Global quadratic estimates for degenerate elliptic operators on cylinders

Abstract

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.

arXiv comment33 pages, 2 figures. Proposition 2.9 replaces former Proposition 2.6 (Poincaré inequality for vertical gradient fields) to fix an error. Proposition 3.11 has also been adapted