Hodge–Dirac, Hodge-Laplacian and Hodge–Stokes operators in $L^p$ spaces on Lipschitz domains

This paper concerns Hodge–Dirac operators D_{{}^\Vert}=d+\underline{\delta} acting in L^p(\Omega, \Lambda) where \Omega is a bounded open subset of {\mathbb{R}}^n satisfying some kind of Lipschitz condition, \Lambda is the exterior algebra of {\mathbb{R}}^n , d is the exterior derivative acting on the de Rham complex of differential forms on \Omega , and \underline{\delta} is the interior derivative with tangential boundary conditions. In L^2(\Omega,\Lambda) , \underline{\delta} = {d}^* and D_{{}^\Vert} is self-adjoint, thus having bounded resolvents \{({\rm I}+itD_{{}^\Vert})^{-1}\}_{t\in{\mathbb{R}}} as well as a bounded functional calculus in L^2(\Omega,\Lambda) . We investigate the range of values p_H < p < p^H about p=2 for which D_{{}^\Vert} has bounded resolvents and a bounded holomorphic functional calculus in L^p(\Omega,\Lambda) . On domains which we call very weakly Lipschitz, we show that this is the same range of values as for which L^p(\Omega,\Lambda) has a Hodge (or Helmholz) decomposition, being an open interval that includes 2. The Hodge-Laplacian \Delta_{{{}^\Vert}} is the square of the Hodge–Dirac operator, i.e., -\Delta_{{}^\Vert}={D_{{}^\Vert}}^2 , so it also has a bounded functional calculus in L^p(\Omega,\Lambda) when p_H < p < p^H . But the Stokes operator with Hodge boundary conditions, which is the restriction of -\Delta_{{}^\Vert} to the subspace of divergence free vector fields in L^p(\Omega,\Lambda^1) with tangential boundary conditions, has a bounded holomorphic functional calculus for further values of p , namely for max \{1,{p_H}_S\} < p < p^H where {p_H}_S is the Sobolev exponent below p_H , given by 1/{{p_H}_S} =1/{p_H}+1/n , so that {{p_H}_S} < 2n/(n+2) . In 3 dimensions, {p_H}_S < 6/5 . We show also that for bounded strongly Lipschitz domains \Omega , p_H < 2n/(n+1) < 2n/(n-1) < p^H , in agreement with the known results that p_H < 4/3 < 4 < p^H in dimension 2, and p_H < 3/2 < 3 < p^H in dimension 3. In both dimensions 2 and 3, {p_H}_S<1 , implying that the Stokes operator has a bounded functional calculus in L^p(\Omega,\Lambda^1) when \Omega is strongly Lipschitz and 1 < p < p^H .