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 $\Omega \subseteq {ℝ}^n$, under the assumption $d\ge divb$ or $d\ge divc$. We show that, in the setting of Lorentz spaces, the assumption $b-c\in L^{n,1}\left(\Omega \right)$ 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 $\Omega \subseteq {ℝ}^n$. 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 $d\ge divc$.

我们为以下形式的椭圆算子构造Green函数 $\mathcal{ℒ}ü=-div（\mathrm{一种}\nabla ü+bü）+C\nabla ü+dü$ 在域中 $\Omega \subseteq {ℝ}^{ñ}$，在假设下 $d\ge divb$ 或者 $d\ge divC$。我们证明，在洛伦兹空间的设定中，假设$b-C\in {\mathrm{大号}}^{ñ，\mathrm{1个}}（\Omega ）$获得格林函数的逐点边界既是必需的又是最优的。我们还显示了格林函数及其梯度的弱型边界。我们的估计是尺度不变的，适用于一般领域$\Omega \subseteq {ℝ}^{ñ}$。而且，在低阶系数的范数上没有小的假设。作为应用程序，我们获得子解决方案的尺度不变全局和局部有界估计$\mathcal{ℒ}ü\le -divF+G$ 在这种情况下 $d\ge divC$。