Hilbert points in Hardy spaces

TL;DR

This paper explores Hilbert points in Hardy spaces on tori, characterizing their properties, especially for 1-homogeneous polynomials, and provides new proofs and examples related to these functions and their associated inequalities.

Contribution

It characterizes Hilbert points in Hardy spaces, especially for 1-homogeneous polynomials, and offers a new proof of the Khintchin inequality, along with studying nonlinear projection dynamics.

Findings

Inner functions are Hilbert points in all Hardy spaces.

Existence of functions that are Hilbert points in some spaces but not others.

New proof of the sharp Khintchin inequality for Steinhaus variables.

Abstract

A Hilbert point in $H^p(\mathbb{T}^d)$, for $d\geq1$ and $1\leq p \leq \infty$, is a nontrivial function $\varphi$ in $H^p(\mathbb{T}^d)$ such that $\| \varphi \|_{H^p(\mathbb{T}^d)} \leq \|\varphi + f\|_{H^p(\mathbb{T}^d)}$ whenever $f$ is in $H^p(\mathbb{T}^d)$ and orthogonal to $\varphi$ in the usual $L^2$ sense. When $p\neq 2$, $\varphi$ is a Hilbert point in $H^p(\mathbb{T})$ if and only if $\varphi$ 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 $d\geq 2$. 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 $\varphi$ 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 $1\leq p<4$.