Functional calculus and harmonic analysis in geometry

Abstract

In this short survey article, we showcase a number of non-trivial geometric problems that have recently been resolved by marrying methods from functional calculus and real-variable harmonic analysis. We give a brief description of these methods as well as their interplay. This is a succinct survey that hopes to inspire geometers and analysts alike to study these methods so that they can be further developed to be potentially applied to a broader range of questions.

arXiv commentFixed some typos, added some extra text, added references

References in corpus (8)

• Quadratic estimates and functional calculi of perturbed Dirac operators
• Boundary Value Problems for Elliptic Differential Operators of First Order
• The Kato Square Root Problem on Submanifolds
• Boundary value problems for general first-order elliptic differential operators
• The Kato square root problem on vector bundles with generalised bounded geometry
• Riesz continuity of the Atiyah-Singer Dirac operator under perturbations of the metric
• Continuity of solutions to space-varying pointwise linear elliptic equations
• Rough metrics on manifolds and quadratic estimates