Regarding our research about harmonic measure, one of the results we have obtained shows that for a very general class of domains with fractal boundaries contained in a hyperplane, the dimension of harmonic measure is strictly smaller than the dimension of the boundary (i.e. there is a dimension drop). This result gives a partial positive answer to an old open problem.
A particularly outstanding connected question that we have solved (in collaboration with other mathematicians) is the L^p regularity problem for the Laplacian and other ellioptic PDE's in rough domains, such as chord-arc domains. This problem was posed around 1990 by Carlos Kenig and it was up to now one of the main open problems in the area of elliptic PDE’s. Roughly speaking, it consists in proving that the gradient of a harmonic function in the interior of a domain is controlled, in the L^p space, by the tangential gradient of the function at the boundary of the domain. We have also obtained very relevant advances in connection with the L^p solvability of the Neumann problem in rough domains (in this case, one wants to control the gradient of a harmonic function in the interior of a domain by the normal gradient of the function at the boundary). However, the full solution of this question still presents important challenges.
In connection with the relationship between rectifiability, square functions, and some free boundary problems, we have already obtained very relevant results in the GAPT project. First, we have fully characterized the L2 boundedness of Riesz transforms in terms of a square function of geometric nature, the so-called Jones-Wolff potential. This result is essential to understand the behavior of Lipschitz harmonic capacity under bilipschitz maps and to describe removable singularities for Lipschitz harmonic functions in a geometric way. This completes a line of investigation which goes back to the formulation of the Painlevé problem for bounded analytic functions more than a hundred years ago. Further, as another application of these techniques, we have obtained a rectifiability criterion for general Radon measures in terms of Riesz transforms which can be used to study the points with vanishing density of harmonic measure.
Our second main result about rectifiability and square functions is the extension of Carleson's epsilon-conjecture to higher dimensions, The conjecture (now a theorem) characterizes rectifiability and existence of tangents in domains in terms of a square function analogous to the Carleson epsilon-square function in the plane. Further, our results show some surprising connections with the inequalities of Faber-Krahn and Friedland-Hayman involving quantitative estimates for the first Dirichlet eigenvalue of the Laplacian.
