The Kato problem for weighted elliptic and parabolic operators of higher order

We solve the Kato square root problem for parabolic operators of arbitrary order $2m$ whose coefficients are allowed to depend on both space and time in a merely measurable way and possess boundedness and ellipticity controlled by a Muckenhoupt $A_2-$weight. Notably, the proof applies to the weighted Kato problem within an elliptic framework.