Vol. 15, No. 4, 2022

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Local maximizers of adjoint Fourier restriction estimates for the cone, paraboloid and sphere

Felipe Gonçalves and Giuseppe Negro

Vol. 15 (2022), No. 4, 1097–1130
Abstract

We show that, possibly after a compactification of spacetime, constant functions are local maximizers of the Tomas–Stein adjoint Fourier restriction inequality for the cone and paraboloid in every dimension, and for the sphere in dimension up to 60. For the cone and paraboloid we work from the PDE framework, which enables the use of the Penrose and the Lens transformations, which map the conjectured optimal functions into constants.

Keywords
Fourier extension, Fourier restriction, Schrödinger equation, wave equation, Strichartz estimates, sharp inequality, local maximizer, cone, paraboloid
Mathematical Subject Classification
Primary: 35A23, 42B10, 42B37
Milestones
Received: 29 March 2020
Revised: 20 October 2020
Accepted: 26 November 2020
Published: 3 September 2022
Authors
Felipe Gonçalves
Hausdorff Center for Mathematics
Universität Bonn
Bonn
Germany
Giuseppe Negro
Departamento de Matemática
Instituto Superior Técnico
Lisbon
Portugal