The authors consider a \(n\)-dimensional Ornstein-Uhlenbeck process \((X_t)_{t\in {\mathbb R}_+}\) written as
\[ X_t = e^{tA} x + \int_0^t e^{(t-s)A} B dZ_s \]
where \((Z_t)_{t\in {\mathbb R}_+}\) is a \(d\)-dimensional Lévy process, \(A\) is a \(n\times n\) matrix, and \(B\) is a \(n\times d\) matrix, \(d \leq n\). They show that under the rank condition
\[ \text{Rank} [ B , AB , \ldots , A^{n-1}B ] = n \]
the law of \(X_t\) is absolutely continuous provided the Lévy measure \(\nu\) of \((Z_t)_{t\in {\mathbb R}_+}\) is infinite and has a density on a ball in \({\mathbb R}^d\) centered around \(0\). In addition they show that the density of \(X_t\) is \({\mathcal C}^\infty\) with all bounded derivatives when \((Z_t)_{t\in {\mathbb R}_+}\) is of \(\alpha\)-stable type in the sense that, for some \(\alpha \in ( 0 , 2 )\) and some \(C>0\), the bound
\[ \int_{ \{z\in {\mathbb R}^d \;: \;| \langle z , h \rangle | \leq r \} } \langle z , h \rangle^2 \nu ( dz ) \geq C r^{2-\alpha} \]
holds for all sufficiently small \(r>0\) and all \(h \in {\mathbb R}^d\) with \(| h | = 1\).