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Mappings of \(BMO\)-bounded distortion. (English) Zbl 0954.30009
In this paper the authors continue developing the theme of mappings of \(BMO\)-bounded distortion, refining and extending previous work, as well as obtaining new results. Let \(\Omega\) be an open subset of \(\mathbb{R}^n\). A function \(f:\Omega\rightarrow\mathbb{R}^n\) is said to have finite distortion if \(f\in W_{\text{loc}}^{1,\phi}(\Omega,\mathbb{R}^n)\) for the Orlicz function \(\phi(t)=t^n\log^{-1}(t+e)\), and there is a function \(K(x), 1\leq K(x) <\infty\) defined a.e. in \(\Omega\) such that \[ |Df(x)|^n = K(x)J(x,f) \] a.e. in \(\Omega\). Here \(|Df(x)|\) stands for the norm of the differential of \(f\) and \(J(x,f)\) is the jacobian. If \(K(x)\) is uniformly bounded then one arrives at the usual class of quasiregular mappings if in addition \(f\in W_{\text{loc}}^{1,n}(\Omega,\mathbb{R}^n)\). A function \(f\in W_{\text{loc}}^{1,\phi}(\Omega,\mathbb{R}^n)\) is said to be of \(BMO\)-bounded distortion if there is \(M\in BMO(\mathbb{R}^n)\) such that \[ |Df(x)|^n \leq M(x)J(x,f) \] a.e. in \(\Omega\). The function \(M\) is called a \(BMO\)-distortion function for \(f\). The authors obtain sharp estimates for the modulus of continuity of a monotone mapping in various Sobolev-Orlicz classes. Briefly, a mapping \(f:\Omega\rightarrow\mathbb{R}^n\) is said to be monotone if for each compact \(G\subset\Omega\) and each \(x,y\in G\) one has \[ |f(x)-f(y)|\leq\max \{|f(z)-f(w)|:z,w\in\partial\Omega \}. \] Mappings of small \(BMO\)-distortion are open and discrete hence monotone. The authors also study the distortion of Hausdorff dimension under mappings of \(BMO\)-bounded distortion. In contrast to the fact that quasiconformal mappings distort the Hausdorff dimension by bounded amounts, no such estimate holds for this wider class, and they need to look at finer measures of dimension, obtained by logarithmic weight functions. This is used in the final section of the paper to prove a theorem analogous to the Painlevé theorem for analytic functions in the plane concerning removable singularities. The results here are qualitatively optimal. In particular, they show that some sets of Hausdorff dimension zero are not removable for bounded mappings of \(BMO\)-bounded distortion, although they are removable for every bounded quasiregular mapping. Finally, they show that there are domains in the plane which admit bounded mappings of \(BMO\)-bounded distortion but no nonconstant bounded quasiregular mapping.

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