Holomorphic functions and quasiconformal mappings with smooth moduli (Q1877873)

An increasing continuous function \(\omega:\;[0,\infty)\rightarrow [0,\infty)\) with \(\omega(0)=0\) is called a majorant if \(\omega(t)/t\) is nonincreasing for \(t>0\). Given a subset \(E\) of \(\mathbb{C}^n\) or \(\mathbb{R}^n\), a function \(f:E\rightarrow C\) is said to belong to the Lipschitz space \(\Lambda_{\omega}(E)\) if there is a constant \(K=K(f)>0\) such that \[ | f(x)-f(y)| \leq K\omega(| x-y| ) \] for all \(x,y\in E\). The norm \(\| f\|_{\Lambda_{\omega}(E)}\) is defined as the smallest such \(K\). The main subject under cinsideration is the implication \[ | f| \in \Lambda_{\omega}(G) \Rightarrow f\in \Lambda_{\omega}(G) \] for holomorphic functions or quasiconformal mappings on a domain \(G\) (the converse implication is trivial). It turns out that this implication holds for large classes of \(f\)'s, \(\omega\)'s and \(G\)'s in the multidimensional setting. On the other hand, the phenomenon has its limits. It can fail either for ``bad'' \(\omega\)'s already for the unit circle as a domain, or for ``bad'' \(G\)'s and nice \(\omega\)'s such as \(\omega(t)=t^\alpha\).