Properties of Besov and $Q_p$ spaces in terms of the Schwarzian derivative of harmonic mappings

TL;DR

This paper characterizes the membership of the logarithm of the Jacobian of harmonic mappings in certain function spaces using the Schwarzian derivative and Carleson measures, and introduces new classes based on the Jacobian operator.

Contribution

It provides new characterizations of harmonic mappings in terms of the Schwarzian derivative and introduces novel classes related to the Jacobian operator.

Findings

Characterization of $ ext{log} J_f$ in $ ilde{ ext{B}}_p$ and $ ilde{ ext{Q}}_p$ spaces.

Introduction of classes $ ext{BT}_p$ and $ ext{QT}_p$ based on the Jacobian.

Use of Carleson measures for characterizations.

Abstract

In this paper we give a characterization of $\log J_f$ belongs to $\widetilde{\mathcal{B}}_p$ or $\widetilde{\mathcal{Q}}_p$ spaces for any locally univalent sense-preserving harmonic mappings $f$ defined in the unit disk, using the Schwarzian derivative of $f$ and Carleson meseaure. In addition, we introduce the classes $\mathcal{BT}_p$ and $\mathcal{QT}_p$, based on the Jacobian operator, and begin a study of these.