#### Vol. 14, No. 1, 2021

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

### Georgios Sakellaris

Vol. 14 (2021), No. 1, 251–299
##### Abstract

We construct Green’s functions for elliptic operators of the form $\mathsc{ℒ}u=-div\left(A\nabla u+bu\right)+c\nabla u+du$ 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 $\mathsc{ℒ}u\le -divf+g$ in the case $d\ge divc$.

##### Keywords
Green's function, fundamental solution, lower-order coefficients, pointwise bounds, Lorentz bounds, maximum principle, Moser-type estimate
##### Mathematical Subject Classification 2010
Primary: 35A08, 35J08, 35J15
Secondary: 35B50, 35J20, 35J86