Joint BCAM-UPV/EHU Analysis and PDE seminar: Global maximizers for spherical restriction

Abstract

We prove that constant functions are the unique real-valued maximizers for all $L^2-L^{2n}$ adjoint Fourier restriction inequalities on the unit sphere $mathbb{S}^{d-1}subsetmathbb{R}^d$, $din{3,4,5,6,7}$, where $ngeq 3$ is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying Euler--Lagrange equation being smooth, a fact of independent interest which we discuss. We further show that complex-valued maximizers coincide with nonnegative maximizers multiplied by the character $e^{ixicdotomega}$, for some $xi$, thereby extending previous work of Christ & Shao (2012) to arbitrary dimensions $dgeq 2$ and general even exponents.

This talk is based on results obtained with René Quilodrán.


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