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 L^{2} -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 L^{2} spaces, under W^{1,1} homeomorphism.