The metric for matrix degenerate Kato square root operators

We prove a Kato square root estimate with anisotropically degenerate matrix coefficients. We do so by doing the harmonic analysis using an auxiliary Riemannian metric adapted to the operator. We also derive L²-solvability estimates for boundary value problems for divergence form elliptic equations with matrix degenerate coefficients. Main tools are chain rules and Piola transformations for fields in matrix weighted L² spaces, under W1,1 homeomorphism.