Variation for riesz transforms and analytic and lipschitz harmonic capacities

Résumé

The topics covered in this dissertation belong to the area of the so-called geometric analysis which, among other things, relates harmonic analysis to geometric measure theory. Most of these topics are related to interesting open problems which have been studied recently by many international mathematicians. More precisely, they are concerned with the Cauchy and Riesz transforms, two fundamental operators in harmonic analysis (in particular in Calderón-Zygmund theory), PDE's, and geometric measure theory. The topics under study are the following ones: 1. Failure of rational approximation on some Cantor type sets. 2. A dual characterization of the C1 harmonic capacity. 3. Variation and oscillation for singular integrals with odd kernel on Lipschitz graphs. 4. Uniform rectifiability and variation and oscillation for the Riesz transforms.