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arXiv · 2302.00356

Exponentials rarely maximize Fourier extension inequalities for cones

Giuseppe Negro·Diogo Oliveira e Silva·Betsy Stovall·James Tautges
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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 $L^2$ Fourier extension inequality on the cone in $\mathbb R^{1+d}$ have been characterized in the lowest-dimensional cases $d\in\{2,3\}$. 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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Giuseppe Negro, Diogo Oliveira e Silva, Betsy Stovall, James Tautges. 2023-02-01. Exponentials rarely maximize Fourier extension inequalities for cones. https://arxiv.org/abs/2302.00356

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