A counterexample for pointwise upper bounds on Green's function with a singular drift at boundary

Abstract: We show an example of a sequence of elliptic operators in the unit ball with drifts that diverge as the inverse distance to the boundary, for which we don't get uniform upper estimates for the Green's function with the pole at the origin. Such drifts have been considered in the literature in the study of the $L^{p}$ Dirichlet problem for both the parabolic and elliptic operators.