  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 $0<c\leq1.$
