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INTERPOLATION OF HILBERT AND SOBOLEV SPACES: QUANTITATIVE ESTIMATES AND COUNTEREXAMPLES

  • S. N. Chandler-Wilde (a1), D. P. Hewett (a2) and A. Moiola (a3)

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.

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1.Adams, R. A., Sobolev Spaces, Academic Press (New York, 1973).
2.Ameur, Y., A new proof of Donoghue’s interpolation theorem. J. Funct. Spaces Appl. 2 2004, 253265.
3.Ameur, Y., Interpolation and operator constructions. Preprint, 2014, arXiv:1401.6090 (accessed 20/10/2014).
4.Bennet, C. and Sharpley, R., Interpolation of Operators, Academic Press (New York, 1988).
5.Bergh, J. and Löfström, J., Interpolation Spaces: An Introduction, Springer (Berlin, 1976).
6.Bramble, J. H., Multigrid Methods, Chapman & Hall (New York, 1993).
7.Calderón, A. P., Lebesgue spaces of differentiable functions and distributions. Proc. Sympos. Pure Math. 4 1961, 3349.
8.Chandler-Wilde, S. N. and Hewett, D. P., Acoustic scattering by fractal screens: mathematical formulations and wavenumber-explicit continuity and coercivity estimates. University of Reading Preprint, 2013, MPS-2013-17; arXiv:1401.2786 (accessed 20/10/2014).
9.Chandler-Wilde, S. N., Hewett, D. P. and Moiola, A., Sobolev spaces on subsets of $\mathbb{R}^{n}$ with application to boundary integral equations on fractal screens (in preparation).
10.Donoghue, W., The interpolation of quadratic norms. Acta Math. 118 1967, 251270.
11.Dunford, N. and Schwarz, J. T., Linear Operators, Part II. Spectral Theory, John Wiley (New York, 1963).
12.Jones, P. W., Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147 1981, 7188.
13.Kato, T., Perturbation Theory for Linear Operators, 2nd edn, Springer (Berlin, 1980).
14.Kress, R., Linear Integral Equations, 2nd edn, Springer (New York, 1999).
15.Lions, J.-L. and Magenes, E., Non-Homogeneous Boundary Value Problems and Applications I, Springer (Berlin, 1972).
16.Maz’ya, V. G., Sobolev Spaces with Applications to Elliptic Partial Differential Equations, 2nd edn, Springer (New York, 2011).
17.McCarthy, J. E., Geometric interpolation between Hilbert spaces. Ark. Mat. 30 1992, 321330.
18.McLean, W., Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press (Cambridge, 2000).
19.Peetre, J., A Theory of Interpolation of Normed Spaces (Notas de Matemática 39), Instituto de Matemática Pura e Aplicada, Conselho Nacional de Pesquisas (Rio de Janeiro, 1968).
20.Rogers, L. G., Degree-independent Sobolev extension on locally uniform domains. J. Funct. Anal. 235 2006, 619665.
21.Rychkov, V. S., On restrictions and extensions of the Besov and Triebel–Lizorkin spaces with respect to Lipschitz domains. J. Lond. Math. Soc. 60 1999, 237257.
22.Stein, E. M., Singular Integrals and Differentiability Properties of Functions, Vol. 1, Princeton University Press (Princeton, NJ, 1970).
23.Tartar, L., An Introduction to Sobolev Spaces and Interpolation Spaces, Springer (Berlin, 2007).
24.Triebel, H., Interpolation Theory, Function Spaces, Differential Operators, North-Holland (Amsterdam, 1978).
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