Let $X$, $Y$ be càdlàg martingales and let $Y^{\#}$ denote the sharp function of $Y$. The paper contains the proof of the estimate ${\Vert {\int }_0^{\infty }\left|d{〈X,Y〉}_t\right|\Vert }_1\le \sqrt{2}{\Vert {〈X〉}^{1∕2}{Y}^{\#}\Vert }_1$ for the total variation between $X$ and $Y$. The constant $\sqrt{2}$ is shown to be the best possible. The proof rests on the construction of an appropriate special function, enjoying certain size and concavity requirements.

中文翻译：

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Pratelli 不等式的扩展

让 $X$, $是$ 成为 càdlàg martingales 并让 ${是}^{\#}$ 表示锐函数 $是$. 该论文包含估计的证明${\Vert {\int }_0^{\infty }\left|d{〈X,是〉}_{吨}\right|\Vert }_1\le \sqrt{2}{\Vert {〈X〉}^{1∕2}{是}^{\#}\Vert }_1$ 之间的总变异 $X$ 和 $是$. 常数$\sqrt{2}$被证明是最好的。证明依赖于适当的特殊函数的构造，具有一定的尺寸和凹度要求。