Equivalent Definitions For Bmo

The space of functions of bounded mean oscillation $BMO$ is defined by the BMO norm

$$||f||_{BMO} = \sup_{\text{cubes }Q} \frac{1}{\|Q\|} \int_Q |u(y) - u_Q| dy$$

But an equivalent definition is to take the sup over balls instead of cubes. Previously I wondered what other shapes gave an equivalent norm.

Proposition: Suppose $D \subset \mathbb{R}^n$ is a open set such that there exists $0 < r_1 < r_2 < \infty$ such that

$$B(0, r_1) \subset D \subset B(0, r_2)$$

then the norm given by

$$||f||_D := \sup_{E \in A_D} \frac{1}{|E|} \int_E |f(y) - f_E| dy$$

is equivalent to the BMO norm. The set $A_D$ is $D$ under any uniform scaling, rotations transtions or composition thereof.

I have not yet proven this, but I think it should be possible by adapting ideas from Stein’s Harmonic Analysis.