On Bogovskiı̆ 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 Bogovskiı̆’s integral operator, acting on differential forms in R. 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 Bogovskiı̆-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 C∞ functions. 2000 Mathematics Subject Classification. Primary 35B65, 35C15; Secondary 58J10, 47G30