Primitive decomposition of Bott-Chern and Dolbeault harmonic $(k,k)$-forms on compact almost Kähler manifolds

We consider the primitive decomposition of $\bar{\mathrm{\partial }},\mathrm{\partial }$, Bott-Chern and Aeppli-harmonic $(k,k)$-forms on compact almost Kähler manifolds $(M,J,\omega )$. For any $D\in \{\bar{\mathrm{\partial }},\mathrm{\partial },BC,A\}$, we prove that the $L^k{P}^0$ component of $\psi \in {\mathcal{H}}_D^{k,k}$, is a constant multiple of ${\omega }^k$. Focusing on dimension 8, we give a full description of the spaces ${\mathcal{H}}_{BC}^{2,2}$ and ${\mathcal{H}}_A^{2,2}$, from which follows ${\mathcal{H}}_{BC}^{2,2}\subseteq {\mathcal{H}}_{\mathrm{\partial }}^{2,2}$ and ${\mathcal{H}}_A^{2,2}\subseteq {\mathcal{H}}_{\bar{\mathrm{\partial }}}^{2,2}$. We also provide an almost Kähler 8-dimensional example where the previous inclusions are strict and the primitive components of an harmonic form $\psi \in {\mathcal{H}}_D^{k,k}$ are not $D$-harmonic, showing that the primitive decomposition of $(k,k)$-forms in general does not descend to harmonic forms.