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INTERPOLATION OF HILBERT AND SOBOLEV SPACES: QUANTITATIVE ESTIMATES AND COUNTEREXAMPLES

Published online by Cambridge University Press:  19 November 2014

S. N. Chandler-Wilde
Affiliation:
Department of Mathematics and Statistics, University of Reading, PO Box 220, Whiteknights, Reading RG6 6AX, U.K. email s.n.chandler-wilde@reading.ac.uk
D. P. Hewett
Affiliation:
Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, U.K. email hewett@maths.ox.ac.uk
A. Moiola
Affiliation:
Department of Mathematics and Statistics, University of Reading, PO Box 220, Whiteknights, Reading RG6 6AX, U.K. email a.moiola@reading.ac.uk

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 normalizations, all the key results hold with equality of norms.) In the final section we apply the Hilbert space results to the Sobolev spaces $H^{s}({\rm\Omega})$ and $\widetilde{H}^{s}({\rm\Omega})$, for $s\in \mathbb{R}$ and an open ${\rm\Omega}\subset \mathbb{R}^{n}$. We exhibit examples in one and two dimensions of sets ${\rm\Omega}$ for which these scales of Sobolev spaces are not interpolation scales. In the cases where they are interpolation scales (in particular, if ${\rm\Omega}$ 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.

Type
Research Article
Copyright
Copyright © University College London 2014 

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INTERPOLATION OF HILBERT AND SOBOLEV SPACES: QUANTITATIVE ESTIMATES AND COUNTEREXAMPLES
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