Pluri-Potential Theory

摘要：本课题第一部分主要研究利用分析盘上泛函包络构造多次谐波函数的方法。最近人们知道，这种方法的理论可以简化为研究某些无限维空间域上函数的盘包络。事实上，完整的理论需要在无限维环境中解决经典的Lelong问题。在构造多次谐波函数时使用灵活方法的可能性将使波列茨基在这些函数的边值理论方面取得进展，即使“边值”的正确定义仍然是一个悬而未决的问题。特别地，Poletsky打算提出这些值的一个自然定义，并建立每一个多次调和函数都可以分解为与给定函数具有相同边值的Dirichlet问题的一个极大解和一个边值为零的势的和。然后，他将分别研究这两部分的分解。艾萨克·牛顿（Isaac Newton）发明的微分学中的主要工具是函数的导数，这个概念提供了比牛顿时代之前更深入、更精确的对函数行为的理解。如果我们看导数的导数，二阶导数的导数，通过任意阶的导数，我们会更好地了解一个函数。然而，并不是所有的函数都适合计算这些高阶导数。这些函数被称为“解析函数”，它们正是微积分应用于其他科学学科时最常遇到的函数。牛顿之后微积分的发展带来了他的原始框架的两个重要扩展：对多变量解析函数的研究和对变量和值都是复数的解析函数的研究。符合后一种描述的函数构成了一个庞大而迷人的函数类，它们非常“敏感”：函数在一个地方的微小变化可能以某种方式波及整个函数，从而在其他地方改变它。这种行为是研究这些功能的一个巨大障碍。为了克服这一困难，人们试图将分析函数分解为“软”和“硬”部分，并分别研究这些部分。这个项目的主要研究对象是所谓的“多次谐波函数”，它是几个复变量解析函数的“软”部分。我们的目标是更好地理解这些函数，希望从长远来看，所获得的知识将转化为对日常经验的许多方面的更好理解，在这些方面，分析函数发挥着重要的作用，但通常是看不见的。

Proposal: DMS-9804755 Principal Investigator: Evgeny A. Poletsky Abstract: The first part of this project is devoted to developing methods for construction of plurisubharmonic functions using envelopes of functionals on analytic disks. It became known recently that the theory of such methods can be reduced to the study of disk envelopes of functions on domains in certain infinite-dimensional spaces. Indeed, the complete theory requires solving the classical Lelong problems in an infinite-dimensional setting. The possibility of using flexible methods in the construction of plurisubharmonic functions will allow Poletsky to make advances in the theory of boundary values of such functions, where even the correct definition of "boundary values" remains an open question. In particular, Poletsky intends to put forward a natural definition of these values and to establish that every plurisubharmonic function can be decomposed into the sum of a maximal solution to the Dirichlet problem with the same boundary values as the given function and a potential with zero boundary values. He will then study separately the two pieces of this decomposition. The main tool in the differential calculus invented by Isaac Newton is the derivative of a function, a concept that furnishes a deeper and more precise understanding of the function's behavior than was possible before Newton's time. One will know a function even better if one looks at the derivative of the derivative, the derivative of that second derivative, and so on through derivatives of arbitrary orders. Not all functions, however, are amenable to the computation of these higher order derivatives. Those which do are called "analytic functions," and they are precisely the functions that are most frequently encountered in the applications of calculus to other scientific disciplines. The development of calculus after Newton ushered in two significant expansions of his original framework: the investigation of analytic functions of more than one variable and the study of analytic functions in which both the variables and the values are complex numbers. Functions fitting the latter description constitute a large and fascinating class of functions that are extremely "sensitive": a small change in the function in one place can somehow ripple through the function to change it everywhere. Such behavior is an enormous obstacle in the study of these functions. To overcome this difficulty, one seeks to decompose analytic functions into "soft" and "rigid" parts and to investigate these parts separately. The main objects of study in this project are the so-called "plurisubharmonic functions," which are the "soft" parts of analytic functions of several complex variables. The goal is to understand these functions better, with the hope that in the long run the knowledge gained will translate into a better understanding of the many aspects of everyday experience in which analytic functions play important, but usually unseen, roles.