### Global maximizers for spherical restriction

- Speaker: Dr. Diogo Oliveira e Silva
- Place: Online (zoom)
- Time: 8.4.2021, 14:00 - 8.4.2021, 15:00
- Invited by: Dr. Rainer Mandel

## 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.

In spite of the seemingly technical abstract, I will make sure to keep at least the first half of the talk suitable for a general audience.