A Sharp Liouville Theorem for Elliptic Operators

We introduce a new condition on elliptic operators $L= {1/2}\triangle + b \cdot \nabla $ which ensures the validity of the Liouville property for bounded solutions to $Lu=0$ on $\R^d$. Such condition is sharp when $d=1$. We extend our Liouville theorem to more general second order operators in non-divergence form assuming a Cordes type condition.