The space of functions of bounded mean oscillation on $\mathbb R^n$ is defined by $$ BMO(\mathbb R^n) = \left\{f\in L^1_{\rm loc}(\mathbb R^n) : \sup_Q\frac 1{|Q|}\int_Q|f(x)-f_Q|\,dx < \infty\right\}. $$ Here, the supremum is taken over all bounded cubes $Q\subset\mathbb R^n$ and $f_Q$ denotes the average of $f$ on $Q$, i.e., $f_Q = \tfrac 1{|Q|}\int_Qf(x)\,dx$.
Now, suppose that $f\in BMO(\mathbb R^n)$ and $A : \mathbb R^n\to\mathbb R^n$ is a linear transformation with $\det A\neq 0$ (actually, I am only interested in the case where $n=2$ and $A = \operatorname{diag}(\alpha,\alpha^{-1})$, $\alpha > 0$). Does then $f\circ A\in BMO(\mathbb R^n)$?
My problem is that (in my example) cubes are transformed into rectangles by $A$ (or $A^{-1}$) and I don't know, whether I can replace cubes by rectangles with fixed side length ratio in the definition of BMO.