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Real interpolation of spaces of differential forms

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

(Report number 2009-23)

Abstract
In this paper we study interpolation of Hilbert spaces of differential forms using the real method of interpolation. We show that the scale of fractional order Sobolev spaces of differential l-forms in $H^{s}$ with exterior derivative in $H^{s}$ can be obtained by real interpolation. Our proof heavily relies on the recent discovery of smoothed Poincaré lifting for differential forms [ M. Costabel and A. McIntosh, On Bogovskii and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains, Math. Z., (2009)]. They enable the construction of universal extension operators for Sobolev spaces of differential forms, which, in turns, pave the way for a Fourier transform based proof of equivalences of K-functionals.

Keywords: Differential forms, fractional Sobolev spaces, real interpolation, K-functional, smoothed Poincaré lifting, universal extension

@Techreport{HLZ09_39,
  author = {R. Hiptmair and J.-Z. Li and J. Zou},
  title = {Real interpolation of spaces of differential forms},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2009-23},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2009/2009-23.pdf },
  year = {2009}
}

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