Sharp Inequalities for the Conditional Square Function of a Martingale

Wang, Gang

Ann. Probab., Tome 19 (1991) no. 4, p. 1679-1688 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 1991-10-14
Classification:  Martingale,  conditionally symmetric martingale,  dyadic martingale,  square-function inequality,  conditional-square-function inequality,  confluent hypergeometric function,  60G42,  60E15,  33A30