Abstract: We consider the Dirichlet problems for second-order linear elliptic equations with the first-order term given by a singular vector field b. The Calderon-Zygmund estimates for weak solutions in are well-known, provided that b is sufficiently regular, e.g., . In this lecture, we derive the Calderon-Zygmund estimates for weak solutions when the drift b belongs to the critical spaces , where is the spatial dimension. Here
denotes the standard weak- space. Moreover, optimality of such results will be discussed by means of concrete examples.