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Idempotent Fourier multipliers acting contractively on $H^p$ spaces

We describe the idempotent Fourier multipliers that act contractively on $H^p$ spaces of the $d$-dimensional torus $\mathbb{T}^d$ for $d\geq 1$ and $1\leq p \leq \infty$. 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 $H^p(\mathbb{T}^\infty)$ for every $1 \leq p \leq \infty$ and that extends to a bounded operator if and only if $p=2,4,\ldots,2(n+1)$.