Non-homogeneous harmonic analysis, geometric measure theory, and fine structures of harmonic measure

One of the goals of harmonic analysis is to study singular integrals. These are ubiquitous in PDEs and mathematical physics and, as it transpired recently, play an important role in geometric measure theory. The simplest ones are called Calderón--Zygmund operators. Their theory was completed in the 1950s. Or so it seemed. The last 20 years saw the need to reconsider CZ operators in very bad environments.

Initially this was needed to solve some outstanding problems in complex analysis, such as Painlevé, Ahlfors, Denjoy, Vitushkin problems. But eventually the non-homogeneous harmonic analysis became fruitful in the part of GMT that deals with rectifiability, as it helps to understand the geometry of harmonic measure. The research on metric properties of harmonic measures was pioneered by Piranian, Carleson, Makarov, Bourgain, Jones, Wolff; most of their results concerned the structure of harmonic measure in planar domains. Now the use of non-homogeneous harmonic analysis allows us to show how to understand very fine properties of harmonic measure in any domain in any dimension.