The ability to form the bounded holomorphic functional calculus, is often the crucial step in the resolution of many problems in PDE. Quadratic estimates, which in PDE arose as square function estimates, provide an equivalent criteria for the existence of such a functional calculus. This is of particular significance in geometric settings where, unlike Euclidean space, Fourier tools are not always available.
However, quadratic estimates can often be established through the use ofreal-variable harmonic analysis techniques. The purpose of this talk is to showcase the significance of these techniques in resolving problems in geometry, starting with the Kato Square Root problem.
