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On convergence analysis of space homeomorphisms. (English) Zbl 1350.58002

The authors present several theorems on convergence of general homeomorphisms, that is one-to-one mappings \(f:D\to D'\) such that \(f\) and \(f^{-1}\) are continuous between the domains \(D\) and \(D'\) in \(\mathbb R^n\), or on the classes of ring \(Q\)-homeomorphisms. Recall that given a Lebesgue measurable function \(Q:D\to (0,\infty)\) and a point \(x_0\in D\), a homeomorphism \(f\) of \(D\) into \(\overline{\mathbb R^n}\) is called a ring \(Q\)-homeomorphism at \(x_0\) whenever \[ M(\Gamma(f(S_1), f(S_2)))\leq \int_A Q(x) \eta^n(|x-x_0|)dm(x) \] for every ring \(A=A(x_0,r_1,r_2)=\{x\in \mathbb R^n: r_1<|x-x_0|<r_2\}\), \( 0<r_1<r_2<d(x_0,\partial D)\), \(S_i=S(x_0,r_i)=\{x\in \mathbb R^n: |x-x_0|=r_i\}\), \(i=1,2\) and every \(\eta:(r_1,r_2)\to [0,\infty]\) such that \(\int_{r_1}^{r_2}\eta(r)dr\geq 1\), where \(\Gamma(C_1,C_2)\) stands for the family of paths \(\gamma\) which join \(C_1\) with \(C_2\), \(\mathrm{adm}\Gamma\) stands for the family of Borel function \(\rho:\mathbb R^n\to [0,\infty]\) such that \(\int_\gamma \rho(x)|dx| \geq 1 \) for all \(\gamma\in \Gamma\) and \(M(\Gamma(C_1,C_2))=\inf \{\int_{\mathbb R^n}\rho^n(x) dm(x): \rho\in \mathrm{adm} \Gamma(C_1,C_2)\}\). The authors give a number of convergence results for general homeormorphisms or those with some modular conditions. They show the completeness for the ring \(Q\)-homeomorphisms and consider normal and compact classes of ring \(Q\)-homeomorphisms. In particular, it is established that the family of all ring \(Q\)-homeomorphims fixing two points is compact provided that the function \(Q\) is of finite mean oscillation. Extensions of these notions to moduli of families of surfaces or refinements such as lower \(Q\)-homeormorphims are also considered. Some applications to mappings in the Sobolev classes and Orlicz-Sobolev classes are provided.

MSC:

58C07 Continuity properties of mappings on manifolds
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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