Inverse mappings for homeomorphisms of class BL
A mapping \(f: G\to \mathbb{R}^ n\) [\textit{Yu. G. Reshetnyak}, ``Spatial mappings with bounded distortion'' (1982; Zbl 0487.30011)] for which the inequality \[ \|[\lambda_ m(x, f)]^ n\mid J(x, f)^{- m}\|_{L_ 1(G)}< \infty\quad (1\leq m\leq n)\tag{1} \] holds, where \(G\) is a bounded domain of the Euclidean \(n\)-space \(\mathbb{R}^ n\), \(J(x,f)\) the Jacobian of \(f\), \(\lambda_ m(x,f)=\sqrt{\sum_ I\sum_ J \Bigl[{(f_{i_ 1},\dots,f_{i_ m})\over (f_{j_ 1},\dots,f_{j_ m})} (x)\Bigr]^ 2}\), \(I= \{i_ 1,\dots,i_ m\}\), \(J=\{j_ 1,\dots,j_ m\}\), \(1\leq i_ 1<\cdots< i_ m\leq n\), \(1\leq j_ 1<\cdots <j_ m\leq n\), is said to be quasiconformal in the mean satisfying (1). In the note under review, the author establishes that if the homeomorphism \(f: G\to D\) belongs to the class BL [\textit{G. D. Suvorov}, ``The generalized `Length and area principle' in mapping theory'' (1985; Zbl 0669.30012)], then its inverse \(g: D\to G\) is a homeomorphism quasiconformal in the mean satisfying (1) for \(m= n-1\).
scientific article; zbMATH DE number 589618