The Mathematical Institute, University of Oxford, Eprints Archive

Interpolation of Hilbert and Sobolev Spaces:
Quantitative Estimates and Counterexamples

Chandler-Wilde, S. N. and Hewett, D. P. and Moiola, A (2014) Interpolation of Hilbert and Sobolev Spaces:
Quantitative Estimates and Counterexamples.
Technical Report. Unspecified. (Submitted)

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Abstract

This paper provides an overview of interpolation of Banach and Hilbert spaces, with a focus on establishing when equivalence of norms is in fact equality of norms in the key results of the theory. (In brief, our conclusion for the Hilbert space case is that, with the right normalisations, all the key results hold with equality of norms.) In the final section we apply the Hilbert space results to the Sobolev spaces $H^s(\Omega)$ and $\tH^s(\Omega)$, for $s\in \R$ and an open $\Omega\subset \R^n$. We exhibit examples in one and two dimensions of sets $\Omega$ for which these scales of Sobolev spaces are not interpolation scales. In the cases when they are interpolation scales (in particular, if $\Omega$ is Lipschitz) we exhibit examples that show that, in general, the interpolation norm does not coincide with the intrinsic Sobolev norm and, in fact, the ratio of these two norms can be arbitrarily large.

Item Type:Technical Report (Technical Report)
Subjects:H - N > Numerical analysis
Research Groups:Numerical Analysis Group
ID Code:1834
Deposited By: Helen Taylor
Deposited On:24 Apr 2014 07:41
Last Modified:29 May 2015 19:31

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