Skip to main content

Geometric analysis in the Euclidean space

Final Report Summary - ANGEOM (Geometric analysis in the Euclidean space)

The summary should be a stand-alone description of the project and its outcomes. This text should be as concise as possible and suitable for dissemination to non specialist audiences. Please notice that this summary will be published.

The major achievement of this research project has been the solution in a joint work by Nazarov, Tolsa and Volberg of the so called Davis-Semmes problem in codimension one. This problem was posed about 25 yeas ago and, roughly speaking, asserts that the L^2 boundedness of codimension 1 Riesz transforms on some subset of the Euclidean space imply the rectifiability of this subset.

Other relevant achievement concerns the study of the connection between harmonic measure and rectifiability. In a joint work by Azzam, Hofmann, Martell, Mayboroda, Mourgoglou, Tolsa and Volberg, it has been shown that the absolute continuity of harmonic measure with respect surface measure in R^n implies the rectifiability of harmonic measure. In the particular case of the plane, this solves an old conjecture by Chris Bishop from 1992.

Another fundamental achievement of this project is the solution of the two phase problem for harmonic measure, first in a work by Azzam, Mourgoglou and Tolsa under the assumption of the capacity density condition and later in full generality together with Volberg. This solves another old conjecture posed by Chris Bishop.

Other relevant results deal with the characterization of rectifiability in terms of the boundedness of certain square functions which involve either the so called beta coefficients of Jones, David and Semmes, or some differences of densities.