Conjugate functions and moduli of continuity (Q1095279)

One denotes by \(\Omega\) a domain properly contained in n-dimensional Euclidean space \({\mathbb{R}}^ n\). A continuous non-decreasing function \[ \lambda (t): [0,\infty)\to [0,\infty)\quad with\quad \lambda (0)=0 \] is called a majorant if \(\lambda (t_ 1+t_ 2)\leq \lambda (t_ 1)+\lambda (t_ 2)\), \(0<t_ 1\), \(t_ 2<\infty\). When \(f: \Omega\to {\mathbb{R}}^ m\) and \(\lambda\) (t) is a majorant, one writes \(f\in Lip_{\lambda}(\Omega)\) if there exists a constant \(M<\infty\) such that \[ (*)\quad | f(x_ 1)-f(x_ 2)| \leq M\lambda (| x_ 1- x_ 2|) \] for all \(x_ 1,x_ 2\in \Omega\). One denotes the smallest such M by \(\| f\|_{\lambda}\). When \(f: \Omega\to {\mathbb{R}}^ m\) one writes \(f\in \log Lip_{\lambda}(\Omega)\) if there exists a constant M such that (*) holds whenever \(x_ 1,x_ 2\in \Omega\) with \(| x_ 1-x_ 2| \leq d(x_ 1,\partial \Omega)/2\). One denotes the smallest such m by \(\| f\|_{\lambda}^{loc}.\) One calls \(\Omega\) a \(Lip_{\lambda}\)-extension domain if there exists a constant b such that \(\| f\|_{\lambda}\leq b\| f\|_{\lambda}^{loc}\) for all \(f: \Omega\to {\mathbb{R}}^ m\). The main result is the following: Theorem. If \(f=(f_ 1,f_ 2,...,f_ n)\) is K-quasiregular in a \(Lip_{\lambda}\)-extension domain \(\Omega\) and if \(f_ j\in Lip_{\lambda}(\Omega)\) for some \(j=1,2,...,n\), then \(f\in Lip_{\lambda}(\Omega)\) with \(\| f\|_{\lambda}\leq C\| f_ j\|_{\lambda}\), where C is a constant which depends only on n, \(\Omega\), \(\lambda\) and K.