Primitive decompositions of Dolbeault harmonic forms on compact almost-Kähler manifolds

Let (X, J, g, ω) be a compact 2n-dimensional almost-Kähler manifold.We prove primitive decompositions of ∂-, ∂-harmonic forms on X in bidegree (1, 1) and (n -1, n -1) (such bidegrees appear to be optimal).We provide examples showing that in bidegree (1, 1) the ∂-and ∂-decompositions differ.