On a quadratic estimate related to the Kato conjecture and boundary value problems

TL;DR

This paper presents a direct proof of a quadratic estimate crucial for understanding elliptic operators and boundary value problems, advancing the functional calculus of Dirac type operators and addressing the Kato conjecture.

Contribution

It provides a new direct proof of a key quadratic estimate and applies it to the Kato conjecture and boundary value problems, expanding the functional calculus framework.

Findings

Established a new quadratic estimate with multiple equivalent forms.

Applied the estimate to solve the Kato conjecture.

Extended results to boundary value problems with $L^2$ boundary data.

Abstract

We provide a direct proof of a quadratic estimate that plays a central role in the determination of domains of square roots of elliptic operators and, as shown more recently, in some boundary value problems with $L^2$ boundary data. We develop the application to the Kato conjecture and to a Neumann problem. This quadratic estimate enjoys some equivalent forms in various settings. This gives new results in the functional calculus of Dirac type operators on forms.