There is a measure preserving map from the unit interval onto the unit cube that is Lipschitz of order 1/2, that is $|f(x)-f(y)| \leq A |x-y|^{1/2}$. By considering the image of small intervals, one can see that one could not have a smoother map.

Now consider maps from $[0,1]^2$ onto $[0,1]^3$ that preserve measure. By looking at the image of small balls we see that f can't be smoother than Lipschiz 2/3.

Does there exist a measure preserving map from $[0,1]^2$ onto $[0,1]^3$ that is Lipschitz 2/3?