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

Abstract

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 $1\le p\le \infty$. 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 $1\le p\le \infty$ and that extends to a bounded operator if and only if $p=2,4,\dots ,2(n+1)$.

This paper has been accepted for publication in Geometric and Functional Analysis

References in corpus (3)

• Linear space properties of $H^p$ spaces of Dirichlet series
• Sharp norm estimates for composition operators and Hilbert-type inequalities
• Composition operators and embedding theorems for some function spaces of Dirichlet series

Cited by in corpus (3)

• Hilbert points in Hardy spaces
• Critical exponents of the Riesz projection
• Contractive projections in Paley-Wiener spaces