Mathematical Sciences: Research on Representing Measures and Boundary Behavior in Potential Theory

作者：loeb@math.uni-frankfurt.de at注日期：3/5/96 4:43 AM优先级：正常TO: jjenkins at nsf11主题：-------------------------------消息内容-------------------------------这封电子邮件是由Peter a . Loeb从德国法兰克福发来的，他是3月份的访问者。拟议的研究将继续Peter Loeb教授在势理论中表示测度和理想边界的工作。特别地，Loeb计划继续研究一个类马丁边界，它在谐波测度方面几乎处处是规则的。这项研究中的一个重要工具是与J. Bliedtner一起开发的技术，该技术可以减少氡-尼科代姆衍生物的产生，作为特殊情况的限制。这一简化对鞅收敛定理有了新的认识，极大地简化了测度微分定理以及概率论和势论中精细极限定理的处理。一个主要的结果是存在简单的边界逼近邻域系统，在确定适当的归一化之后，就产生Radon-Nikodym导数作为边界的极限而言，是“最佳”可能的。这种“最佳”系统的存在，即使是在单位圆盘上的调和函数也不为人所知。事实上，先前文献中的结果表明，这样的系统不可能存在。在相关的、正在进行的研究中，Loeb通过加强结果和简化证明，改进了著名的Besicovitch和Morse覆盖定理，包括证明对于任何范数，Besicovitch结果的最佳常数（就所有已知的证明而言）是一个填充常数。在他所有的研究中，勒布教授将继续使用非标准模型。这些是具有无限大和无限小数字的数学结构。即使这些结构的使用从结果的最终证明中删除，它们在发现过程中也非常有帮助。Loeb最近使用非标准模型来优化Besicovitch定理的覆盖，并且与Bliedtner一起使用这些方法将径向极限的概念扩展到一般的势理论设置。迄今为止已经获得的这项研究的结果，以及将在拟议的研究中建立的结果，都是数学分析和概率论的核心。这些是数学的基本领域，被科学家和工程师用来形成他们所研究的现象的概念模型。思维过程本身只能用数学来表达，而所使用的数学反过来又产生了对所研究现象的新见解。因此，所提出的研究所实现的基本数学工具的简化和扩展将在使用这些工具的所有科学和工程领域产生共鸣。对未来发展特别重要的将是成功地应用无限大和无限小数的数学结构。这些结构在科学和工程的所有领域都有很大的用处，因为它们的使用简单而强大。与这些结构相关的一个重要构造是度量空间，现在在文献中称为“Loeb空间”，它允许使用适用于有限集合的方法来处理无限概率现象。包括Loeb在内的研究人员用这些方法在偏微分方程、概率论、经济学研究和气体运动规律的物理学研究等领域取得了新的成果。

Author: loeb@math.uni-frankfurt.de at NOTE Date: 3/5/96 4:43 AM Priority: Normal TO: jjenkins at nsf11 Subject: ------------------------------- Message Contents ------------------------------- This e-mail letter is sent by Peter A. Loeb from Frankfurt, Germany where he is a visitor for the month of March. Abstract of Peter A. Loeb's project The proposed research will continue Professor Peter Loeb's work on representing measures and ideal boundaries in potential theory. In particular, Loeb plans to continue research on a Martin-like boundary which is regular almost everywhere with respect to harmonic measure. An important tool in this research is the technique developed with J. Bliedtner that reduces the production of Radon-Nikodym derivatives as limits to a special case. This reduction casts new light on the martingale convergence theorem and significantly simplifies measure differentiation theorems as well as the treatment of fine limit theorems in probability and potential theory. A major consequence is the existence of simple boundary approach neighborhood systems which, after fixing a suitable normalization, are the "best" possible in terms of producing Radon-Nikodym derivatives as limits at the boundary. The existence of such "best" systems had not been known even for harmonic functions on the unit disk. Indeed, previous results in the literature suggested that no such system could exist. In related, ongoing research, Loeb has improved the celebrated Besicovitch and Morse covering theorems from geometric measure theory by strengthening the results and simplifying the proofs, including a demonstration that for any norm, the best constant (in terms of all known proofs) for the Besicovitch result is a packing constant. In all of his research, Professor Loeb will continue to employ nonstandard models. These are mathematical structures with infinitely large and infinitely small numbers. Even when the use of such structures is removed from the final proofs of results, they are extremely helpful in the discovery process. Loeb has recently used nonstandard models to optimize the coverings of Besicovitch's theorem, and with Bliedtner he is now using these methods to extend the notion of radial limits to general potential theoretic settings. The results of this research that have been obtained to date, and the results that will be established in the proposed research all lie at the heart of mathematical analysis and probability theory. These are fundamental areas of mathematics used by scientists and engineers to form the conceptual models of the phenomena with which they work. The thought process itself can only be couched in terms of mathematics, and the mathematics being used in turn yields new insight into the phenomena being studied. Therefore, the simplification and extension of the basic mathematical tools achieved by the proposed research will resonate in all of the areas of science and engineering that use these tools. Of particular importance for future developments will be the successful employment of mathematical structures with infinitely large and infinitely small numbers. These structures have a great deal to offer in all areas of science and engineering because of the simplicity and power inherent in their use. An important construction associated with these structures are measure spaces, now called "Loeb spaces" in the literature, which allow the treatment of infinite probability phenomena using methods appropriate for finite sets. Researchers including Loeb have obtained new results with these methods in areas such as the study of partial differential equations, probability theory, research in economics, and the study in physics of laws governing the motion of gases.