Geometric Measure Theory and fine structure of harmonic measure

Singular integrals are ubiquitous objects and play an important part in Geometric Measure Theory. The simplest ones are called CalderonCZygmund operators. Their theory was completed in the 50’s by Zygmund and Calderon. Or it seemed like that. The last 20 years saw the need to consider CZ operators in very bad environment, so kernels are still very good, but the ambient set/measure has no regularity whatsoever.
Initially such situations appeared from the wish to solve some outstanding problems in complex analysis: such as problems of Painlevé, Ahlfors’, Denjoy’s and Vitushkin’s.
The analysis of CZ operators on very bad sets is also very fruitful in the part of Geometric Measure Theory that deals with rectifiability. It can be viewed as the study of very low regularity free boundary problems. As such this analysis helps to understand the geometry of harmonic measure. Lennart Carleson, Nikolai Makarov, Jean Bourgain, Peter Jones and Tom Wolff obtained important results on metric properties of harmonic measure in the 80’s and 90’s. But most of the results concerned the structure of harmonic measure of planar domains. As an example of the use of non-homogeneous harmonic analysis, we will show how it allows us to understand very fine property of harmonic measure of any domain in any dimension and to find the answer to several problems of C. Bishop dated from 1991.