Bott–Chern Laplacian on almost Hermitian manifolds

Let $(M,J,g,\omega)$ be a $2n$-dimensional almost Hermitian manifold. We extend the definition of Bott-Chern Laplacian on $(M,J,g,\omega)$, proving that it is still elliptic. On compact K\"ahler manifold, kernels Dolbeault and coincide. show such property does not hold when providing an explicit structure Kodaira-Thurston Furthermore, if connected $4$-manifold, denoting by $h^{1,1}_{BC}$ dimension space harmonic $(1,1)$-forms, we prove either $h^{1,1}_{BC}=b^-$ or $h^{1,1}_{BC}=b^-+1$. In particular, $g$ K\"ahler, then $h^{1,1}_{BC}=b^-+1$, extending result Holt Zhang for kernel Laplacian. also dimensions spaces $(1,1)$-forms behave differently complex 4-manifolds endowed with strictly locally conformally metrics. Finally, relate some forms to cohomology groups manifolds, recently introduced Coelho, Placini Stelzig.