Jordan domains and the universal Teichmüller space

Abstract:

Let $L$ denote the lower half plane and let $B(L)$ denote the Banach space of analytic functions $f$ in $L$ with ${\left \| f \right \|_L} < \infty$, where ${\left \| f \right \|_L}$ is the suprenum over $z \in L$ of the values $\left | {f(z)} \right |{(text{Im} z)^2}$. The universal Teichmüller space, $T$, is the subset of $B(L)$ consisting of the Schwarzian derivatives of conformal mappings of $L$ which have quasiconformal extensions to the extended plane. We denote by $J$ the set \[ \left \{ {{S_f}:f{\text {is conformal in }}L{\text {and }}f(L){\text {is a Jordan domain}}} \right \},\] which is a subset of $B(L)$ contained in the Schwarzian space $S$. In showing $S - \bar T \ne \emptyset$, Gehring actually proves $S - \bar J \ne \emptyset$. We give an example which demonstrates that $J - \bar T \ne \emptyset$.