Asymptotic behaviour of densities of stable semigroups of measures (Q910083)

We prove that densities of the measures in a strictly stable semigroup \((h_ t)\) of symmetric measures on a homogeneous group, if they exist, have the following asymptotic behavior: \[ \lim_{| x| \to \infty}| x|^{Q+\alpha}\cdot h_ 1(x)=k(\bar x), \] where \(\alpha\) is the characteristic exponent, \(\bar x=| x|^{-1}x\), and k is the density of the Lévy measure associated to the semigroup. Moreover, if \(k(\bar x)=0\) a more precise description is given.