Distribution density of the norm of a stable vector

Let $B$ be a Banach space, $X$ be a stable $B$-valued random vector with exponent $\alpha\in(0,2)$, a $p(\cdot)$, and $p(\cdot)$ be the distribution density of the norm of $X$. In this paper we study the question of the boundedness of $p$. In particular, we construct examples of a space $B$ with a symmetric stable vector $X$ with exponent $\alpha\in(1,2)$ with unbounded $p$ and prove that if $X$ is a nondegenerate strictly stable vector with exponent $\alpha\in(0,1)$, then $p$ is bounded.