Hilbert points in Hardy spaces

Abstract

A Hilbert point in H p ( T d ) H^p(\mathbb {T}^d) , for d ≥ 1 d\geq 1 and 1 ≤ p ≤ ∞ 1\leq p \leq \infty , is a nontrivial function φ \varphi in H p ( T d ) H^p(\mathbb {T}^d) such that ‖ φ ‖ H p ( T d ) ≤ ‖ φ + f ‖ H p ( T d ) \| \varphi \|_{H^p(\mathbb {T}^d)} \leq \|\varphi + f\|_{H^p(\mathbb {T}^d)} whenever f f is in H p ( T d ) H^p(\mathbb {T}^d) and orthogonal to φ \varphi in the usual L 2 L^2 sense. When p ≠ 2 p\neq 2 , φ \varphi is a Hilbert point in H p ( T ) H^p(\mathbb {T}) if and only if φ \varphi is a nonzero multiple of an inner function. An inner function on T d \mathbb {T}^d is a Hilbert point in any of the spaces H p ( T d ) H^p(\mathbb {T}^d) , but there are other Hilbert points as well when d ≥ 2 d\geq 2 . The case of 1 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 > ∞ 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 φ \varphi that is a Hilbert point in H p ( T 3 ) H^p(\mathbb {T}^3) for p = 2 , 4 p=2, 4 , but not for any other p p ; this is verified rigorously for p > 4 p>4 but only numerically for 1 ≤ p > 4 1\leq p>4 .