Given a complex manifold $M$ equipped with a hermitian metric $g$, one can define a Laplace operator on it by $\Delta u = g^{i \bar j} \partial_i \partial_{\bar j} u$. The claim is that in real coordinates this is an elliptic equation. I wonder why this is true.
Looking at the simplest case where $dim M = 2$: $g^{i \bar j} = \frac{1}{4}g(\frac{\partial}{\partial x} - i \frac{\partial}{\partial y}, \frac{\partial}{\partial x} + i \frac{\partial}{\partial y}) = \frac{1}{4}(g(\frac{\partial}{\partial x}, \frac{\partial}{\partial x}) - g(\frac{\partial}{\partial y}, \frac{\partial}{\partial y}) - 2i g(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}))$ and $\partial \bar \partial u = \frac{1}{4}(\frac{\partial }{\partial x}\frac{\partial }{\partial x}+ \frac{\partial }{\partial y}\frac{\partial }{\partial y})$. How is this elliptic?
Maybe another way to do this is to show that the complex laplacian is half of the real laplacian, but I do not know how to do that either.