Oscillation Classes: An Interpolation Approach

Abstract

Let \(φ\) be an admissible concave function on \((0,1)\) and let \(X\) be a rearrangement-invariant space. We study the classes determined by the oscillation functional \[ \mathcal N_{φ,X}(f) = \left\| \frac{f^{**}-f^*}φ \right\|_X+\|f\|_1. \] We develop an interpolation method, based on the Aronszajn--Gagliardo extremal construction, which allows the normability problem and the determination of the optimal rearrangement-invariant Banach exterior to be treated in a unified way. A recovery principle shows that the oscillation construction reflects the inclusion order of the underlying rearrangement-invariant spaces. This makes it possible to transfer the corresponding Aronszajn--Gagliardo extremal structure to the oscillation classes. In particular, the upper extremal generates the least rearrangement-invariant Banach space containing the class, while normability occurs precisely when the lower and upper extremals collapse. At the critical fundamental scale, normability is rigid and forces the underlying space to be the corresponding Lorentz endpoint. Applications to Lorentz, limiting Lorentz, and Orlicz scales illustrate the scope of the method, including limiting normable examples for which the classical Copson absorption mechanism fails.