Maximal inequalities and weighted BMO processes

For a general adapted integrable right-continuous with left limits (RCLL) process $(X_t{)}_{t\in [0,\tau ]}$ taking values in a metric space $(\mathcal{E},d)$, we show (among other things) that for every $m\in (1,\mathrm{\infty })$ $$\frac{m-1}{2m-1}\mathrm{\parallel }\underset{t\in [0,\tau ]}{\sup }\mathbb{E}(d(X_{t-},X_{\tau })\mathrm{\mid }{\mathcal{F}}_t){\mathrm{\parallel }}_m\le \mathrm{\parallel }\underset{t\in [0,\tau ]}{\sup }d(X_0,X_t){\mathrm{\parallel }}_m\le c\frac{m^2}{m-1}\mathrm{\parallel }\underset{t\in [0,\tau ]}{\sup }\mathbb{E}(d(X_{t-},X_{\tau })\mathrm{\mid }{\mathcal{F}}_t){\mathrm{\parallel }}_m$$ 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.