An Extension of the Chen-Beurling-Helson-Lowdenslager Theorem

TL;DR

This paper extends the Beurling-Helson-Lowdenslager Theorem for Hardy spaces on the unit circle to a broader class of continuous gauge norms, using measure-theoretic techniques to relate these norms to Lebesgue measure.

Contribution

It generalizes Chen's extension of the theorem to a larger class of continuous gauge norms on $L^{ty}$, introducing a measure-theoretic approach to establish norm inequalities.

Findings

Established the existence of a probability measure absolutely continuous with respect to Lebesgue measure.

Proved that continuous normalized gauge norms dominate a scaled $L^1$ norm with respect to this measure.

Extended the classical theorem to a broader setting of gauge norms.

Abstract

Yanni Chen extended the classical Beurling-Helson-Lowdenslager Theorem for Hardy spaces on the unit circle $\mathbb{T}$ defined in terms of continuous gauge norms on $L^{\infty}$ that dominate $\Vert\cdot\Vert_{1}$. We extend Chen's result to a much larger class of continuous gauge norms. A key ingredient is our result that if $\alpha$ is a continuous normalized gauge norm on $L^{\infty}$, then there is a probability measure $\lambda$, mutually absolutely continuous with respect to Lebesgue measure on $\mathbb{T}$, such that $\alpha\geq c\Vert\cdot\Vert_{1,\lambda}$ for some $0<c\leq1.$