Abstract
Let $f$ be a real martingale and $s(f)$ its conditional square function. Then the following inequalities are sharp: $\|f\|_p \leq \sqrt{\frac{2}{p}}\|s(f)\|_p,\quad 0 < p \leq 2,$ $\sqrt{\frac{2}{p}}\|s(f)\|_p \leq \|f\|_p,\quad p \geq 2.$ The second inequality is still sharp if $f$ is replaced by the maximal function $f^\ast$. Let $S(f)$ denote the square function of $f$. Then the following inequalities are also sharp: $\|S(f)\|_p \leq \sqrt{\frac{2}{p}}\|s(f)\|_p,\quad 0 < p \leq 2,$ $\sqrt{\frac{2}{p}}\|s(f)\|_p \leq \|S(f)\|_p,\quad p \geq 2.$ These inequalities hold for Hilbert-space-valued martingales and are strict inequalities in all of the nontrivial cases.
Citation
Gang Wang. "Sharp Inequalities for the Conditional Square Function of a Martingale." Ann. Probab. 19 (4) 1679 - 1688, October, 1991. https://doi.org/10.1214/aop/1176990229
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