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(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?

===========

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.

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

===========

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?

===========

It follows immediately from the main result of

Tamás Keleti, András Máthé, Ondřej Zindulka: Hausdorff dimension of metric spaces and Lipschitz maps onto cubes, https://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.

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

===========

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?

============

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.

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

===========

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?

===========

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.

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

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.

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

===========

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?

============

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.