The space $BMO$

Introducing $BMO$

The space of functions of bounded mean oscillation or $BMO$ arises when studying the space of functions whos deviation from the mean over cubes is bounded. In ways $BMO$ is similar to $L^\infty$, and it is often used as a replacement, however functions in $BMO$ may be unbounded. The classic example of this is $\log(x)$.

The space $BMO$ naturally arises in other situations too such as when studying singular integral operators, \begin{equation} f(x) \mapsto \int k(x,y) f(y)\, dy. \end{equation} These operators map $L^\infty$ to $BMO$.

For a complex-valued locally integrable function on $\mathbb{R}^n$ we define $$ ||f||_{BMO} = \sup_Q \frac{1}{|Q|} \int_Q |f(x) - \text{Avg}_Q f| dx, $$ where the supremium is taken over all cubes $Q \subset \mathbb{R}^n$. This defines a norm when we take an equivalance class of functions that differ a.e. by a constant. We denote $BMO = \Big\lbrace f : \mathbb{R}^n \to \mathbb{C} \Big| ||f||_{BMO} < \infty \Big\rbrace$. In fact, $BMO$ is a Banach space that contains $L^\infty$.

Our definition for $BMO$ used cubes, but the definition using balls instead is equivalent. Question: What other shapes could we use? How about $L^p$ balls for $1 \le p \le \infty$?

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