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

## TL;DR

This paper develops scale-invariant bounds for Green's functions associated with second order elliptic equations with lower order coefficients, establishing optimal conditions in Lorentz spaces and applying results to subsolution estimates.

## Contribution

It introduces scale-invariant bounds for Green's functions with optimal Lorentz space conditions, without smallness assumptions on coefficients, and applies these to subsolution estimates.

## Key findings

- Green's functions bounds are scale invariant.
- Optimal Lorentz space condition $b-c
otin L^{n,1}$ for pointwise bounds.
- No smallness assumption needed on lower order coefficients.

## Abstract

We construct Green's functions for elliptic operators of the form $\mathcal{L}u=-\text{div}(A\nabla u+bu)+c\nabla u+du$ in domains $\Omega\subseteq\mathbb R^n$, under the assumption $d\geq\text{div}b$, or $d\geq\text{div}c$. We show that, in the setting of Lorentz spaces, the assumption $b-c\in L^{n,1}(\Omega)$ is both necessary and optimal to obtain pointwise bounds for Green's functions. We also show weak type bounds for Green's functions and their gradients. Our estimates are scale invariant and hold for general domains $\Omega\subseteq\mathbb R^n$. 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 $\mathcal{L}u\leq -\text{div}f+g$ in the case $d\geq\text{div}c$.