The Kato Square Root Problem for Divergence Form Operators with Potential

Date & time

1.30–2.30pm 3 April 2018


John Dedman Building 27 Room G35


Julian Bailey (MSI)

The Kato square root problem for divergence form elliptic operators is the equivalence statement $\norm{\sqrt{- \mathrm{div} \br{A \nabla}}u} \simeq \norm{\nabla u}$, where  $A$ is a complex matrix-valued function. In 2006, a few years after the first proof of this statement, A. Axelsson, S. Keith and A. McIntosh developed a general framework for proving square function estimates associated to Dirac-type operators and they showed that the Kato problem followed as an immediate application. In this talk I will discuss a generalisation of the Kato problem to include positive potentials $V$, namely $\norm{\sqrt{- \mathrm{div} \br{A \nabla} + V}u} \simeq \norm{\nabla u} + \norm{V^{\frac{1}{2}} u}$. I will discuss how the Axelsson-Keith-McIntosh framework can be altered to allow for dependence of the Dirac-type operator on the potential. The Kato problem for certain potentials will then follow as a result.

Updated:  27 April 2018/Responsible Officer:  Director/Page Contact:  School Manager