On the weak Lipschitz condition for quasiconformal mappings.
It is known that the continuity of the Beltrami coefficient \(\mu=\mu(z)\) of a quasiconformal mapping \(f\) of the plane does not guarantee that \(f\) is continuously differentiable. In this paper, the authors study the problem of the weak Lipschitz condition for quasiconformal mapping and, as a strengthening of a result of \textit{P. P. Belinskii} [The general properties of quasiconformal mappings, (1974; Zbl 0292.30019), p. 98], establish the following theorem. Theorem 1: Let \(f\) be a q-quasiconformal mapping of the plane with a continuous or approximately continuous Beltrami coefficient \(\mu(z)\). Then for every compact set \(F\) and every \(\alpha <1\) there exists a constant \(M<\infty\) such that \[ | f(z)-f(w)| \leq M| z-w| ^{\alpha} \] for all \(z,w\in F\). Moreover, for any such \(\alpha\) \(| f(z)-f(w)| /| z-w| ^{\alpha}\rightarrow 0\) as \(z\rightarrow w\). An equivalent statement is established for quasiconformal mappings in the \(n\)-dimensional Euclidean space, when \(\mu(z)\) is replaced by either the dilation matrix \(M_f(x)=Df (x)/J_f(x)^{1/n}\) or the dilation tensor \(G_f(x)=M_f(x)^{*} M_f(x)\), where \(Df(x)\) stands for the differential at \(x\), \(J_f\) its determinant, and \(M_f(x)^{*}\) the transpose. The authors study other related distortion properties of quasiregular mappings in dimension \(n\geq 2\). Let \[ L(x,f,r)=\sup\{| f(x)-f(y)| : | y-x| <r\}, \] where \(x\in B^n\) (the unit ball) and \(r<1-| x| \). They establish the following result. Theorem 3: Let \(f:B^n\rightarrow \mathbb{R}^n\), \(f(0)=0\) \((n\geq 2)\) be a quasiregular mapping such that \[ \int_{B^n}\left(\frac{| f(x)| }{| x| }\right)^pdx \leq \infty, \tag{1} \] for some \(p>n\). Then \[ \lim_{r\rightarrow 0}\frac{L(0,f,r)}{r^{(p-n)/p}} =0. \] The authors finally draw other related corollaries, showing, in particular, that condition (1) is necessary for any \(p>n\) if the dilation matrix or the dilation tensor of a quasiconformal mapping \(f\) of the ball with \(f(0)=0\) is continuous at the origin.
scientific article; zbMATH DE number 2074608