AI 中文总结
该研究确定了区间上有界下振荡函数的单侧John-Nirenberg不等式的最佳常数，给出两种证明方法并阐述其推论。
AI 中文摘要
我们确定区间上有界下振荡（BLO）函数的单侧John-Nirenberg不等式中的最佳常数。更准确地说，若$f\in\BLO(I_0)$，则对任意$\lambda>0$，有$\sup_{I\subseteq I_0}\frac{1}{|I|}\left|\left\{x\in I: f(x)-\essinf_I f>\lambda\right\}\right| \le c_1\exp\left(-\frac{c_2\lambda}{\\|f\\|_{\BLO(I_0)}}\right)$，当$\\|f\\|_{\BLO(I_0)}=0$时按常规解释，最佳常数为$c_1^*=e$、$c_2^*=1$。我们基于Riesz上升引理和独立的Bellman函数给出证明，最后给出该最佳估计的若干推论。
英文摘要
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 $\|f\|_{\mathrm{BLO}(I_0)}=0$. 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.
