A note on $\text{tmd}$-operator

In almost Kähler manifolds, one of the challenges is to construct an elliptic operator on functions that plays a role analogous to the $\mathrm{\partial }\bar{\mathrm{\partial }}$ operator in complex or Kähler manifolds. One of the aims of this paper is to revisit the $\text{tmd}$-operator introduced in \cite{TWZZ}. We will provide some local analysis estimates and highlight several difficulties that remain to be addressed. Additionally, we use the Atiyah-Hitchin-Singer operator to demonstrate that every $d$-exact $(1,1)$-form is globally $\text{tmd}$-exact for any compact taming symplectic $4$-manifold.