Weak Type Inequality for the Square Function of a Nonnegative Submartingale

Volume 57 / 2009

Adam Os/ekowski Bulletin Polish Acad. Sci. Math. 57 (2009), 81-89 MSC: Primary 60G42; Secondary 60G48. DOI: 10.4064/ba57-1-9

Abstract

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

Authors

  • Adam Os/ekowskiInstitute of Mathematics
    University of Warsaw
    Banacha 2
    02-097 Warszawa, Poland
    e-mail

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