A reverse quasiconformal composition problem for \(Q_\alpha(\mathbb{R}^n)\).

*(English)*Zbl 1436.30018The main topic of this paper is to give sufficient conditions for homeomorphisms to be quasiconformal. More precisely, given a homeomorphism \(f\) with certain properties, the authors look at the operator \(C_f\) mapping a function \(u\) to \(u\circ f\), especially they investigate its boundedness on certain spaces to deduce the quasiconformality.

Previous results established a sort of converse statements.

Let me be more precise. The authors are interested in homeomorphisms that are absolutely continuous on almost all lines parallel to the coordinates axes and differentiable almost everywhere. First, the authors recall the so-called Essén-Janson-Peng-Xiao spaces, [M. Essén et al., Indiana Univ. Math. J. 49, No. 2, 575–615 (2000; Zbl 0984.46020)]. These spaces depend on a parameter. Additionally to the above, the authors require that \(C_f\) and \(C_{f^{-1}}\) are bounded on these spaces for certain parameters. This implies that \(f\) is quasiconformal.

One of the main tasks of the authors consists in proving various properties of the Essén-Janson-Peng-Xiao spaces. The employed strategy is actually to show that \(f^{-1}\) is quasiconformal, which then implies that \(f\) is quasiconformal as well.

The second main result looks at homeomorphisms \(f\) in the plane; in this case the quasiconformality is obtained under weaker assumptions on \(f\). In the proof of this theorem, the metric definition of quasiconformality is verified. This approach uses the concept of capacity and covering theorems.

Previous results established a sort of converse statements.

Let me be more precise. The authors are interested in homeomorphisms that are absolutely continuous on almost all lines parallel to the coordinates axes and differentiable almost everywhere. First, the authors recall the so-called Essén-Janson-Peng-Xiao spaces, [M. Essén et al., Indiana Univ. Math. J. 49, No. 2, 575–615 (2000; Zbl 0984.46020)]. These spaces depend on a parameter. Additionally to the above, the authors require that \(C_f\) and \(C_{f^{-1}}\) are bounded on these spaces for certain parameters. This implies that \(f\) is quasiconformal.

One of the main tasks of the authors consists in proving various properties of the Essén-Janson-Peng-Xiao spaces. The employed strategy is actually to show that \(f^{-1}\) is quasiconformal, which then implies that \(f\) is quasiconformal as well.

The second main result looks at homeomorphisms \(f\) in the plane; in this case the quasiconformality is obtained under weaker assumptions on \(f\). In the proof of this theorem, the metric definition of quasiconformality is verified. This approach uses the concept of capacity and covering theorems.

Reviewer: Thomas Zürcher (Katowice)

##### MSC:

30C65 | Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations |