Conformal operators on weighted forms; their decomposition and null space on Einstein manifolds

There is a class of Laplacian like conformally invariant differential operators on differential forms $L_k^{\mathrm{\ell }}$ which may be considered the generalisation to differential forms of the conformally invariant powers of the Laplacian known as the Paneitz and GJMS operators. On conformally Einstein manifolds we give explicit formulae for these as explicit factored polynomials in second order differential operators. In the case the manifold is not Ricci flat we use this to provide a direct sum decomposition of the null space of the $L_k^{\mathrm{\ell }}$ in terms of the null spaces of mutually commuting second order factors.