The Kato problem and extensions for degenerate elliptic operators of higher order in weighted spaces

Abstract

We consider the Kato problem and extensions for degenerate elliptic operators of arbitrary order $2m$ ($m\ge 1$), whose coefficients are measurable, complex-valued and satisfy the G$\mathring{a}$rding inequality with respect to a Muckenhoupt $A_2$-weight; this generalizes the work of [Cruz-Uribe, Martell and Rios 2018]. As an application, the unweighted $L^p$-Dirichlet, regularity and Neumann boundary value problems associated to such an operator are solved when $p$ is sufficiently close to $2.$