On scale-invariant bounds for the Green’sfunction for second-order elliptic equations with lower-order coefficients andapplications

We construct Green’s functions for elliptic operators of the form ℒ u = − div ( A ∇ u + b u ) + c ∇ u + d u in domains Ω ⊆ 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 ∈ L n , 1 ( Ω ) 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 Ω ⊆ 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 ℒ u ≤ − div f + g in the case d ≥ div c .