BCAM Working Seminar APDE: Approximation and Coincidence: Corona Decompositions vs. Big Pieces

Abstract
Corona-type decompositions are ubiquitous in harmonic analysis. Given a dyadic grid these decompositions say in a quantitative way that some "bad behavior" does not occur frequently and even more that some "good behavior" persists in a strong, quantitative manner. This talk will focus on the role of corona decompositions in quantitative (uniform) rectifiability and, more generally, quantitative geometry. In this context, the ideas began with the remarkable work of David and Semmes, who used corona-type decompositions to characterize the boundedness of "nice" CalderÓn Zygmund operators on d-dimensional sets in R^n. Here their corona decomposition is with respect to approximation by Lipschitz graphs.

David and Semmes introduced a related notion called "big pieces". Here we fix a collection of sets S and say a set "E is big pieces of S" if at every scale and location (on E) there is a set from S that coincides with E in an "ample" way. In many ways this coincidence is much easier to work with. For instance, if "nice2 CalderÓn-Zygmund operators are (uniformly) bounded on S then the same is true for E. The main result presented is that if E admits a corona-type approximation by a family of sets S, then E is "big pieces squared of S". While this was already shown by Azzam and Schul in the context of uniform rectifiability, we demonstrate that this is a very general phenomenon (in metric spaces, with possibly non-integer dimension, an arbitrary family of approximating sets...). In particular, this theorem has direct applications to parabolic uniform rectifiability. Time permitting I will discuss some future work on parabolic uniform rectifiability and some (perhaps difficult) open problems.

Link to the session: https://zoom.us/j/93605194162?pwd=T0lVbGM3L0dINDY5Y0ZRNzVPbDh0QT09