## The Annals of Probability

### Sharp Inequalities for the Conditional Square Function of a Martingale

Gang Wang

#### 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.

#### Article information

Source
Ann. Probab., Volume 19, Number 4 (1991), 1679-1688.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176990229

Digital Object Identifier
doi:10.1214/aop/1176990229

Mathematical Reviews number (MathSciNet)
MR1127721

Zentralblatt MATH identifier
0744.60046

JSTOR