Vol. 13, No. 2, 2020

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Sharp Strichartz inequalities for fractional and higher-order Schrödinger equations

Gianmarco Brocchi, Diogo Oliveira e Silva and René Quilodrán

Vol. 13 (2020), No. 2, 477–526
Abstract

We investigate a class of sharp Fourier extension inequalities on the planar curves s = |y|p , p > 1. We identify the mechanism responsible for the possible loss of compactness of nonnegative extremizing sequences, and prove that extremizers exist if 1 < p < p0 for some p0 > 4. 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 n-fold convolutions of certain singular measures in d , 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 L2-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 s = y|y|p1 , p > 1.

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Keywords
sharp Fourier restriction theory, extremizers, Strichartz inequalities, fractional Schrödinger equation, convolution of singular measures
Mathematical Subject Classification 2010
Primary: 35B38, 35Q41, 42B37
Milestones
Received: 3 August 2018
Accepted: 7 March 2019
Published: 19 March 2020
Authors
Gianmarco Brocchi
School of Mathematics
University of Birmingham
Birmingham
United Kingdom
Diogo Oliveira e Silva
School of Mathematics
University of Birmingham
Birmingham
United Kingdom
Hausdorff Center for Mathematics
Bonn
Germany
René Quilodrán