Vol. 13, No. 2, 2020

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 10, 3371–3670
Issue 9, 2997–3369
Issue 8, 2619–2996
Issue 7, 2247–2618
Issue 6, 1871–2245
Issue 5, 1501–1870
Issue 4, 1127–1500
Issue 3, 757–1126
Issue 2, 379–756
Issue 1, 1–377

Volume 16, 10 issues

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
Editors' interests
 
Subscriptions
 
ISSN 1948-206X (online)
ISSN 2157-5045 (print)
 
Author index
To appear
 
Other MSP journals
This article is available for purchase or by subscription. See below.
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.

PDF Access Denied

We have not been able to recognize your IP address 18.97.14.84 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

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