I want to define "the difference quotient of a differential form $ \omega $ in the direction of the vector field $ X $" on a Riemannian manifold. Let's call this object, if it can be defined, $ \nabla_X^h \omega $, where $ h > 0 $ is the "step size" and $ \omega $ is a tensor or differential form. My questions are, can this object be defined in such a way that it has the properties one would want: $ \omega \in W^{1,2}(\bigwedge^k M) $ if and only if $ \| \nabla^h_X \omega \|_{L^2} $ remains bounded for small $ h $ and all smooth vector fields $ X $ with $ \| X \|_{L^\infty} < 1 $ (say), and hopefully an integration by parts formula such as
$ \int_M{ \nabla^h_X \omega \wedge * \varphi} = -\int_M{ \omega \wedge * \nabla^{-h}_X \varphi } $
My question is, does any such operation $ \nabla^h_X \omega $ exist?
In case the reader is curious, my goal is, I'd like to write a "quick" and "easy" argument that the Poisson Equation $ \Delta \omega = f $ on a Riemannian Manifold $ M $ (where $ \Delta $ is the Laplace-Beltrami operator) is solvable for $ f \in L^2(\bigwedge^k M) $ with $ \omega \in W^{2,2} $. I've read the proof by Warner; and I've read proofs of elliptic regularity in places like Evans; however, I believe it can be done more elegantly in the case of the Laplace-Beltrami operator on a closed compact riemannian manifold. This seems to be the object I need to complete the proof of regularity.