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

TL;DR

This paper extends the Kato problem to higher-order degenerate elliptic operators with complex coefficients in weighted spaces, establishing solvability of boundary value problems near $p=2$.

Contribution

It generalizes previous work to arbitrary order operators with measurable coefficients satisfying Gårding inequality in weighted spaces.

Findings

Solved unweighted $L^{p}$ boundary value problems near $p=2$

Extended Kato problem to higher-order degenerate elliptic operators

Established solvability for complex-valued coefficients satisfying Gårding inequality

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 $2.$