Schauder bases in Lipschitz free spaces over nets of \(\mathcal{L}_\infty\)-spaces (Q2681977)

The paper discusses two explicit constructions of a Schauder basis for the Lipschitz-free space \(\mathcal{F}(N)\) over certain uniformly discrete metric spaces \(N\). The authors present explicit constructions of Lipschitz retractions for nets in finite-dimensional Banach spaces and grids in Banach spaces with an FDD. For this the author introduce the notion of a \(K\)-retractional basis of a countable metric space \(M\), which is a sequence of retractions \(\{\varphi_n\}_{n=1}^\infty\), \(\varphi_n\colon M\to M\), satisfying the following conditions: i. \(\varphi_n(M)=M_n:=\bigcup_{j=1}^{n}\{x_j\}\), ii. \(\bigcup_{j=1}^\infty\{x_j\}=M\), iii. \(\varphi_n\) is \(K\)-Lipschitz for every \(n\in\mathbb N\), iv. \(\varphi_m\circ\varphi_n=\varphi_{\min(m,n)}\) for every \(m,n\in\mathbb N\). The authors construct an explicit family of Lipschitz retractions for nets in finite-dimensional Banach spaces, which results in a Schauder basis for their Lipschitz free spaces whose constant is independent of the dimension. Furthermore, the construction of an explicit family of Lipschitz retractions for grids in Banach spaces with an FDD is presented. The retractions are then used to extend Lipschitz maps from a finite number of points to the whole net by composition, keeping the Lipschitz constant under control independently of the number of points from which the extension is made.