A sharp Liouville theorem for elliptic operators

We introduce a new condition on elliptic operators $ L = \triangle + b \cdot \nabla $, which ensures the validity of the Liouville property, i.e., all smooth bounded solutions to $Lu = 0$ on $R^d$ are constant. 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