The $p$-integrable Teichm\"uller space for $p \geqslant 1$

TL;DR

This paper establishes that the $p$-integrable Teichmüller space $T_p$ has a canonical complex Banach manifold structure for all $p \, \geq 1$, and characterizes associated quasisymmetric homeomorphisms via $p$-Besov spaces.

Contribution

It proves the complex Banach manifold structure of $T_p$ for all $p \, \geq 1$ and links elements of $T_p$ to $p$-Besov spaces, extending previous understanding.

Findings

$T_p$ admits a canonical complex Banach manifold structure for all $p \, \geq 1$

Characterization of quasisymmetric homeomorphisms in $T_p$ via $p$-Besov spaces for $p > 1$

Extension of the structure and characterization results to the full range $p \, \geq 1$

Abstract

We verify that the $p$-integrable Teichm\"uller space $T_p$ admits the canonical complex Banach manifold structure for any $p \geq 1$. Moreover, we characterize a quasisymmetric homeomorphism corresponding to an element of $T_p$ in terms of the $p$-Besov space for any $p>1$.