AI 中文总结
该研究基于Aronszajn–Gagliardo极值构造开发插值方法，统一处理振荡类的可赋范性与最优重排不变巴拿赫外空间确定，在多尺度上验证了方法的有效性。
AI 中文摘要
设φ是(0,1)上的一个容许凹函数，X是一个重排不变空间。我们研究由振荡泛函N_{φ,X}(f)=‖(f^{**}-f^*)/φ‖_X + ‖f‖_1所确定的类。我们基于Aronszajn–Gagliardo极值构造开发了一种插值方法，该方法可统一处理可赋范性问题及最优重排不变巴拿赫外空间的确定。一个恢复原理表明，振荡构造反映了基础重排不变空间的包含序，这使得能将相应的Aronszajn–Gagliardo极值结构转移到振荡类中。特别地，上极值生成包含该类的最小重排不变巴拿赫空间；当上下极值重合时，恰好出现可赋范性。在临界基本尺度下，可赋范性是刚性的，迫使基础空间为对应的Lorentz端点空间。该方法在Lorentz、极限Lorentz及Orlicz尺度上的应用展示了其适用范围，包括经典Copson吸收机制失效的极限可赋范示例。
英文摘要
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.
