Uniform Upper Bound for a Stable Measure of a Small Ball

Abstract

P. Hitczenko, S.Kwapien, W.N.Li, G.Schechtman, T.Schlumprecht and J.Zinn stated the following conjecture. Let $\mu$ be a symmetric $\alpha$-stable measure on a separable Banach space and $B$ a centered ball such that $\mu(B)\le b$. Then there exists a constant $R(b)$, depending only on $b$, such that $\mu(tB)\le R(b)t\mu(B)$ for all $0 \lt t \lt 1$. We prove that the above inequality holds but the constant $R$ must depend also on $\alpha$.