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Sharp Strichartz
inequalities for fractional and higher-order Schrödinger
equations
Gianmarco Brocchi, Diogo Oliveira e Silva and René Quilodrán |
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Vol. 13 (2020), No. 2, 477–526
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Abstract |
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We investigate a class of sharp Fourier extension inequalities on the planar curves , . We identify the mechanism responsible for the possible loss of compactness of nonnegative extremizing sequences, and prove that extremizers exist if for some . In particular, this resolves the dichotomy of Jiang, Pausader, and Shao concerning the existence of extremizers for the Strichartz inequality for the fourth-order Schrödinger equation in one spatial dimension. One of our tools is a geometric comparison principle for -fold convolutions of certain singular measures in , developed in the companion paper of Oliveira e Silva and Quilodrán (Math. Proc. Cambridge Philos. Soc., (2019)). We further show that any extremizer exhibits fast -decay in physical space, and so its Fourier transform can be extended to an entire function on the whole complex plane. Finally, we investigate the extent to which our methods apply to the case of the planar curves , . |
Keywords
sharp Fourier restriction theory, extremizers, Strichartz
inequalities, fractional Schrödinger equation, convolution
of singular measures
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Mathematical Subject Classification 2010
Primary: 35B38, 35Q41, 42B37
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Milestones
Received: 3 August 2018
Accepted: 7 March 2019
Published: 19 March 2020
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Authors |
| Gianmarco Brocchi | |
| School of Mathematics University of Birmingham Birmingham United Kingdom |
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| Diogo Oliveira e Silva | |
| School of Mathematics University of Birmingham Birmingham United Kingdom |
Hausdorff Center for
Mathematics Bonn Germany |
| René Quilodrán | |
