relationship between BMO norm and $L_p$ norm

Question

Are there any relationship between the BMO norm of a function and its $L_p$ norms? For example, one norm is controlled by the other for functions of some special class.

Answer 1

BMO is a bit like the space $L_\infty$, and also a bit like the Orlicz norm corresponding to $e^x$. So if you are on a bounded domain, then BMO is a subspace of $L_p$ for every $1 \le p < \infty$, with $\|f\|_p \le c_p \|f\|_{\text{BMO}}$ where $c_p$ grows like $p$. However $L_\infty$ is a subspace of BMO, and there are functions in BMO that are not in $L_\infty$, for example, $-\log |x|$ on $[-1,1]$.

Also BMO has the following property: there exist functions $0 \le g \le f$ such that $f \in \text{BMO}$, but $g \notin \text{BMO}$.