On scale-invariant bounds for the Green's function for second-order elliptic equations with lower-order coefficients and applications

Abstract

We construct Green’s functions for elliptic operators of the form $\mathcal{ℒ}u=-div\left(A\nabla u+bu\right)+c\nabla u+du$ in domains $\mathrm{\Omega }\subseteq {ℝ}^n$, under the assumption $d\ge divb$ or $d\ge divc$. We show that, in the setting of Lorentz spaces, the assumption $b-c\in L^{n,1}\left(\mathrm{\Omega }\right)$ 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 $\mathrm{\Omega }\subseteq {ℝ}^n$. 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 $\mathcal{ℒ}u\le -divf+g$ in the case $d\ge divc$.