John–Nirenberg inequality

Remarks

The proof applies the Calderón–Zygmund lemma to $f - f_Q$ on the cube $Q$, at a height of about $2\norm{f}_{\mathrm{BMO}}$. On the selected subcubes $Q_k$ the mean $f_{Q_k}$ differs from $f_Q$ by at most a constant times $\norm{f}_{\mathrm{BMO}}$, and their total measure is at most half of $\abs{Q}$. Repeating the argument inside each $Q_k$ shows that the set where $\abs{f - f_Q}$ exceeds $C m \norm{f}_{\mathrm{BMO}}$ has measure at most $2^{-m}\abs{Q}$, which is the exponential decay.

Consequences: for $c < c_2$ the function $\exp(c \abs{f - f_Q}/\norm{f}_{\mathrm{BMO}})$ has average over $Q$ bounded by a constant depending on $n$ and $c$. Also, for every $1 \le p < \infty$, the quantity $\sup_Q \left( \frac{1}{\abs{Q}} \int_Q \abs{f - f_Q}^p \right)^{1/p}$ is a norm on BMO equivalent to $\norm{f}_{\mathrm{BMO}}$.

Example: $f(x) = \log \abs{x}$ on $\R$ and $Q = [-1, 1]$, where $f_Q = -1$. The set $\set{\abs{f + 1} > \lambda}$ contains $\set{\abs{x} < e^{-1 - \lambda}}$, of measure $2 e^{-1 - \lambda}$. So the exponential decay in $\lambda$ cannot be improved to a faster decay for all BMO functions.

Going deeper

For a student

The inequality shows that BMO functions are locally integrable to every power, and even exponentially integrable after subtracting means. It is used to show that $\log w$ is in BMO for every Muckenhoupt weight $w$, and conversely that $e^{\delta b}$ is a Muckenhoupt weight for $b \in$ BMO and $\delta$ small. It enters the proof of Fefferman's duality theorem and of the boundedness of commutators with BMO functions.