# Wydawnictwa / Czasopisma IMPAN / Bulletin Polish Acad. Sci. Math. / Wszystkie zeszyty

## 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 $\|f\|_1\neq 0$.