The estimate of the inner dilatation of the ring \(Q\)-homeomorphisms (Q2901661)

It is proved that the inner dilatation \(K_I(x,f)\) of a ring \(Q\)-homeomorphism is majorized by the function \(Q\) with some multiplicative constant depending only on the dimension \(n\) of the space provided that \(Q\) is locally integrable and \(J(x,f)\neq 0\) almost everywhere. In particular it follows that the inner dilatation of such a mapping is locally integrable if \(Q\) is. Another useful consequence is local integrability of the linear dilatation \(H(x,f)\) under the above conditions. Finally, the author states that the inverse mapping belongs to the class \(W_{\text{loc}}^{1,2}\) if \(Q\) is locally integrable.