The Kato Square Root Problem for Divergence Form Operators with Potential

The Kato square root problem for divergence form elliptic operators with potential $V : \mathbb{R}^{n} \rightarrow \mathbb{C}$ is the equivalence statement $\left\Vert (L + V)^{\frac{1}{2}} u\right\Vert_{2} \simeq \left\Vert \nabla u \right\Vert_{2} + \left\Vert V^{\frac{1}{2}} u \right\Vert_{2}$, where $L + V := - \mathrm{div} A \nabla + V$ and the perturbation $A$ is an $L^{\infty}$ complex matrix-valued function satisfying an accretivity condition. This relation is proved for any potential with range contained in some positive sector and satisfying $\left\Vert |V|^{\frac{\alpha}{2}} u\right\Vert_{2} + \left\Vert (-\Delta)^{\frac{\alpha}{2}} \right\Vert_{2} \lesssim \left\Vert ( |V| - \Delta)^{\frac{\alpha}{2}}u \right\Vert_{2}$ for all $u \in D(|V| -\Delta)$ and some $\alpha \in (1,2]$. The class of potentials that will satisfy such a condition is known to contain the reverse H\"{o}lder class $RH_{2}$ and $L^{\frac{n}{2}}(\mathbb{R}^{n})$ in dimension $n > 4$. To prove the Kato estimate with potential, a non-homogeneous version of the framework introduced by A. Axelsson, S. Keith and A. McIntosh for proving quadratic estimates is developed. In addition to applying this non-homogeneous framework to the scalar Kato problem with zero-order potential, it will also be applied to the Kato problem for systems of equations with zero-order potential.