  On Riemannian signature conformal 4-manifolds we give a conformally invariant
extension of the Maxwell operator on 1-forms. We show the extension is in an
appropriate sense injectively elliptic, and recovers the invariant gauge
operator of Eastwood and Singer. The extension has a natural compatibility with
the de Rham complex and we prove that, given a certain restriction, its
conformally invariant null space is isomorphic to the first de Rham cohomology.
General machinery for extending this construction is developed and as a second
application we describe an elliptic extension of a natural operator on
perturbations of conformal structure. This operator is closely linked to a
natural sequence of invariant operators that we construct explictly. In the
conformally flat setting this yields a complex known as the conformal
deformation complex and for this we describe a conformally invariant Hodge
theory which parallels the de Rham result.
