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Traces of differential forms on Lipschitz domains, the boundary De Rham complex, and Hodge decompositions

We study the extent to which classical trace and extension theorems for scalar-valued functions can be extended to differential forms of higher-degree. For maximum applicability, this is done in the context of Lipschitz subdomains of Riemannian manifolds, and on the scale of Besov and Triebel-Lizorkin spaces. A key ingredient in this regard is the boundary De Rham complex, which we consider in the geometric and analytic setting above. Applications to Hodge-decompositions and interpolation with constraints are also presented.