Real-analytic realization of universal Teichm\"uller space via complex-structures on $H^{1/2}$

Abstract

Let $H$ be the Hilbert transform, let $h$ be a quasisymmetric homeomorphism of the unit circle $S^1$, and set $V_hu=u\circ h$ and $J_h=V_hHV_h^{-1}$, defined on the Sobolev space $H^{1/2}(S^1)$. We prove that $h\mapsto V_h$ is nowhere continuous in operator norm, although it is continuous in the strong operator topology. By contrast, the induced map $h\mapsto J_h$ is a real-analytic diffeomorphism onto its image in the operator-norm topology. Based on this, we further compute the differential at the identity and show that it is precisely the Calder\'on commutator. In graph coordinates, the tangent map admits a weighted Hankel matrix representation, whose Hilbert-Schmidt norm recovers the Weil-Petersson tangent quadratic form.