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 $\mathbb{S}^{d-1}\subset\mathbb{R}^d$, $d\in\{3,4,5,6,7\}$, where $n\geq 3$ 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 $e^{i\xi\cdot\omega}$, for some $\xi$, thereby extending previous work of Christ & Shao (2012) to arbitrary dimensions $d\geq 2$ and general even exponents. This talk is based on results obtained with René Quilodrán.

analysis of PDEsclassical analysis and ODEsfunctional analysis

Audience: researchers in the topic

Lisbon webinar in analysis and differential equations

Series comments: The Lisbon Webinar in Analysis in Differential Equations is a joint iniciative of CAMGSD, CMAFcIO and GFM, three research centers of the University of Lisbon. It is aimed at filling the absence of face-to-face seminars and wishes to be a meeting point of mathematicians working in the field. All seminars will take place in the Zoom platform. The links for the planned seminars can be found at: wade.ulisboa.pt/ In order to get the password to access the seminars, please subscribe the announcements in the registration menu wade.ulisboa.pt/registration, or contact one of the organizers.

To see some of the past lectures, please check the webinar youtube channel at www.youtube.com/channel/UCkKS4OJdK1Tq50kWJWaifBg