The solution of the Kato problem for degenerate elliptic operators with Gaussian bounds

TL;DR

This paper proves the Kato conjecture for a class of degenerate elliptic operators with Gaussian bounds, establishing the equivalence of the operator's square root and the associated energy form.

Contribution

It demonstrates the Kato conjecture for degenerate elliptic operators with weights and Gaussian bounds, extending previous results to more general settings.

Findings

Kato square root estimate holds for weighted degenerate elliptic operators

Operators with Gaussian bounds satisfy the Kato conjecture

Extension of Kato conjecture to complex-valued, weighted operators

Abstract

We prove the Kato conjecture for degenerate elliptic operators in R^n. More precisely, we consider the divergence form operator L_w = -1/w div (wA) grad, where w is a Muckenhoupt A_2 weight and A is a complex valued n x n matrix which is bounded and uniformly elliptic. We show that if the associated semigroup satisfies Gaussian upper bounds, then the Kato square root estimate holds.