Mapping Analytic sets onto cubes by little Lipschitz functions

Abstract

A mapping $f:X\to Y$ between metric spaces is called \emph{little Lipschitz} if the quantity $$\mathrm{lip}(f(x)=\underset{r\to 0}{\mathrm{liminf}}\frac{\mathrm{diam}f(B(x,r))}{r}$$ 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 $X$ can be mapped onto an $n$-dimensional cube by a little Lipschitz function. The result requires two facts that are interesing in their own right. First, an analytic metric space $X$ contains, for any $\epsilon >0$, a compact subset $S$ that embeds into an ultrametric space by a Lipschitz map, and ${\dim }_{P}S\ge {\dim }_{P}X-\epsilon$. Second, a little Lipschitz function on a closed subset admits a little Lipschitz extension.