Given two real functions $f, h$ on a Kähler manifold $(X, \omega)$, I am trying to make sense of the following equality:
$\Delta_{\omega}(fh) = (\Delta_{\omega} f) h + f (\Delta_{\omega}h) + 2 \nabla_{w} f \cdot \nabla_{w}h$.
In local coordinates $\Delta_{\omega}(fh) = g^{i\bar j} \partial_i \partial_{\bar j} (fh) = g^{i\bar j}\partial_i f \partial_{\bar j}h +g^{i \bar j} (\partial_i \partial_{\bar j} f) h + g^{i \bar j} \partial_{\bar j} f \partial_{i} h + g^{i \bar j} (\partial_i \partial_{\bar j} h) f = (\Delta_{\omega} f) h + f (\Delta_{\omega}h) + g^{i\bar j}\partial_i f \partial_{\bar j}h + g^{i \bar j} \partial_{\bar j} f \partial_{i} h$
How are the remaining two terms assembled into forming the dot product of the gradient? How is the gradient define on a kahler manifold?
Thoughts: Since we are dealing with real functions, can potentially use the induced Riemannian metric of a Kähler metric to define the gradient in the usual sense. Is $\nabla_{\omega}$ intepreted like so? What would be a local expression in terms of coordinates?