Generalized Beurling-Ahlfors' theorem (Q1906584)

The theorem referred to in the title states that a self-homeomorphism \(h\) of \(R\) is the boundary value of a \(K\)-quasiconformal self-mapping of the upper half plane if and only if \(h\) is quasisymmetric. The author has previously defined a generalization of quasiconformal mappings: Let \(f\) be a homeomorphism between domains \(D\) and \(D'\) in \(\overline{\mathbb{R}^n}\). Let \[ H(x, f)= \sup_{r\to 0} {\max_{|y- x|= r} |f(y)- f(x)|\over \min_{|y- x|= r} |f(y)- f(x)|} \] be the spherical dilatation of \(f\) at \(x\) and suppose that \(f\) is a.e. differentiable, absolutely continuous on lines, and \(H\) is locally integrable. Then \(f\) is called a locally integrable homeomorphism or IDH. (In another paper of the author, they are labelled LDH.) A condition which is analogous to quasisymmetry is defined and results corresponding to the Beurling-Ahlfors theorem are proved.