A Compactness Characterization of Strongly Symmetric Homeomorphisms

Abstract

A self-homeomorphism $h$ of the unit circle $\mathbb{T}$ is strongly symmetric if it is absolutely continuous and $\log h^{\prime }\in \text{VMO}(\mathbb{T})$. Let $P_h^{-}$ be the anti-analytic component of the pullback operator $P_h:F\mapsto F\circ h$ on $\text{BMOA}(\mathbb{D})$, where $\mathbb{D}$ is the unit disk. P. Jones proved $P_h$ is bounded on BMO if and only if $h$ is strongly quasisymmetric. Fan, Hu, and Shen showed that the strong symmetry of $h$ yields the compactness of $P_h^{-}$, and raised the question of whether the converse is true. We answer this question affirmatively, establishing that $P_h^{-}$ is compact if and only if $\log h^{\prime }\in \text{VMO}(\mathbb{T}))$, which completes the VMO theory of the universal Teichmüller space.