Weil-Petersson Teichm\"{u}ller space II: smoothness of flow curves of $H^{\frac{3}{2}}$-vector fields

Given a continuous vector field $\lambda (t,\cdot )$ of Sobolev class $H^{\frac{3}{2}}$ on the unit circle $S^1$, the flow maps $\eta =g(t,\cdot )$ of the differential equation $$\text{cases}\frac{d\eta }{dt}=\lambda (t,\eta )\eta (0,\zeta )=\zeta \text{endcases}$$ are known to be quasisymmetric homeomorphisms. Very recently, Gay-Balmaz-Ratiu [GR] conjectured that the flow curve $g(t, \cdot)$isintheWeil-PeterssonclassWP$(S^1)$ and is continuously differentiable with respect to the Hilbert manifold structure of WP$(S^1)$ introduced by Takhtajan-Teo [TT]. The first assertion had already been demonstrated in our previous paper [Sh2]. In this sequel to [Sh2], we will continue to deal with the Weil-Petersson class WP$(S^1)$ and completely solve this conjecture in the affirmative.