Pointwise Lipschitz functions on metric spaces (Q1046504)

Let \(f\) be a real-valued function on a metric space \((X,d)\). The pointwise Lipschitz constant of \(f\) at a point \({x\in X}\) is defined as \[ \text{Lip} f(x)= \limsup_{y\to x, y\neq x}\frac{|f(x)-f(y)|}{d(x,y)}. \] The pointwise Lipschitz function space is defined as \[ D(X)=\{f: X\to\mathbb R: \|\text{Lip} f\|_\infty <\infty\}. \] It is obvious that always \({\text{LIP}(X)\subset D(x)}\), where \({\text{LIP}(X)}\) is the space of all real-valued Lipschitz functions on \(X\). The authors introduce a geometric condition on \(X\) called quasi-convexity and show that, for quasi-convex \(X\), the equality \({\text{LIP}(X)=D(X)}\) holds. Then the authors introduce local radially quasi-convex metric spaces. For such \(X\), the space \({(D^\infty(X), \|\cdot\|_{D^\infty})}\) is a Banach space. Here, \({D^\infty(X)}\) denotes the space of all bounded functions in \({D(X)}\), and \({\| f \|_{D^\infty}=\max\{\|f\|_\infty, \|\text{Lip} f\|_\infty \}}\). A Banach-Stone theorem for the space \({(D^\infty(X), \|\cdot\|_{D^\infty})}\) is proved. In the last section, real-valued functions on a metric measure space \({(X,d,\mu)}\) are considered. The space \({D^\infty(X)}\) is compared with the Newtonian-Sobolev space \({N^{1,\infty}(X)}\), introduced by \textit{N.\,Shanmugalingam} [Rev.\ Mat.\ Iberoam.\ 16, No.\,2, 243--279 (2000; Zbl 0974.46038)]. In particular, if \(X\) supports a doubling measure and satisfies a local Poincaré inequality, then \({D^\infty(X)=N^{1,\infty}(X)}\).