**Backround**

In several complex variables, an essential tool is Hormander's machinery for solving the $\overline{\partial}$ problem with $L^2$ estimates.

If $\alpha$ is a $(p,q+1)$ form on a domain $\Omega \subset \mathbb{C}^n$, and one is attempting to solve $\overline{\partial} u = \alpha$ with a weight function $\phi$ one of the terms which must be estimated to get the method to work is

$$ \int_\Omega \left[ \sum_{|I|=p,|J|=q} \sum_{j,k=1}^n \frac{\partial^2 \phi}{\partial z_j \partial \overline{z_k}} \alpha_{I,jJ}\overline{\alpha_{I,kJ}} \right]e^{-\phi} dV $$

where $I,J$ are multi-indices, $kJ$ means "stick the index $j$ to the front of $J$", and $\alpha_{I,K}$ is the coefficient of $dz^I\wedge d\overline{z}^K$

This is a term in the integration by parts identity called the "Morrey-Kohn-Hormander" identity.

Let's give the notation

$$ H_\phi^{(p,q)}(\alpha,\alpha) = \sum_{|I|=p,|J|=q} \sum_{j,k=1}^n \frac{\partial^2 \phi}{\partial z_j \partial \overline{z_k}} \alpha_{I,jJ}\overline{\alpha_{I,kJ}} $$

$H_\phi^{(0,1)}$ is a quadratic form on $(0,1)$ forms which is called the "complex Hessian" of $\phi$. It seems like the rest of the $H_\phi^{(p,q)}$ are "built out" of the complex hessian, in the sense that the entries are all sums of entries from the complex hessian in a structured way.

**Question**

While the most satisfying explanation of this term might just be "it is what shows up when you integrate by parts", I am hoping for a more conceptual explanation. Is $H_\phi^{(p,q)}$ "natural" in any way? Is it the "unique way to lift a quadratic form on $\Lambda^{(0,1)}$ to $\Lambda^{(p,q)}$ such that..."?

I would also be interested in the corresponding question in the real case. Given a quadratic form $Q$ on $\Lambda^1(\mathbb{R^n})$, say $Q(dx^i,dx^j) = Q_{i,j}$ and define $Q^k$ on $\Lambda^k$ by

$$ Q^k(\alpha,\alpha) = \sum_{|I|=k-1} \sum_{j,k=1}^n Q_{j,k} \alpha_{jI}\alpha_{kI} $$

does $Q^k$ have a nice characterization? Also, does anyone have any pointers to this stuff written out for manifolds? The formulas I am working with in the case of $\mathbb{C}^n$ or $\mathbb{R}^n$ do not look like they transform in a particularly nice way under coordinate transformations.