Quadratic estimates for degenerate elliptic systems on manifolds with lower Ricci curvature bounds and boundary value problems

Weighted quadratic estimates are proved for certain bisectorial first-order differential operators with bounded measurable coefficients which are (not necessarily pointwise) accretive, on complete manifolds with positive injectivity radius. As compared to earlier results, Ricci curvature is only assumed to be bounded from below, and the weight is only assumed to be locally in $A_2$.
The Kato square root estimate is proved under this weaker assumption. On compact Lipschitz manifolds we prove solvability estimates for solutions to degenerate elliptic systems with not necessarily self-adjoint coefficients, and with Dirichlet, Neumann and Atiyah--Patodi--Singer boundary conditions.