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

In this paper we give a characterization of $\log J_f$ belongs to ${\tilde{\mathcal{B}}}_p$ or ${\tilde{\mathcal{Q}}}_p$ 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 ${\mathcal{B}\mathcal{T}}_p$ and ${\mathcal{Q}\mathcal{T}}_p$, based on the Jacobian operator, and begin a study of these.