The Kato Square Root Problem for Divergence Form Operators with Potential

Abstract

The Kato square root problem for divergence form elliptic operators with potential $V:{\mathbb{R}}^n\to \mathbb{C}$ is the equivalence statement ${\Vert (L+V{)}^{\frac{1}{2}}u\Vert }_2\simeq {\Vert \nabla u\Vert }_2+{\Vert V^{\frac{1}{2}}u\Vert }_2$, 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 ${\Vert |V{|}^{\frac{\alpha }{2}}u\Vert }_2+{\Vert (-\Delta {)}^{\frac{\alpha }{2}}\Vert }_2\lesssim {\Vert (|V|-\Delta {)}^{\frac{\alpha }{2}}u\Vert }_2$ 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 Hölder class $R{H}_2$ 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.

arXiv admin note: text overlap with arXiv:1902.01101