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arXiv · 2608.10892

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

Chao Zhang
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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 $$ \sup_{I\subseteq I_0}\frac{1}{|I|} \left|\left\{x\in I: f(x)-\essinf_I f>\lambda\right\}\right| \le c_1\exp\left(-\frac{c_2\lambda}{\norm{f}_{\BLO(I_0)}}\right), \qquad \lambda>0, $$ with the usual interpretation when $\|f\|_{\mathrm{BLO}(I_0)}=0$. The sharp constants are $c_1^*=e$ and $c_2^*=1$. 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.

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Chao Zhang. 2026-08-11. Sharp constants in the one-sided John-Nirenberg inequality for functions of bounded lower oscillation. https://arxiv.org/abs/2608.10892

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