Sharp Strichartz inequalities for fractional and higher-order Schrödinger equations

Brocchi, Gianmarco; Oliveira e Silva, Diogo; Quilodrán, René

doi:10.2140/apde.2020.13.477

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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

DOI: 10.2140/apde.2020.13.477

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 $1 < p < p_{0}$ for some $p_{0} > 4$ . 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 $ℝ^{d}$ , 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 $L^{2}$ -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 $s = y | y |^{p - 1}$ , $p > 1$ .

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

Vol. 13, No. 2, 2020