Hilbert points in Hardy spaces

Abstract:

A Hilbert point in $H^p(\mathbb {T}^d)$, for $d\geq 1$ 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 $d\geq 2$. 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 $H^p(\mathbb {T}^3)$ for $p=2, 4$, but not for any other $p$; this is verified rigorously for $p>4$ but only numerically for $1\leq p<4$.