Exponentials rarely maximize Fourier extension inequalities for cones

2023/02/01 by Negro, Giuseppe, Silva, Diogo Oliveira e, Stovall, Betsy +1
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2302.00356

Abstract

We prove the existence of maximizers and the precompactness of L^p-normalized maximizing sequences modulo symmetries for all valid scale-invariant Fourier extension inequalities on the cone in \mathbb R^1+d. In the range for which such inequalities are conjectural, our result is conditional on the boundedness of the extension operator. Global maximizers for the L² Fourier extension inequality on the cone in \mathbb R^1+d have been characterized in the lowest-dimensional cases d∈\2,3\. We further prove that these functions are critical points for the L^p to L^q Fourier extension inequality if and only if p = 2.

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