The Kato square root problem for weighted parabolic operators

TL;DR

This paper provides a simplified proof of the Kato square root estimate for weighted parabolic operators with measurable coefficients, allowing for degenerate ellipticity and nearly complete separation of space and time variables.

Contribution

It introduces a new, streamlined proof technique that handles degenerate weights and non-autonomous operators in the parabolic setting, extending previous results.

Findings

Established Kato square root estimates for weighted parabolic operators

Allowed for coefficients depending measurably on space and time

Handled degenerate ellipticity via spatial A2-weight

Abstract

We give a simplified and direct proof of the Kato square root estimate for parabolic operators with elliptic part in divergence form and coefficients possibly depending on space and time in a merely measurable way. The argument relies on the nowadays classical reduction to a quadratic estimate and a Carleson-type inequality. The precise organization of the estimates is different from earlier works. In particular, we succeed in separating space and time variables almost completely despite the non-autonomous character of the operator. Hence, we can allow for degenerate ellipticity dictated by a spatial $A_2$-weight, which has not been treated before in this context.