Toward the theory of ring \(Q\)-homeomorphisms
The authors investigate so-called ring \(Q\)-homeomorphisms, including, in particular, \(Q\)-homeo-morphisms, where a homeomorphism \(f : D \rightarrow {\mathbb R}^N \cup {\infty}\) is said to be a \(Q\)-homeomorphism if the modulus \(M\) of \(f \Gamma\) satisfies the following inequality \[ M(f \Gamma) \leq \int\int_{D} Q(x) \rho^n(x) d m(x) \] for every path \(\Gamma\) in \(D\) and any admissible function \(\rho\) with \(Q(x)\) being a measurable function. In terms of the majorant \(Q(x)\), a series of criteria for normality is given based on estimates of the distortion of the spherical distance under ring \(Q\)-homeomorphisms. In particular, it is shown that the class \(\Re_{Q, \triangle}\) of all ring \(Q\)-homeomorphisms \(f\) of a domain \(D \subset {\mathbb R}^N\) into \(\overline{{\mathbb R}^N}\), \(N\geq 2\), with \(h\big(\overline{{\mathbb R}^N}\setminus f(D)\big) \geq \triangle > 0\) forms a normal family if \(Q(x)\) has finite mean oscillation in \(D\).
scientific article; zbMATH DE number 5508717
scientific article