Geodesic
Hilbert points in Hardy spaces
Algebra i analiz, Tome 34 (2022) no. 3, pp. 131-158

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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 Khinchin inequality for Steinhaus variables in the range $2$. 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 $1\leq p4$.
Keywords: Hardy spaces, inner functions, Hilbert points, $1$-homogeneous polynomials, Khinchin inequality for Steinhaus variables. 
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O. F. Brevig; J. Ortega-Cerdá; K. Seip. Hilbert points in Hardy spaces. Algebra i analiz, Tome 34 (2022) no. 3, pp. 131-158. http://geodesic.mathdoc.fr/item/AA_2022_34_3_a5/

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