Stability of the Lipschitz extension property under metric transforms
Let \((\mathcal M, d)\) be a metric space and let \(X\) denote a Banach space with the norm \(||\cdot||\). The Lipschitz space \(\text{Lip}(\mathcal M, X)\) is the linear space of mappings \(F:\mathcal M\to X\) defined by the finiteness of the seminorm \[ |f|_{\text{Lip}(\mathcal M, X)}:=\sup_{x\neq y}\frac{||f(x)-f(y)||}{d(x,y)}. \] (a) A metric space \(\mathcal M\) satisfies the Lipschitz extension property with respect to \(X\), if there is a constant \(\lambda >0\) such that for every \(\mathcal M'\subset\mathcal M\) there exists an extension operator \(E_{\mathcal M'}:\text{Lip}(\mathcal M',X)\to\text{Lip}(\mathcal M, X)\) with norm \(\leq\lambda\). (b) \(\mathcal M\) satisfies the linear Lipschitz extension property with respect to \(X\), if the operator \(E_{\mathcal M'}\) can be chosen to be linear. The optimal choice of \(\lambda\) is denoted by \(e(\mathcal M,X)\) in case (a) and by \(e_l(\mathcal M,X)\) in case (b). Let \(w:\mathbb R_+\to\mathbb R_+\) be a concave non-decreasing function with \(\omega(0)=0\). The operation of superposition \(d_{\omega}:=\omega\circ d\) defines a new metric space \((\mathcal M, d_{\omega})\). This space is called the metric transform of \(\mathcal M\) by \(\omega\) and denoted by the symbol \(\omega(\mathcal M)\). The main theorem of the paper, which implies that linear and nonlinear Lipschitz extension properties of \(\mathcal M\) do not change when \(d\) is replaced by a new metric \(d_{\omega}\), is the following: Theorem. (a) \(e(\omega(\mathcal M), X)\leq C e(\mathcal M,X)^2\). (b) The same inequality holds for \(e_l\). \(C\) means an absolute constant.
scientific article; zbMATH DE number 1756798