On Bogovskiǐ and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains

We study integral operators related to a regularized version of the classical Poincaré path integral and the adjoint class generalizing Bogovskiǐ's integral operator, acting on differential forms in ℝⁿ. We prove that these operators are pseudodifferential operators of order -1. The Poincaré-type operators map polynomials to polynomials and can have applications in finite element analysis. For a domain starlike with respect to a ball, the special support properties of the operators imply regularity for the de Rham complex without boundary conditions (using Poincaré-type operators) and with full Dirichlet boundary conditions (using Bogovskiǐ-type operators). For bounded Lipschitz domains, the same regularity results hold, and in addition we show that the cohomology spaces can always be represented by {script}^∞ functions.