Interpolation Hilbert spaces between Sobolev spaces

TL;DR

This paper explicitly characterizes all Hilbert function spaces that serve as interpolation spaces between Sobolev spaces, showing they are a subclass of H"ormander spaces parametrized by specific radial functions.

Contribution

It provides a complete description of interpolation Hilbert spaces between Sobolev spaces, including explicit examples of intermediate but non-interpolation spaces.

Findings

Interpolation spaces form a subclass of H"ormander spaces

Parametrization by OR-varying radial functions at infinity

Explicit examples of non-interpolation intermediate spaces

Abstract

We explicitly describe all Hilbert function spaces that are interpolation spaces with respect to a given couple of Sobolev inner product spaces considered over $\mathbb{R}^{n}$ or a half-space in $\mathbb{R}^{n}$ or a bounded Euclidean domain with Lipschitz boundary. We prove that these interpolation spaces form a subclass of isotropic H\"ormander spaces. They are parametrized with a radial function parameter which is OR-varying at $+\infty$ and satisfies some additional conditions. We give explicit examples of intermediate but not interpolation spaces.