BMO spaces on the torus

Question

I was reading BMO spaces (John-Nirenberg) on wikipidia http://en.wikipedia.org/wiki/Bounded_mean_oscillation. There they define BMO norm as $$sup_{Q}\frac{1}{Q}\int_Q |u(y) - u_Q|dy$$ where $u_Q$ is the average of $u(y)$ over $Q$ and the supremum is taken over all cubes of arbitrary diameter.

Questions: 1. I think what can be the definition of BMO spaces on a torus $T^n = S^1 \times ...\times S^1$? We cannot have cubes of arbitrary diameter. Maybe we can look at the supremum over cubes whose diameter is smaller than or equal to that of torus?

2 What will happen if we take the supremum over cubes $Q$ whose diameter is less than or equal $r$, say, r being very small? Will that give the same norm? Thanks please.

Edit: Question 3. Please also advise a little about non-quotient spaces?

Edit:Joonas Ilmavirta answers Q1 and 3 below. Someone please look at Question 2.

Answer 1

A function $T^n\to\mathbb R$ can be thought of as a periodic function $\mathbb R^n\to\mathbb R$. (Periodic in each variable, that is.) This way you can use the BMO norm of $\mathbb R^n$ on $T^n$ as well, and suddenly the size of the cube is no problem.

More explicitly, identify $T^n=\mathbb R^n/\mathbb Z^n$ and let $q:\mathbb R^n\to T^n$ be the quotient map. Then simply set $$ ||u||_{BMO(T^n)} = ||q^*u||_{BMO(\mathbb R^n)}, $$ where $q^*u(x)=u(q(x))$ is the pullback of $u$ over $q$.

This may not be the only possible choice, but at least it seems natural.

Edit (to answer the edit in the question): For a more general space $X$, we could define the BMO norm of $u:X\to\mathbb R$ as $$ \sup_{x\in X,r>0}|B(x,r)|^{-1}\int_{B(x,r)}|u-u_{B(x,r)}|. $$ To make sense of this, you only need $X$ to have a metric (or some kind of balls) and a measure. If $X$ is bounded, then it just happens that $B(x,r)=X$ for $r$ large enough, which is no issue.