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

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 $\alpha \ge c\Vert \cdot {\Vert }_{1,\lambda }$ for some $0<c\le 1.$

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