Duality and Distance Formulas in Spaces Defined by Means of Oscillation

TL;DR

This paper explores duality and distance formulas in function spaces defined by oscillation, extending classical results from BMO and VMO to Bloch and other invariant spaces using pure functional analysis.

Contribution

It provides new duality results and distance formulas for various function spaces, including Bloch, Q_K, weighted, Lipschitz-Hölder, and rectangular BMO spaces.

Findings

Established duality between BMO and VMO.

Derived distance formulas for functions in BMO to VMO.

Extended results to M"obius invariant and multi-variable spaces.

Abstract

For the classical space of functions with bounded mean oscillation, it is well known that VMO** = BMO and there are many characterizations of the distance from a function f in BMO to VMO. When considering the Bloch space, results in the same vein are available with respect to the little Bloch space. In this paper such duality results and distance formulas are obtained by pure functional analysis. Applications include general M\"obius invariant spaces such as Q_K-spaces, weighted spaces, Lipschitz-H\"older spaces and rectangular BMO of several variables.