\(\Theta\)-mappings and generalized local Lipschitz transformations. (English) Zbl 0689.30016

Caraman, Petru

Rev. Roum. Math. Pures Appl. 34, No. 8, 705-714 (1989).

Let X be a normed space and h: [0,\(\infty)\to {\mathbb{R}}^ a \)differentiable modulus of continuity. Set \(h(x,y)=h(| x-y|)\). A mapping f: \(D\to X\), D a domain in X, is said to belong to the local Lipschitz class loc \(Lip_ h(D)\) if for all \(x,y\in D\) with \(| y- x| \leq b dist(x,\partial D)\), \(| f(x)-f(y)| \leq mh(x,y)\) for some constants \(b\in (0,1]\) and \(m<\infty\) independent of x and y.
For \(X={\mathbb{R}}^ n\) and \(h(x,y)=| x-y|^{\alpha}\), \(0<\alpha \leq 1\), the classes log \(Lip_ h(D)\), extension properties of functions in these classes and their applications to quasiconformal mappings were considered in [F. W. Gehring, and the reviewer, Ann. Acad. Sci. Fenn., Ser. A I 10, 203-219 (1985; Zbl 0584.30018)]; generalizations to arbitrary h can be found in [V. Lappalainen, Ann. Acad. Sci. Fenn., Ser. A I, Diss. 56, 1-52 (1985; Zbl 0584.30019)]. Here some of these results are generalized to normed spces. Connections between the class loc \(Lip_ h(D)\) and \(\theta\)-mappings (a quasisymmetric version for quasiconformality which works in normed spaces) are studied and Hadamard differentiability properties of quasiconformal (in a very restrictive sense) mappings are considered.

Reviewer: O.Martio

MSC:

30C62

Quasiconformal mappings in the complex plane

Citations:

Zbl 0584.30018; Zbl 0584.30019