Sharp constants in the one-sided John-Nirenberg inequality for functions of bounded lower oscillation

Abstract

We determine the sharp constants in the one-sided John-Nirenberg inequality for functions of bounded lower oscillation on an interval. More precisely, if $f\in\BLO(I_0)$, then $$ \sup_{I\subseteq I_0}\frac{1}{|I|} \left|\left\{x\in I: f(x)-\essinf_I f>λ\right\}\right| \le c_1\exp\left(-\frac{c_2λ}{\norm{f}_{\BLO(I_0)}}\right), \qquad λ>0, $$ with the usual interpretation when $\Vert f{\Vert }_{\mathrm{BLO}(I_0)}=0$. The sharp constants are $c_1^{\ast }=e$ and $c_2^{\ast }=1$. We give a proof based on the Riesz rising sun lemma and an independent Bellman function proof. Finally, we present several consequences of the sharp estimate.