Hilbert points in Hardy spaces

Abstract

A Hilbert point in $H^p\left(\mathbb{T}^d\right)$, for $d \geq 1$ and $1 \leq p \leq \infty$, is a nontrivial function $\varphi$ in $H^p\left(\mathbb{T}^d\right)$ such that $\|\varphi\|_{H^p\left(\mathbb{T}^d\right)} \leq\|\varphi+f\|_{H^p\left(\mathbb{T}^d\right)}$ whenever $f$ is in $H^p\left(\mathbb{T}^d\right)$ 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 $H^p\left(\mathrm{~T}^d\right)$, but there are other Hilbert points as well when $d \geq 2$. The case of 1 -homogeneous polynomials is studied in depth and, as a byproduct, a new proof is given for the sharp Khinchin inequality for Steinhaus variables in the range $2<p<\infty$. Briefly, the dynamics of a certain nonlinear projection operator is treated. This operator characterizes Hilbert points as its fixed points. An example is exhibited of a function $\varphi$ that is a Hilbert point in $H^p\left(\mathbb{T}^3\right)$ for $p=2,4$, but not for any other $p$; this is verified rigorously for $p>4$ but only numerically for $1 \leq p<4$.