论文标题
最大不平等和加权BMO过程
Maximal inequalities and weighted BMO processes
论文作者
论文摘要
适用于左极限（rcll）进程$（x_t）_ {t \ in [0，τ]} $在公制空间$（\ Mathcal e，d）$中取值的一般适应性右连续（rcll）进程$（x_t）_ {t \ in [0，τ]} $$ \ frac {m-1} {2M-1} \ | \ | \ sup_ {t \ in [0，τ]} \ Mathbb {e}（d（x_ {t - }，x___________ \ m m马理F_T）\ | _mm \ le \ | \ sup_ {t \ in [0，τ]} d（x_0，x_t）\ | _m \ le c \ frac {m^2} {m^2} {m-1} \ | \ | \ | \ sup_ {t { f_t）\ | _M $$ 带有通用常数$ c $。这是Fefferman的概率版本 - 尖锐功能的Stein估计。虽然以前的不平等很容易从Doob的Martingale不平等中得出，但后来的不平等是John-Nirenberg加权BMO过程的不等式的结果，这是在本注中获得的。我们解释了如何利用约翰 - 尼伦贝格的不平等，以统一的方式获得新老式的不平等现象。
For a general adapted integrable right-continuous with left limits (RCLL) process $(X_t)_{t\in[0,τ]}$ taking values in a metric space $(\mathcal E,d)$, we show (among other things) that for every $m\in(1,\infty)$ $$ \frac{m-1}{2m-1}\|\sup_{t\in[0,τ]}\mathbb{E}(d(X_{t-},X_τ)|\mathcal F_t)\|_m\le \|\sup_{t\in[0,τ]}d(X_0,X_t)\|_m\le c\frac{m^2}{m-1} \|\sup_{t\in[0,τ]}\mathbb{E}(d(X_{t-},X_τ)|\mathcal F_t)\|_m $$ with a universal constant $c$. This is a probabilistic version of Fefferman--Stein estimate for the sharp maximal functions. While the former inequality is derived easily from Doob's martingale inequality, the later inequality is a consequence of John--Nirenberg inequalities for weighted BMO processes, which are obtained in this note. We explain how John--Nirenberg inequalities can be utilized to obtain inequalities for martingales, both old and new alike in a unified way.
