Characterizations of circle homeomorphisms of different regularities in the universal Teichmüller space

In this survey, we first give a summary of characterizations of circle homeomorphisms of different regularities (quasisymmetric, symmetric, or $C^{1+\alpha}$) in terms of Beurling–Ahlfors extension, Douady–Earle extension, and Thurston's earthquake representation of an orientation-preserving circle homeomorphism. Then we provide a brief account of characterizations of the elements of the tangent spaces of these sub-Teichmüller spaces at the base point in the universal Teichmüuller space. We also investigate the regularity of the Beurling–Ahlfors extension $BA(h)$ of a $C^{1+\mathrm{Zygmund}}$ orientation-preserving diffeomorphism $h$ of the real line, and show that the Beltrami coefficient $\mu (BA(h))(x+iy)$ vanishes as $O(y)$ uniformly on $x$ near the boundary of the upper half plane if and only if $h$ is $C^{1+\mathrmtarted with any homeomorphism of the real line that is a lifting map of an orientation-preserving circle homeomorphism.{Lipschitz}}$. Finally, we show this criterion is indeed true when $h$ is started with any homeomorphism of the real line that is a lifting map of an orientation-preserving circle homeomorphism.