Admissible changes of variables in the class of BMO functions (Q803856)

Let \(\phi\) be a function (not necessarily monotone) which maps an interval P into itself. The author gives sufficient conditions for \(\phi\) to guarantee that \((*)\;\{f\in BMO(P)\Rightarrow f\circ \phi \in BMO(P)\}.\) The conditions are complicated and not easy to verify for a concrete function. Earlier, \textit{P. Jones} [Ark. Mat. 21, 229-231 (1983; Zbl 0527.42007)] gave necessary and sufficient conditions implying (*) when \(\phi\) is a homeomorphism of the real line. The author shows that for monotone convex functions \(\phi\) on the real line his condition is also necessary and is equivalent to \(\phi\) being quasisymmetric.