There are such surjections with critical Hölder exponent for any pair of dimensions k < n. Stong showed that there is a bijection $\mathbb Z^k \to \mathbb Z^n$ that is Hölder continuous with exponent $k/n$:
R. Stong, Mapping $\mathbb Z^r$ into $\mathbb Z^s$ with Maximal Contraction, Discrete Comput Geom 20:131–138 (1998)R. Stong, Mapping $\mathbb Z^r$ into $\mathbb Z^s$ with Maximal Contraction, Discrete Comput Geom 20:131–138 (1998)
A limit construction can then be used to obtain surjections from $\mathbb R^k$ to $\mathbb R^n$ of the same regularity, which also implies the surjection result for cubes. Some details and further interesting discussions about such maps are contained in section 9.1 of the following notes by Semmes:
