The Kato Square Root Problem for Divergence Form Operators with Potential

The Kato square root problem for divergence form elliptic operators with potential V:Rn→C is the equivalence statement ∥∥(L+V)12u∥∥2≃∥∇u∥2+∥∥V12u∥∥2, where L+V:=−div(A∇)+V and the perturbation A is an L∞ complex matrix-valued function satisfying an accretivity condition. This relation is proved for any potential with range contained in some positive sector and satisfying ∥∥|V|α2u∥∥2+∥∥(−Δ)α2u∥∥2≲∥∥(|V|−Δ)α2u∥∥2 for all u∈D(|V|−Δ) and some α∈(1,2]. The class of potentials that will satisfy such a condition is known to contain the reverse Hölder class RH2 and Ln2(Rn) in dimension n>4. To prove the Kato estimate with potential, a non-homogeneous version of the framework introduced by Axelsson, Keith and 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.