Bounded functions of vanishing mean oscillation on compact metric spaces
Consider a metric space \((X,\,\rho)\) with a doubling measure \(\mu\). Generalizing a classical result due to \textit{T. H. Wolff} [Duke Math. J. 49, 321--328 (1982; Zbl 0494.30042)], the author proves that for every \(f\in L^{\infty}(\mu)\) there exists \(\eta\in VMO(X,d,\mu)\) such that \(0\leq\eta\leq 1\), \(\log\eta\in VMO(X,d,\mu)\), and \(\eta f\in VMO\cap L^{\infty}\) (the last condition can even be improved to \(qf\in VMO\cap L^{\infty}\) for every \(q\in VMO\cap L^{\infty}\) with \(| q| \leq\eta\)). The main motivating example of the metric space \(X\) is the boundary \(S^{2n-1}\) of the complex \(n\)-ball. The author shows that, unlike the case of \(n=1\) treated by Wolff, for \(n\geq 2\) there exists an \(L^{\infty}\)-function \(f\) on \(S^{2n-1}\) for which there is no nonzero \(H^{\infty}\)-function \(q\) that multiplies \(f\) to \(VMO\).
scientific article; zbMATH DE number 2091735
scientific article