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

TL;DR

This paper establishes a biholomorphic correspondence between $p$-Weil-Petersson Teichmüller spaces and $p$-Besov spaces for $p>1$, clarifying the analytic structure of $p$-Weil-Petersson curves and their parameterizations.

Contribution

It proves a fundamental biholomorphic equivalence between $p$-Weil-Petersson embeddings and $p$-Besov spaces, extending classical results to a broader class of curves.

Findings

Biholomorphic correspondence between $p$-Weil-Petersson spaces and $p$-Besov spaces.

Homeomorphism with bi-real-analytic dependence for Riemann mapping parameters.

Clarification of the analytic structure of $p$-Weil-Petersson curves.

Abstract

Similarly to the Bers simultaneous uniformization, the product of the $p$-Weil-Petersson Teichm\"uller spaces for $p \geq 1$ 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 $u=\log \gamma'$ 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.