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2020 Sharp Strichartz inequalities for fractional and higher-order Schrödinger equations
Gianmarco Brocchi, Diogo Oliveira e Silva, René Quilodrán
Anal. PDE 13(2): 477-526 (2020). 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<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.

Citation

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Gianmarco Brocchi. Diogo Oliveira e Silva. René Quilodrán. "Sharp Strichartz inequalities for fractional and higher-order Schrödinger equations." Anal. PDE 13 (2) 477 - 526, 2020. https://doi.org/10.2140/apde.2020.13.477

Information

Received: 3 August 2018; Accepted: 7 March 2019; Published: 2020
First available in Project Euclid: 25 June 2020

zbMATH: 07181508
MathSciNet: MR4078234
Digital Object Identifier: 10.2140/apde.2020.13.477

Subjects:
Primary: 35B38 , 35Q41 , 42B37

Keywords: convolution of singular measures , extremizers , fractional Schrödinger equation , sharp Fourier restriction theory , Strichartz inequalities

Rights: Copyright © 2020 Mathematical Sciences Publishers

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Vol.13 • No. 2 • 2020
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