Smoothness Properties of Harmonic Measures and the Polynomial Inverse Image Method

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DMS 0406450V TotikU of South FloridaSmoothness properties of harmonic measures and the polynomial inverse image methodAbstract:The principal investigator proposes a three-year program toexplore several problems in harmonic analysis and potentialtheory with applications to approximation theory andpolynomial inequalities. The questions to be consideredare at the core of classical harmonic analysis,for they deal with smoothness properties of thefundamental 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 ofhow smoothness of harmonic measures and Green'sfunctions 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 relevantto other branches of mathematics, physics and engineering.