Abstract
Let H be a selfadjoint operator and A a closed operator on a Hilbert space \({\mathcal{H}}\) . If A is H-(super)smooth in the sense of Kato-Yajima, we prove that \({AH^{-\frac{1}{4}}}\) is \({\sqrt{H}}\) -(super)smooth. This allows us to include wave and Klein-Gordon equations in the abstract theory at the same level of generality as Schrödinger equations.
We give a few applications and in particular, based on the resolvent estimates of Erdogan, Goldberg and Schlag (Forum Mathematicum 21:687–722, 2009), we prove Strichartz estimates for wave equations perturbed with large magnetic potentials on \({\mathbb{R}^{n}}\) , n ≥ 3.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ben-Artzi, Matania, Klainerman, Sergiu: Decay and regularity for the Schrödinger equation. J. Anal. Math. 58. Festschrift on the occasion of the 70th birthday of Shmuel Agmon, pp. 25–37 (1992)
Burq N. et al.: Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay. Indiana Univ. Math. J. 53(6), 1665–1680 (2004)
Chihara H.: Smoothing effects of dispersive pseudodifferential equations. Commun. Partial Differ. Equ. 27(9-10), 1953–2005 (2002)
D’Ancona P., Fanelli L.: L p-boundedness of the wave operator for the one dimensional Schrödinger operator. Commun. Math. Phys. 268(2), 415–438 (2006)
D’Ancona P., Fanelli L.: Strichartz and smoothing estimates of dispersive equations with magnetic potentials. Commun. Partial Differ. Equ. 33(4–6), 1082–1112 (2008)
D’Ancona P., Racke R.: Evolution equations on non-flat waveguides. English. Arch. Ration. Mech. Anal. 206(1), 81–110 (2012)
D’Ancona P., Selberg S.: Dispersive estimates for the 1D Schrödinger equation with a steplike potential. J. Differ. Equ. 252, 1603–1634 (2012)
Erdogan, B., Goldberg, M., Green, W.: Dispersive estimates for four dimensional Schrödinger and wave equations with obstructions at zero energy. Commun. Partial. Differ. Equ., arXiv:1310.6302
Erdoğan M.B., Goldberg M., Schlag W.: Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic potentials in three and higher dimensions. Forum Math. 21, 687–722 (2009)
Fang D., Wang C.: Weighted Strichartz estimates with angular regularity and their applications. Forum Math. 23(1), 181–205 (2011)
Green, W.: Time decay estimates for the wave equation with potential in dimension two (2013). arXiv:1307.2219v4 [math.AP]
Hille, E., Phillips, R.S.: Functional analysis and semi-groups. Third printing of the revised edition of 1957, American Mathematical Society Colloquium Publications, vol. XXXI. Providence, R. I.: American Mathematical Society, pp. xii+808 (1974)
Hoshiro T.: On weighted L 2 estimates of solutions to wave equations. J. Anal. Math. 72, 127–140 (1997)
Ionescu A., Kenig C.: Well-posedness and local smoothing of solutions of Schrödinger equations. Math. Res. Lett. 12, 193–205 (2005)
Journé J.-L., Soffer A., Sogge Christopher D.: Decay estimates for Schrödinger operators. Commun. Pure Appl. Math. 44(5), 573–604 (1991)
Kato T.: Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162, 258–279 (1965/1966)
Kato T., Yajima K.: Some examples of smooth operators and the associated smoothing effect. Rev. Math. Phys. 1(4), 481–496 (1989)
Marzuola J., Metcalfe J., Tataru D.: Strichartz estimates and local smoothing estimates for asymptotically flat Schrödinger equations. J. Funct. Anal. 255(6), 1497–1553 (2008)
Mochizuki K.: Uniform resolvent estimates for magnetic Schrödinger operators and smoothing effects for related evolution equations. Publ. Res. Inst. Math. Sci. 46(4), 741–754 (2010)
Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. New York: Academic Press [Harcourt Brace Jovanovich Publishers], pp. xv+396 (1978)
Rodnianski I., Schlag W.: Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. Invent. Math. 155(3), 451–513 (2004)
Ruzhansky, M., Sugimoto, M.: A New Proof of Global Smoothing Estimates for Dispersive Equations. In: Advances in pseudo-differential operators. vol. 155. Oper. Theory Adv. Appl. Basel: Birkhäuser, pp. 65–75 (2004)
Walther Björn G.: A sharp weighted L 2-estimate for the solution to the time-dependent Schrödinger equation. Ark. Mat. 37(2), 381–393 (1999)
Watanabe K.: Smooth perturbations of the selfadjoint operator \({|\Delta|^{\alpha/2}}\) . Tokyo J. Math. 14(1), 239–250 (1991)
Yajima K.: Existence of solutions for Schrödinger evolution equations. Commun. Math. Phys. 110(3), 415–426 (1987)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by W. Schlag
Rights and permissions
About this article
Cite this article
D’Ancona, P. Kato Smoothing and Strichartz Estimates for Wave Equations with Magnetic Potentials. Commun. Math. Phys. 335, 1–16 (2015). https://doi.org/10.1007/s00220-014-2169-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-014-2169-8