Global quadratic estimates for degenerate elliptic operators on cylinders

2026/07/16 by Gianmarco Brocchi, Andreas Rosén
Mathematics · #math.AP #math.CA

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Abstract

On d-dimensional cylinders C= ℝ^k× N, with a closed manifold N as base and large scale dimension k∈[1,d), we prove quadratic estimates in weighted L² space for Dirac operators perturbed by bounded, measurable and accretive coefficients. This gives in particular homogeneous Kato square root estimates on C for Riesz transforms associated with second order divergence form elliptic operators, having measurable coefficients with degeneracy governed by a Muckenhoupt A₂ 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.

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