Metric definition of \(\mu\)-homeomorphisms. (Q1396348)

The David class [\textit{G. David}, Ann. Acad. Sci. Fenn., Ser. A, Math. 13, 25--70 (1988; Zbl 0619.30024)] consists of homeomorphisms \(f\in W^{1,1}_{\text{loc}}(D, D')\) of a plane domain \(D\) onto a plane domain \(D'\) such that \[ | Df(x)|^2\leq K(x) J(x,f)\tag{\(*\)} \] a.e. in \(D\) with \(\exp(aK)\) in \(L^1_{\text{loc}}(D)\) for some constant \(a> 0\). The authors give a sufficient condition in terms of the ``lim inf'' metric distortion \(h= h(f, x)\) that a homeomorphism \(f\) is in the David class: If \(h< \infty\) outside a set \(E\) of sigma-finite length, then there is a constant \(b\) such that \(\exp(bh^2)\in L^1_{\text{loc}}(D)\) implies that \(f\in W^{1,2}_{\text{loc}}(D, D')\) and that \((*)\) holds with \(K= h^2\). Related problems in higher dimension are also considered.