Résumé : We describe the idempotent Fourier multipliers that act contractively on $H^p$ spaces of the $d$-dimensional torus ${\mathbb{T}}^d$ for $d\ge 1$ and $1\le p\le \mathrm{\infty }$. When $p$ is not an even integer, such multipliers are just restrictions of contractive idempotent multipliers on $L^p$ spaces, which in turn can be described by suitably combining results of Rudin and Andô. When $p=2(n+1)$, with $n$ a positive integer, contractivity depends in an interesting geometric way on $n,d$, and the dimension of the set of frequencies associated with the multiplier. Our results allow us to construct a linear operator that is densely defined on $H^p\left({\mathbb{T}}^{\mathrm{\infty }}\right)$ for every $1\le p\le \mathrm{\infty }$ and that extends to a bounded operator if and only if $p=2,4,\dots ,2(n+1)$. The talk is based on joint work with Ole Fredrik Brevig and Joaquim Ortega-Cerdà.

Keywords : Idempotent Fourier multipliers; Hardy spaces; contractivity