Mapping analytic sets onto cubes by little Lipschitz functions

Abstract

A mapping $$f:X\rightarrow Y$$ between metric spaces is called little Lipschitz if the quantity $$\begin{aligned} \mathrm{lip}\,f(x)=\liminf _{r\rightarrow 0}\,\frac{\mathrm{diam}\,f(B(x,r))}{r} \end{aligned}$$ is finite for every $$x\in X$$ . We prove that if a compact (or, more generally, analytic) metric space has packing dimension greater than n, then it can be mapped onto an n-dimensional cube by a little Lipschitz function. The result requires two facts that are interesting in their own right. First, an analytic metric space X contains, for any $$\varepsilon >0$$ , a compact subset S that embeds into an ultrametric space by a Lipschitz map, and . Second, a little Lipschitz function on a closed subset admits a little Lipschitz extension.