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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Uniform resolvent and Strichartz estimates for Schrödinger equations with critical singularities


Authors: Jean-Marc Bouclet and Haruya Mizutani
Journal: Trans. Amer. Math. Soc. 370 (2018), 7293-7333
MSC (2010): Primary 35Q41; Secondary 35B45
DOI: https://doi.org/10.1090/tran/7243
Published electronically: May 9, 2018
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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.


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Additional Information

Jean-Marc Bouclet
Affiliation: Institut de Mathématiques de Toulouse (UMR CNRS 5219), Université Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse FRANCE
Email: jean-marc.bouclet@math.univ-toulouse.fr

Haruya Mizutani
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
Email: haruya@math.sci.osaka-u.ac.jp

DOI: https://doi.org/10.1090/tran/7243
Keywords: Strichartz estimates, resolvent estimates, Schr\"odinger operator, critical singularities
Received by editor(s): July 12, 2016
Received by editor(s) in revised form: March 8, 2017, and March 23, 2017
Published electronically: May 9, 2018
Additional Notes: The first author is partially supported by ANR Grant GeRaSic, ANR-13-BS01-0007-01.
The second author is partially supported by JSPS Grant-in-Aid for Young Scientists (B), No. 25800083, and by Osaka University Research Abroad Program, No. 150S007.
Article copyright: © Copyright 2018 American Mathematical Society

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