Maximal inequalities and weighted BMO processes

Abstract

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 $m\in (1,\infty )$ $$\frac{m-1}{2m-1}\Vert \underset{t\in [0,\tau ]}{\sup }\mathbb{E}(d(X_{t-},X_{\tau })|{\mathcal{F}}_t){\Vert }_m\le \Vert \underset{t\in [0,\tau ]}{\sup }d(X_0,X_t){\Vert }_m\le c\frac{m^2}{m-1}\Vert \underset{t\in [0,\tau ]}{\sup }\mathbb{E}(d(X_{t-},X_{\tau })|{\mathcal{F}}_t){\Vert }_m$$ 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.

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