Suppose that $E$ is a complete separable real metric vector space. It is proved that if $X$ is a symmetric $E$-valued $p$-stable random vector, $0 < p < 2$, and $q$ is a lower semicontinuous, a.s. finite seminorm, then the distribution of $q(X)$ is absolutely continuous apart from a possible jump. If, additionally, $q$ is strictly convex or $0 < p < 1$, then the distribution of $q(X)$ is either absolutely continuous or degenerate at 0. This result settles, in particular, the problem of absolute continuity of the supremum of stable sequences, extending thus Tsirel'son's theorem.
"Absolute Continuity of Stable Seminorms." Ann. Probab. 14 (1) 299 - 312, January, 1986. https://doi.org/10.1214/aop/1176992629