Bounded Poincaré operators for twisted and BGG complexes

Abstract

We construct bounded Poincaré operators for twisted complexes and BGG complexes with a wide class of function classes (e.g., Sobolev spaces) on bounded Lipschitz domains. These operators are derived from the de Rham versions using BGG diagrams and, for vanishing cohomology, satisfy the homotopy identity dP+Pd=I in degrees >0. The operators preserve polynomial classes if the de Rham versions do so. Nontrivial cohomology and the complex property P∘P=0 can be incorporated. We present applications to polynomial exact sequences.