The restriction conjecture is one of the cornerstones of modern

harmonic analysis. We will here focus on the

restriction conjecture for the sphere $\mathbb{S}^{d-1}$ of

$\mathbb{R}^{d}$ ($d\geq2$) where the conjecture states that

the Fourier transform extends continuously from $L^p(\mathbb{R}^d)$ to

$L^q(\mathbb{S}^{d-1})$

if and only if $1\leq p\leq\frac{2d}{d+1}$ (< 2) and

$\frac{d+1}{p'}\leq\frac{d-1}{q}$ ($\frac{1}{p}+\frac{1}{p'}=1$).

In particular, if $f\in L^p(\mathbb{R}^d)$ for those $p$'s, one may

restrict its Fourier transform to the sphere, despite the fact that

this set

has measure $0$ (the same is not possible if one replaces the sphere

by the boundary of a cube).

The aim of these 2 talks is

-- to explain the meaning of the conjecture (where do those strange

limitations come from)

-- prove it in dimension 2 (Feffermann and Zygmund in the early 70s)

-- prove it when $q=2$ (Thomas-Stein theorem).

-- give a recent application to Nazarov's uncertainty principle (joint

work with A. Iosevich and A. Mayeli)