The Peano's square-filling curve $p:I\to I^2$ turn's out to be Hölder continuous with exponent $1/2$ on the unit interval $I$ (a quick way to see it, is to note that $p$ is a fixed point of a suitable contraction $T:C(I,I^2)\to C(I,I^2)$, and the non-empty, closed subset of curves with modulus of continuity $\omega(t):=ct^{1/2}$ is $T$-invariant, for a suitable choice of $c$, so that $p$ is therein).

For the same reason, the more general analogous $n$-cube-filling curves $I\to I^n$ (e.g. described in the same Peano paper) are Hölder continuous with exponent $1/n$.

On the other hand, for any $1\le k\le n$, by elementary considerations on Hausdorff measures, no $\alpha$-Hölder continuous map $I^k\to I^n$ with exponent $\alpha > k/n $ can be surjective. The natural questions are therefore:

Given $1\le k\le n$, does there exist an $\alpha$-Hölder continuous map $I^k\to I^n$ with exponent $\alpha=k/n$? Otherwise, what is the best exponent $\alpha$ obtainable for such a surjective map? In particular, is there a simple construction for the case $I^2\to I^3$? (Actually, we may focus on this last question, which appears to be the simpler non-trivial case).

Summing up the above remarks, the answer is affirmative if $k=1$ or if $k=n$, and we may also note that if for a pair $(k,n)$ there is such a surjective map $q:I^k\to I^n$, then the map $(x_1,\dots,x_m)\mapsto (q(x_1), q(x_2),\dots ,q(x_m))$ is also a surjective map $I^{mk}\to I^{mn}$ with the same exponent $k/n$ of $q$. Also, we may consider compositions of maps, so that affirmative answers for $(k,n)$ and $(n,m)$ imply the affirmative answer for $(k,m)$.

Update 08.01.16. *The only answer received so far suggests a nice article, yet not related with this question (the only theorem in that paper that deals with Hölder maps is Thm 2.1, but has nothing or very little to do with the present problem, since it is about $\mathbb{R}$-valued functions, that is $n=1$ (existence of Hölder functions on a metric space which map surjectively onto an interval is a non-trivial problem only for totally disconnected spaces).*