On hypoellipticity of generators of Lévy processes (Q711470)

The paper under review provides a condition on a Lévy measure \(\nu\) which ensures the hypoellipticity of the generator \(L\) of the corresponding pure-jump Lévy process. An operator \(L\) is called hypoelliptic with respect to \(H=\bigcup_{s\in \mathbb R} H^s(\mathbb R^d)\) if for every \(f,u\in H\) such that \(Lu=f\) the infinite differentiability of \(f\) on any open set \(U\) implies the infinite differentiability of \(u\) on \(U\). Here, \(H^s(\mathbb R^d)\) is the Sobolev space of order \(s\in\mathbb R\). The authors prove that a generator \(L\) of a pure-jump \(\mathbb R^d\)-valued Lévy process with Lévy measure \(\nu\) is hypoelliptic provided that \(\nu\) has a density \(n\) which is smooth on \(\mathbb R^d\backslash \{0\}\) with derivatives which are square-integrable away from \(0\) and some lower bound near \(0\) is satisfied. Also, the authors show that if \(\nu\) is compactly supported, then the smoothness of \(n\) on \(\mathbb R^d\backslash \{0\}\) is a necessary condition for the hypoellipticity of \(L\).