Lipschitz classes and the Hardy--Littlewood property
A proper subdomain \(D\) of \(\mathbb{C}\) has the Hardy-Littlewood property if there is a constant \(k\) such that for any \(\beta\in(0,1]\) and any \(f\) analytic in \(D\) with \(| f'(z)|\leq m d(z,D)^{\beta-1}\) in \(D\) we have the Hölder condition (*) \(| f(z_ 1)-f(z_ 2)|\leq M| z_ 1-z_ 2|^ \beta\) in \(D\) with \(M=km/\beta\). If \(D\) satisfies (*) with a fixed \(\beta\in (0,1]\), then \(f\) is said to have the Hardy-Littlewood property of order \(\beta\in(0,1]\). A function \(f\) (not necessarily analytic) is said to belong to the local Lipschitz class \(\text{loc Lip}_ \beta(D)\) if there exists a constant \(M\) such that (*) holds whenever \(z_ 1\) and \(z_ 2\) lie in any open disk contained in \(D\). Finally, a domain \(D\) is called a \(\text{Lip}_ \beta\)-extension domain if every function \(f\) in \(\text{loc Lip}_ \beta(D)\) is \(\beta\)- Lipschitz in \(D\) with the bound for the Lipschitz constant which is independent of \(f\). Combining the ideas in [\textit{F. W. Gehring} and \textit{O. Martio}, Ann. Acad. Sci. Fenn., Ser. A 10, 203-219 (1985; Zbl 0584.30018)] and in [\textit{R. Kaufman} and \textit{J. M. Wu}, Complex Variables, Theory Appl. 4, 1-5 (1984; Zbl 0561.30018)] the authors first observe that a simply connected domain \(D\) has the Hardy-Littlewood property of order \(\beta\) iff \(D\) is a \(\text{Lip}_ \beta\)-extension domain. They then produce several interesting examples: For instance, there exists a simply connected domain which has not the Hardy-Littlewood property but which is a \(\text{Lip}_ \beta\)-extension domain for each \(\beta\in(0,1]\).
scientific article; zbMATH DE number 438881
scientific article