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

We determine the sharp constants in the one-sided John-Nirenberg inequality for functions of bounded lower oscillation on an interval. More precisely, if $f\in \text{BLO}(I_0)$, then $$\underset{I\subseteq I_0}{\sup }\frac{1}{\mathrm{\mid }I\mathrm{\mid }}\mid \{x\in I:f(x)-{\text{essinf}}_{I}f>\lambda \}\mid \le c_1\exp (-\frac{c_2\lambda }{\text{norm}{f}_{\text{BLO}(I_0)}}),\lambda >0,$$ with the usual interpretation when $\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{B}\mathrm{L}\mathrm{O}(I_0)}=0$. 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.