# Integrable Teichm\"uller spaces for analysis on Weil-Petersson curves

## TL;DR

This paper characterizes integrable Teichmüller spaces using Besov spaces, establishing a biholomorphic correspondence and analyzing Cauchy transforms on Weil-Petersson curves, advancing the understanding of complex structures in Teichmüller theory.

## Contribution

It introduces a real-analytic equivalence between $T_p$ and the $p$-Besov space, and links Cauchy transforms to holomorphic maps in the context of Weil-Petersson curves.

## Key findings

- Characterization of $T_p$ via $p$-Besov space membership.
- Biholomorphic correspondence between $T_p$ and $p$-Besov space.
- Holomorphic dependence of Cauchy transforms on embeddings.

## Abstract

The integrableTeichm\"uller space $T_p$ for $p \geq 1$ is defined by the $p$-integrability of Beltrami coefficients. We characterize a quasisymmetric homeomorphism $h$ in $T_p$ by the condition that $\log h'$ belongs to the real $p$-Besov space, with a certain modification applied in the case $p=1$. This is done as part of the arguments for establishing a biholomorphic correspondence $\Lambda$ from the product of $T_p$ for simultaneous uniformization of $p$-Weil-Petersson curves into the $p$-Besov space. In particular, this proves the real-analytic equivalence between $T_p$ and the real $p$-Besov space. Moreover, the Cauchy transform of Besov functions on Weil-Petersson curves can be expressed by the derivative of this holomorphic map $\Lambda$, and from this, the Calder\'on theorem in this setting is straightforward. It also follows that the Cauchy transforms on $p$-Weil-Petersson curves holomorphically depend on their embeddings as they vary in the Bers coordinates.