On the density of the distribution of $p$-stable seminorms, $0<p<1$

Abstract:

Let $\mu$ be a symmetric $p$-stable measure, $0 < p < 1$, on a locally convex separable linear metric space $E$ and let $q$ be a lower semicontinuous seminorm on $E$. It is known that $F(t) = \mu \{ x:q(x) < t\}$ is absolutely continuous with respect to the Lebesgue measure. We prove an explicit formula for the density $F’(t)$ and give an asymptotic estimate of it at infinity.