Weak Type Inequality for the Square Function of a Nonnegative Submartingale
Tom 57 / 2009
Bulletin Polish Acad. Sci. Math. 57 (2009), 81-89
MSC: Primary 60G42; Secondary 60G48.
DOI: 10.4064/ba57-1-9
Streszczenie
Let $f$ be a nonnegative submartingale and $S(f)$ denote its square function. We show that for any $\lambda>0$, $$ \lambda \mathbb{P}(S(f)\geq \lambda) \leq \frac{\pi}{2}\,\|f\|_1,$$ and the constant $\pi/2$ is the best possible. The inequality is strict provided $\|f\|_1\neq 0$.
