Best Hölder exponents of surjective maps from the unit square to the unit cube

(This is not a complete answer, but I cannot comment.)

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Edit: To clarify, I should first observe, as Willie Wong points out in the comments, that there is a typo in the original question. Simple Hausdorff dimension arguments show that there can be no $\alpha$-Hölder map from $I^k$ onto $I^n$ with $\alpha>k/n$. (The question says the opposite.) The question is therefore: what is the largest value of $\alpha$ for which we can find such a map?

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It follows immediately from the main result of

http://arxiv.org/pdf/1203.0686.pdf

that for every $\alpha<k/n$ there is an $\alpha$-Hölder map from $I^k$ onto $I^n$.

In the general metric space case of that theorem, a map with optimal Hölder exponent need not exist.

In this very specific case, I would bet that it does, but I don't have a specific construction in mind.

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)

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:

S. Semmes, Where the Buffalo Roam: Infinite Processes and Infinite Complexity, arXiv:math/0302308v1 (2003)