Global quadratic estimates for degenerate elliptic operators on cylinders

On $d$-dimensional cylinders $\mathcal{C}= \mathbb{R}^k\times N$, with a closed manifold $N$ as base and large scale dimension $k\in[1,d)$, we prove quadratic estimates in weighted $L^2$ space for Dirac operators perturbed by bounded, measurable and accretive coefficients. This gives in particular homogeneous Kato square root estimates on $\mathcal{C}$ for Riesz transforms associated with second order divergence form elliptic operators, having measurable coefficients with degeneracy governed by a Muckenhoupt $A_2$ weight. By localisation and scaling, it also yields local quadratic estimates for perturbed Dirac operators on general manifolds with locally thin cylindrical geometry, and possibly with zero injectivity radius.