Parametrization of the $p$-Weil-Petersson curves: holomorphic dependence

Similarly to the Bers simultaneous uniformization, the product of the $p$-Weil-Petersson Teichm"uller spaces for $p\ge 1$ provides the coordinates for the space of $p$-Weil-Petersson embeddings $\gamma$ of the real line $\mathbb{R}$ into the complex plane $\mathbb{C}$. We prove the biholomorphic correspondence from this space to the $p$-Besov space of $u=\log {\gamma }^{\mathrm{\prime }}$ on $\mathbb{R}$ for $p>1$. From this fundamental result, several consequences follow immediately which clarify the analytic structures concerning parameter spaces of $p$-Weil-Petersson curves. In particular, it follows that the correspondence of the Riemann mapping parameters to the arc-length parameters keeping the images of curves is a homeomorphism with bi-real-analytic dependence of change of parameters. This is a counterpart to a classical theorem of Coifman and Meyer for chord-arc curves.