Hilbert points in Hardy spaces

Abstract

A Hilbert point in $H^p({\mathbb{T}}^d)$, for $d\ge 1$ and $1\le p\le \infty$, is a nontrivial function $\phi$ in $H^p({\mathbb{T}}^d)$ such that $\Vert \phi {\Vert }_{H^p({\mathbb{T}}^d)}\le \Vert \phi +f{\Vert }_{H^p({\mathbb{T}}^d)}$ whenever $f$ is in $H^p({\mathbb{T}}^d)$ and orthogonal to $\phi$ in the usual $L^2$ sense. When $p\ne 2$, $\phi$ is a Hilbert point in $H^p(\mathbb{T})$ if and only if $\phi$ is a nonzero multiple of an inner function. An inner function on ${\mathbb{T}}^d$ is a Hilbert point in any of the spaces $H^p({\mathbb{T}}^d)$, but there are other Hilbert points as well when $d\ge 2$. We investigate the case of $1$-homogeneous polynomials in depth and obtain as a byproduct a new proof of the sharp Khintchin inequality for Steinhaus variables in the range $2<p<\infty$. We also study briefly the dynamics of a certain nonlinear projection operator that characterizes Hilbert points as its fixed points. We exhibit an example of a function $\phi$ that is a Hilbert point in $H^p({\mathbb{T}}^3)$ for $p=2,4$, but not for any other $p$; this is verified rigorously for $p>4$ but only numerically for $1\le p<4$.

This paper has been accepted for publication in Algebra and Analysis

References in corpus (2)

• Linear functions and duality on the infinite polytorus
• Idempotent Fourier multipliers acting contractively on $H^p$ spaces

Cited by in corpus (2)

• Norm attaining vectors and Hilbert points
• Minimal norm Hankel operators