inverse and algebraic quantum scattering theory (Lake Balaton, 1996) Lecture Notes in Phys., vol. 488, Springer, Berlin, 1997, pp. 204–226. MR 1603530, DOI 10.1007/BFb0104937
Nicolas Burq, Fabrice Planchon, John G. Stalker, and A. Shadi Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal. 203 (2003), no. 2, 519–549. MR 2003358, DOI 10.1016/S0022-1236(03)00238-6
Nicolas Burq, Fabrice Planchon, John G. Stalker, and A. Shadi Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay, Indiana Univ. Math. J. 53 (2004), no. 6, 1665–1680. MR 2106340, DOI 10.1512/iumj.2004.53.2541
Thierry Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. MR 2002047, DOI 10.1090/cln/010
Daniele Cassani, Bernhard Ruf, and Cristina Tarsi, Optimal Sobolev type inequalities in Lorentz spaces, Potential Anal. 39 (2013), no. 3, 265–285. MR 3102987, DOI 10.1007/s11118-012-9329-2
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
Piero D’Ancona, Kato smoothing and Strichartz estimates for wave equations with magnetic potentials, Comm. Math. Phys. 335 (2015), no. 1, 1–16. MR 3314497, DOI 10.1007/s00220-014-2169-8
Piero D’Ancona and Luca Fanelli, Strichartz and smoothing estimates of dispersive equations with magnetic potentials, Comm. Partial Differential Equations 33 (2008), no. 4-6, 1082–1112. MR 2424390, DOI 10.1080/03605300701743749
Piero D’Ancona, Luca Fanelli, Luis Vega, and Nicola Visciglia, Endpoint Strichartz estimates for the magnetic Schrödinger equation, J. Funct. Anal. 258 (2010), no. 10, 3227–3240. MR 2601614, DOI 10.1016/j.jfa.2010.02.007
J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964, DOI 10.1090/surv/015
Thomas Duyckaerts, A singular critical potential for the Schrödinger operator, Canad. Math. Bull. 50 (2007), no. 1, 35–47. MR 2296623, DOI 10.4153/CMB-2007-004-3
T. Duyckaerts, private communication.
M. Burak Erdoğan, Michael Goldberg, and Wilhelm Schlag, Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in $\Bbb R^3$, J. Eur. Math. Soc. (JEMS) 10 (2008), no. 2, 507–531. MR 2390334, DOI 10.4171/JEMS/120
M. Burak Erdoğan, Michael Goldberg, and Wilhelm Schlag, Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic potentials in three and higher dimensions, Forum Math. 21 (2009), no. 4, 687–722. MR 2541480, DOI 10.1515/FORUM.2009.035
Luca Fanelli, Non-trapping magnetic fields and Morrey-Campanato estimates for Schrödinger operators, J. Math. Anal. Appl. 357 (2009), no. 1, 1–14. MR 2526801, DOI 10.1016/j.jmaa.2009.03.057
Luca Fanelli, Veronica Felli, Marco A. Fontelos, and Ana Primo, Time decay of scaling invariant electromagnetic Schrödinger equations on the plane, Comm. Math. Phys. 337 (2015), no. 3, 1515–1533. MR 3339184, DOI 10.1007/s00220-015-2291-2
Charles L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 2, 129–206. MR 707957, DOI 10.1090/S0273-0979-1983-15154-6
S. Fournais and E. Skibsted, Zero energy asymptotics of the resolvent for a class of slowly decaying potentials, Math. Z. 248 (2004), no. 3, 593–633. MR 2097376, DOI 10.1007/s00209-004-0673-9
Rupert L. Frank, Eigenvalue bounds for Schrödinger operators with complex potentials, Bull. Lond. Math. Soc. 43 (2011), no. 4, 745–750. MR 2820160, DOI 10.1112/blms/bdr008
Michael Goldberg, Strichartz estimates for the Schrödinger equation with time-periodic $L^{n/2}$ potentials, J. Funct. Anal. 256 (2009), no. 3, 718–746. MR 2484934, DOI 10.1016/j.jfa.2008.11.005
Michael Goldberg, Luis Vega, and Nicola Visciglia, Counterexamples of Strichartz inequalities for Schrödinger equations with repulsive potentials, Int. Math. Res. Not. , posted on (2006), Art. ID 13927, 16. MR 2211154, DOI 10.1155/IMRN/2006/13927
M. Goldberg and W. Schlag, A limiting absorption principle for the three-dimensional Schrödinger equation with $L^p$ potentials, Int. Math. Res. Not. 75 (2004), 4049–4071. MR 2112327, DOI 10.1155/S1073792804140324
Loukas Grafakos, Classical Fourier analysis, 2nd ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2008. MR 2445437
Colin Guillarmou and Andrew Hassell, Uniform Sobolev estimates for non-trapping metrics, J. Inst. Math. Jussieu 13 (2014), no. 3, 599–632. MR 3211800, DOI 10.1017/S1474748013000273
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, DOI 10.2140/apde.2016.9.151
Arne Jensen and Tosio Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J. 46 (1979), no. 3, 583–611. MR 544248
Arne Jensen and Gheorghe Nenciu, A unified approach to resolvent expansions at thresholds, Rev. Math. Phys. 13 (2001), no. 6, 717–754. MR 1841744, DOI 10.1142/S0129055X01000843
Tosio Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann. 162 (1965/66), 258–279. MR 190801, DOI 10.1007/BF01360915
Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617
Tosio Kato and Kenji Yajima, Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys. 1 (1989), no. 4, 481–496. MR 1061120, DOI 10.1142/S0129055X89000171
Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955–980. MR 1646048, DOI 10.1353/ajm.1998.0039
C. E. Kenig, A. Ruiz, and C. D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J. 55 (1987), no. 2, 329–347. MR 894584, DOI 10.1215/S0012-7094-87-05518-9
Rowan Killip, Changxing Miao, Monica Visan, Junyong Zhang, and Jiqiang Zheng, The energy-critical NLS with inverse-square potential, Discrete Contin. Dyn. Syst. 37 (2017), no. 7, 3831–3866. MR 3639442, DOI 10.3934/dcds.2017162
Rowan Killip, Jason Murphy, Monica Visan, and Jiqiang Zheng, The focusing cubic NLS with inverse-square potential in three space dimensions, Differential Integral Equations 30 (2017), no. 3-4, 161–206. MR 3611498
Seonghak Kim and Youngwoo Koh, Strichartz estimates for the magnetic Schrödinger equation with potentials $V$ of critical decay, Comm. Partial Differential Equations 42 (2017), no. 9, 1467–1480. MR 3717441, DOI 10.1080/03605302.2017.1377229
H. Kovařík and F. Truc, Schrödinger operators on a half-line with inverse square potentials, Math. Model. Nat. Phenom. 9 (2014), no. 5, 170–176. MR 3264315, DOI 10.1051/mmnp/20149511
Jeremy Marzuola, Jason Metcalfe, and Daniel Tataru, Strichartz estimates and local smoothing estimates for asymptotically flat Schrödinger equations, J. Funct. Anal. 255 (2008), no. 6, 1497–1553. MR 2565717, DOI 10.1016/j.jfa.2008.05.022
Giorgio Metafune, Motohiro Sobajima, and Chiara Spina, Weighted Calderón-Zygmund and Rellich inequalities in $L^p$, Math. Ann. 361 (2015), no. 1-2, 313–366. MR 3302622, DOI 10.1007/s00208-014-1075-x
H. Mizutani, Eigenvalue bounds for non-self-adjoint Schrödinger operators with the inverse-square potential, preprint. http://arxiv.org/abs/1607.01727
Haruya Mizutani, Remarks on endpoint Strichartz estimates for Schrödinger equations with the critical inverse-square potential, J. Differential Equations 263 (2017), no. 7, 3832–3853. MR 3670038, DOI 10.1016/j.jde.2017.05.006
Kiyoshi Mochizuki, Uniform resolvent estimates for magnetic Schrödinger operators and smoothing effects for related evolution equations, Publ. Res. Inst. Math. Sci. 46 (2010), no. 4, 741–754. MR 2791005, DOI 10.2977/PRIMS/24
Shu Nakamura, Low energy asymptotics for Schrödinger operators with slowly decreasing potentials, Comm. Math. Phys. 161 (1994), no. 1, 63–76. MR 1266070, DOI 10.1007/BF02099413
Benoit Perthame and Luis Vega, Morrey-Campanato estimates for Helmholtz equations, J. Funct. Anal. 164 (1999), no. 2, 340–355. MR 1695559, DOI 10.1006/jfan.1999.3391
Fabrice Planchon, John G. Stalker, and A. Shadi Tahvildar-Zadeh, $L^p$ estimates for the wave equation with the inverse-square potential, Discrete Contin. Dyn. Syst. 9 (2003), no. 2, 427–442. MR 1952384, DOI 10.3934/dcds.2003.9.427
Fabrice Planchon, John G. Stalker, and A. Shadi Tahvildar-Zadeh, Dispersive estimate for the wave equation with the inverse-square potential, Discrete Contin. Dyn. Syst. 9 (2003), no. 6, 1387–1400. MR 2017672, DOI 10.3934/dcds.2003.9.1387
M. Reed and B. Simon, Methods of Modern Mathematical Physics I, IV, Academic Press, 1972, 1978.
Serge Richard, Some improvements in the method of the weakly conjugate operator, Lett. Math. Phys. 76 (2006), no. 1, 27–36. MR 2223761, DOI 10.1007/s11005-006-0079-1
Igor Rodnianski and Wilhelm Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math. 155 (2004), no. 3, 451–513. MR 2038194, DOI 10.1007/s00222-003-0325-4
Igor Rodnianski and Terence Tao, Effective limiting absorption principles, and applications, Comm. Math. Phys. 333 (2015), no. 1, 1–95. MR 3294943, DOI 10.1007/s00220-014-2177-8
Keith M. Rogers, Strichartz estimates via the Schrödinger maximal operator, Math. Ann. 343 (2009), no. 3, 603–622. MR 2480704, DOI 10.1007/s00208-008-0283-7
E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), no. 4, 813–874. MR 1175693, DOI 10.2307/2374799
Hart F. Smith and Christopher D. Sogge, Global Strichartz estimates for nontrapping perturbations of the Laplacian, Comm. Partial Differential Equations 25 (2000), no. 11-12, 2171–2183. MR 1789924, DOI 10.1080/03605300008821581
Giorgio Talenti, Inequalities in rearrangement invariant function spaces, Nonlinear analysis, function spaces and applications, Vol. 5 (Prague, 1994) Prometheus, Prague, 1994, pp. 177–230. MR 1322313
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, DOI 10.1090/cbms/106
Björn G. Walther, A sharp weighted $L^2$-estimate for the solution to the time-dependent Schrödinger equation, Ark. Mat. 37 (1999), no. 2, 381–393. MR 1714761, DOI 10.1007/BF02412222
Junyong Zhang and Jiqiang Zheng, Scattering theory for nonlinear Schrödinger equations with inverse-square potential, J. Funct. Anal. 267 (2014), no. 8, 2907–2932. MR 3255478, DOI 10.1016/j.jfa.2014.08.012