Universal extension for Sobolev spaces of differential forms and applications

This article is devoted to the construction of a family of universal extension operators for the Sobolev spaces Hk(d,Ω,Λl) of differential forms of degree l (0⩽l⩽d) in a Lipschitz domain Ω⊂Rd (d∈N, d⩾2) for any k∈N0. It generalizes the construction of the first universal extension operator for standard Sobolev spaces Hk(Ω), k∈N0, on Lipschitz domains, introduced by Stein [E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, NJ, 1970, Theorem 5, p. 181]. We adapt Steinʼs idea in the form of integral averaging over the pullback of a parametrized reflection mapping. The new theory covers extension operators for Hk(curl;Ω) and Hk(div;Ω) in R3 as special cases for l=1,2, respectively. Of considerable mathematical interest in its own right, the new theoretical results have many important applications: we elaborate existence proofs for generalized regular decompositions.