Smoothness Properties of Harmonic Measures and the Polynomial Inverse Image Method

DMS 0406450V TotikU南佛罗里达调和测度的光滑性和多项式逆镜像方法摘要：主要研究者提出了一个为期三年的计划，以探索调和分析和势理论中的几个问题，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调和分析的核心，因为它们涉及基本对象的光滑性：调和测度和绿色函数。这些直接与Dirichlet问题解的光滑性有关，本文的工作将拓宽我们对调和测度和绿色函数的光滑性如何与函数的光滑性相联系的理解。基础域的几何形状。它还将拓宽逼近理论中潜在的理论方法的使用，从而在后一领域提供新的工具。 本文采用了与经典分析不同的分析工具，但主要采用了势理论的方法。所得结果对数学、物理学和工程学的其它分支也有一定的参考价值。

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.