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Universal extension for sobolev spaces of differential forms and applications

by R. Hiptmair and J.-Z. Li and J. Zou

(Report number 2009-22)

Abstract
This article is devoted to the construction of a family of universal extension operators for the Sobolev spaces Hk(d, ,l) of differential forms of degree l (0  l  d) in a Lipschitz domain  Rd (d 2 N, d  2) for any k 2 N0. It generalizes the construction of the first universal extension operator for standard Sobolev spaces Hk( ), k 2 N0, on Lipschitz domains, introduced by Stein [Theorem 5, pp. 181, E. M. STEIN, Singular integrals and differentiability properties of functions, Princeton University Press, N. J., 1970]. This corresponds to the case l = 0 of our theory. We adapt Stein’s idea in the form of integral averaging over the pullback of a parametrized reflection mapping. The new theory covers extension operators for Hk(curl; ) and Hk(div; ) 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.

Keywords: Universal (Stein) extension, Sobolev spaces of differential forms, Lipschitz domains, integral averaging, parametrized reflection mapping, generalized regular decomposition

@Techreport{HLZ09_40,
  author = {R. Hiptmair and J.-Z. Li and J. Zou},
  title = {Universal extension for sobolev spaces of differential forms and applications},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2009-22},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2009/2009-22.pdf },
  year = {2009}
}

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