Boundary behaviour of BMO-qc automorphisms
A homeomorphism \(f\) of a plane domain \(D\) into \(\mathbb R^2\) is said to be BMO-quasiconformal (qc) if \(f\) is ACL and the maximal dilatation \(K_f\) of \(f\) satisfies \(K_f(x)\leq Q(x)\) a.e. in \(D\) where \(Q\) is a BMO-function in \(D\). These mappings have been studied by G. David and P. Tukia and contributions to their boundary behavior can be found in [\textit{V. Ryazanov, U. Srebro, E. Yakubov}, J. Anal. Math. 83, 1--20 (2001; Zbl 1011.30029)]. The author shows that the boundary map \(\varphi\) of a BMO-qc map of the upper half space \(H\) satisfies a quasisymmetry condition of the form \((\varphi(x+t)-\varphi(x))/(\varphi(x)-\varphi(x-t)) \leq\alpha\exp(\beta Q_B)\) with a similar lower bound. Here \(Q_B\) is the mean value of \(Q\) over the ball \(B(x,2t)\). Note that a BMQ function in \(H\) can be extended by reflection to a BMO function in \(\mathbb R^2\). The author also gives a condition that a boundary mapping can be extended to a BMQ-qc mapping of \(H\). The construction is based on \textit{L. Carleson}'s extension method [Contributions to Analysis, L. V. Ahlfors et al. (eds.), Academic Press, 39--47 (1974; Zbl 0304.30016)].
scientific article; zbMATH DE number 1760723