Idempotent Fourier multipliers acting contractively on Hp spaces

Abstract

We describe the idempotent Fourier multipliers that act contractively on Hp spaces of the d-dimensional torus Td for d≥1 and 1≤p≤∞. When p is not an even integer, such multipliers are just restrictions of contractive idempotent multipliers on Lp 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 Hp(T∞) for every 1≤p≤∞ and that extends to a bounded operator if and only if p=2,4,…,2(n+1).