An extended Hilbert scale and its applications

TL;DR

This paper introduces an extended Hilbert scale using all interpolation spaces between original scale spaces, providing explicit descriptions and applications to inequalities, Sobolev spaces, and spectral theory.

Contribution

It extends traditional Hilbert scales via interpolation spaces and offers explicit descriptions using OR-varying functions, enhancing their analytical utility.

Findings

Explicit description of the extended Hilbert scale using OR-varying functions.

The extended scale is closed under quadratic interpolation.

Applications to inequalities, Sobolev spaces, and spectral expansions.

Abstract

We propose a new viewpoint on Hilbert scales extending them by means of all Hilbert spaces that are interpolation ones between spaces on the scale. We prove that this extension admits an explicit description with the help of $\mathrm{OR}$-varying functions of the operator generating the scale. We also show that this extended Hilbert scale is obtained by the quadratic interpolation (with function parameter) between the above spaces and is closed with respect to the quadratic interpolation between Hilbert spaces. We give applications of the extended Hilbert scale to interpolational inequalities, generalized Sobolev spaces, and spectral expansions induced by abstract and elliptic operators.