ON SCALE-INVARIANT BOUNDS FOR THE GREEN'S FUNCTION FOR SECOND-ORDER ELLIPTIC EQUATIONS WITH LOWER-ORDER COEFFICIENTS AND APPLICATIONS

We construct Green's functions for elliptic operators of the form Lu = -div(A del u + bu) + c del u + du in domains Omega subset of R-n, under the assumption d >= div b or d >= div c. We show that, in the setting of Lorentz spaces, the assumption b - c is an element of L-n,L-1(Omega) is both necessary and optimal to obtain pointwise bounds for Green's functions. We also show weak-type bounds for the Green's function and its gradients. Our estimates are scale-invariant and hold for general domains Omega subset of R-n. Moreover, there is no smallness assumption on the norms of the lower-order coefficients. As applications we obtain scale-invariant global and local boundedness estimates for subsolutions to Lu <= -div f + g in the case d >= div c.