The Hodge--Dirac operator $D_H = d_\Omega + {d_\Omega}^*$ on a bounded domain $\Omega\subset R^n$ is
self-adjoint in $L^2(\Omega)$, and so has bounded resolvents and bounded functional calculi. I shall discuss the question as to when the resolvents and functional calculi are bounded in $L^p(\Omega)$ for some range of values of $p \in (p_H,p^H)$, where $p_H < 2 < p^H$.

The square of the Hodge--Dirac operator is the Hodge--Laplacian $\Delta_H={D_H}^2$, so it has a bounded functional calculus for the same range of values as $D_H$. But an operator such as the Hodge--Stokes operator, which is the restriction of $\Delta_H$ to a subspace such as the divergence free vector fields, can have a bounded functional calculus on a larger range of values of $p$.
I shall discuss recent joint research with Sylvie Monniaux (Marseille) on this topic.