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

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
MathSciNet review: 3841849
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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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