Interpolation of Hilbert and Sobolev Spaces: Quantitative Estimates and Counterexamples

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 (Ω) and H s (Ω), for s ∈ R and an open Ω ⊂ R n . We exhibit examples in one and two dimensions of sets Ω for which these scales of Sobolev spaces are not interpolation scales. In the cases when they are interpolation scales (in particular, if Ω 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.A main result of the paper is to exhibit one-and two-dimensional counterexamples that show that H s (Ω) and H s (Ω) are not in general interpolation scales. It is well-known that these Sobolev spaces are interpolation scales for all s ∈ R when Ω is Lipschitz. In that case we demonstrate, via a number of counterexamples that, in general (we suspect, in fact, whenever Ω R n ), H s (Ω) and H s (Ω) are not exact interpolation scales. Indeed, we exhibit simple examples where the ratio of interpolation norm to intrinsic Sobolev norm may be arbitrarily large. Along the way we give explicit formulas for some of the interpolation norms arising that may be of interest in their own right. We remark that our investigations, which are inspired by applications arising in boundary integral equation methods (see [9]), in particular are inspired by McLean [18], and by its appendix on interpolation of Banach and Sobolev spaces. However a result of §4 is that one result claimed by McLean ( [18, Theorem B.8]) is false.Much of the Hilbert space Section 3 builds strongly on previous work. In particular, our result that, with the right normalisations, the norms in the K-and J-methods of interpolation coincide in the Hilbert space case is a (corrected version of) an earlier result of Ameur [2] (the normalisations proposed and the definition of the J-method norm seem inaccurate in [2]). What is new in our Theorem 3.3 is the method of proof-all of our proofs in this section are based on the spectral theorem that every bounded normal operator is unitarily equivalent to a multiplication operator on L 2 (X , M, µ), for some measure space (X , M, µ), this coupled with an elementary explicit treatment of interpolation on weighted L 2 spaceswhich deals seamlessly with the general Hilbert space case without an assumption of separability or that H 0 ∩ H 1 is dense in H 0 and H 1 . Again, our result in Theorem 3.5 that there is only one (geometric) interpolation space of exponent θ, when interpolating Hilbert spaces, is a version of McCarthy's [17] uniqueness theorem. What is new is that we treat the general Hilbert space case by a method of proof based on the aforementioned spectral theorem. O...

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