A characterization of vanishing mean oscillation (Q863492)

Let \(\mu\) be a positive, finite Borel measure on the unit circle \(T\). The space of functions of vanishing mean oscillation with respect to \(\mu\) was introduced by \textit{D. Sarason} [Trans. Am. Math. Soc. 207, 391--405 (1975; Zbl 0319.42006)]. \textit{D. S. Jerison} and \textit{C. E. Kenig} [in: Studies in Partial Differential Equations, MAA Stud. Math. 23, 1--68 (1982; Zbl 0529.31007)] proved that \(f\in L^1_\mu(T)\) has this property iff \(e^f\) satisfies an asymptotic reverse Cauchy-Schwarz inequality. In the present paper it is shown that, if \(\mu\) is a positive, finite, nonatomic Borel measure, then \(f\) has a vanishing mean oscillation iff it satisfies an asymptotic reverse Jensen inequality. The results, however, are too complicated to be stated here.