Hausdorff dimension of metric spaces and Lipschitz maps onto cubes

We prove that a compact metric space (or more generally an analytic subset of a complete separable metric space) of Hausdorff dimension bigger than $k$ can be always mapped onto a $k$-dimensional cube by a Lipschitz map. We also show that this does not hold for arbitrary separable metric spaces. As an application we essentially answer a question of Urba\'nski by showing that the transfinite Hausdorff dimension (introduced by him) of an analytic subset $A$ of a complete separable metric space is the integer part of $\dim_H A$ if $\dim_H A$ is finite but not an integer, $\dim_H A$ or $\dim_H A-1$ if $\dim_H A$ is an integer and at least $\omega_0$ if $\dim_H A=\infty$.