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Transactions of the American Mathematical Society

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

 
 

 

On extremizers for Strichartz estimates for higher order Schrödinger equations


Authors: Diogo Oliveira e Silva and René Quilodrán
Journal: Trans. Amer. Math. Soc. 370 (2018), 6871-6907
MSC (2010): Primary 42B10
DOI: https://doi.org/10.1090/tran/7223
Published electronically: February 26, 2018
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Abstract: For an appropriate class of convex functions $ \phi $, we study the Fourier extension operator on the surface $ \{(y,\vert y\vert ^2+\phi (y)):y\in \mathbb{R}^2\}$ equipped with projection measure. For the corresponding extension inequality, we compute optimal constants and prove that extremizers do not exist. The main tool is a new comparison principle for convolutions of certain singular measures that holds in all dimensions. Using tools of concentration-compactness flavor, we further investigate the behavior of general extremizing sequences. Our work is directly related to the study of extremizers and optimal constants for Strichartz estimates of certain higher order Schrödinger equations. In particular, we resolve a dichotomy from the recent literature concerning the existence of extremizers for a family of fourth order Schrödinger equations and compute the corresponding operator norms exactly where only lower bounds were previously known.


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Additional Information

Diogo Oliveira e Silva
Affiliation: Hausdorff Center for Mathematics 53115 Bonn, Germany
Email: dosilva@math.uni-bonn.de

René Quilodrán
Affiliation: Departamento de Ciencias Exactas Universidad de Los Lagos Avenida Fuchslocher 1305, Osorno, Chile
Email: rene.quilodran@ulagos.cl

DOI: https://doi.org/10.1090/tran/7223
Keywords: Fourier extension theory, extremizers, optimal constants, convolution of singular measures, concentration-compactness, Strichartz inequalities.
Received by editor(s): June 13, 2016
Received by editor(s) in revised form: December 26, 2016
Published electronically: February 26, 2018
Additional Notes: The first author was partially supported by the Hausdorff Center for Mathematics.
Article copyright: © Copyright 2018 American Mathematical Society

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