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 $2m$ ($m\geq 1$), 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 $2.$