On the imbeddings \(C^ 1_ B(\Omega)\), \(C^ 1({\bar \Omega})\to C^{0,\lambda}(\Omega;d)\) (Q923327)

Let \(\Omega\) be any arcwise connected open subset of \({\mathbb{R}}^ n\) and let \(C^ 1_ B(\Omega)\) and \(C^{0,\lambda}(\Omega,d)\) be the spaces of all functions with bounded first derivates and of \(\lambda\)-Hölder continuous functions on \(\Omega\), respectively. Here d is any given distance function on \(\Omega\). The author studies the validity of a continuous embedding of \(C^ 1_ B(\Omega)\) into \(C^{0,\lambda}(\Omega,d)\). As the main result he proves that this holds true iff the geodesic distance function g on \(\Omega\) is \(\lambda\)- Hölder continuous with respect to d, more precisely, if there is some constant \(c>0\) and some \(\delta >0\) such that \(g(x,y)\leq cd(x,y)^{\lambda}\) for all x, y with \(d(x,y)<\delta\). Here g(x,y) is the distance of x and y measured inside of \(\Omega\). Similar results are valid for the embedding of \(C^ 1({\bar \Omega})\) into \(C^{0,\lambda}(\Omega,d)\).