The principal investigator proposes a three-year program to explore several problems in harmonic analysis and potential theory with applications to approximation theory and polynomial inequalities. The questions to be considered are at the core of classical harmonic analysis, for they deal with smoothness properties of the fundamental objects: harmonic measures and Green's functions. These are directly linked to smoothness of solutions of Dirichlet problems, as well as to different problems in approximation theory and polynomial inequalities. The proposed work will widen our understanding of how smoothness of harmonic measures and Green's functions is connected with the geometry of the underlying domains. It will also broadens the use of potential theoretical methods in approximation theory, thereby offering new tools in the latter field. The research uses different tools from classical analysis, but the main method is potential theoretic. The results are relevant to other branches of mathematics, physics and engineering.
