  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.
