Smoothness Properties of Harmonic Measures and the Polynomial Inverse Image Method

摘要：本课题拟开展为期三年的项目，研究谐波分析和位势理论中的若干问题，并将其应用于近似理论和多项式不等式。要考虑的问题是经典谐波分析的核心，因为它们处理基本对象的平滑特性：谐波测度和格林函数。这些直接关系到狄利克雷问题解的光滑性，以及近似理论和多项式不等式中的不同问题。所提出的工作将扩大我们对谐波测度和格林函数的平滑度如何与底层域的几何形状相联系的理解。它还将拓宽近似理论中潜在理论方法的应用，从而为后一个领域提供新的工具。该研究采用了不同于经典分析的工具，但主要方法是势理论。这些结果与数学、物理和工程的其他分支有关。

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.