Function Theory on Varieties

摘要：该项目的主要目标是找到的条件，保证良好的估计，在各种解析函数的价值观在真实的点。 特别是，它关注的估计类似于那些的Phragmen-Lindelof定理，给出了统一的上界为多次调和函数，有界以上的真实的点的复欧几里德空间，并满足一个渐近线性增长率。 提出了一个新的几何条件，即簇是近双曲的，来刻画这类性质。 新的条件给出了一个精确的意义，其中的品种有许多realpoints，是中间的强度之间有一个全维集的真实的点，并承认投影映射与真实的纤维在真实的点。 该项目的目的是确定几何性质的代数和解析簇在欧几里得空间，决定当解析函数的品种有类似的性质，整个功能的欧几里得空间。 它的动机与常系数偏微分算子和卷积算子系统的解算子的性质的整体光滑函数的连接。 目标是给出算法，以决定是否特定的运营商承认的解决方案和解决方案的运营商空间的无穷可微函数和分布。 本文的研究借鉴了多元位势理论的方法，并为多元位势理论提供了新的结果，多元位势理论与多复变解析函数的关系类似于经典位势理论与单复变解析函数理论的关系。

Abstract: The main goal of the project is to find the conditions on analgebraic variety that guarantee good estimates for analytic functionson the variety in terms of their values at real points. Particularly,it is concerned with estimates analogous to those of thePhragmen-Lindelof theorem which gives uniform upper bounds forpluri-subharmonic functions that are bounded above at the real pointsof the complex Euclidean space and satisfy an asymptotic linear growthrate. A new geometric condition, that the variety is nearlyhyperbolic, is proposed to characterize such properties. The newcondition gives a precise sense in which the variety has many realpoints, and is intermediate in strength between having a fulldimensional set of real points and admitting a projection map withreal fibers over real points. The project aims to determine the geometric properties ofalgebraic and analytic varieties in Euclidean space that determinewhen the analytic functions on the variety have properties similar tothose of entire functions on Euclidean space. It is motivated byconnections with the properties of solution operators for systems ofconstant coefficient partial differential operators and convolutionoperators on global smooth functions. The goal is to give algorithmsfor deciding whether or not specific operators admit solutions andsolution operators on spaces of infinitely differentiable functionsand distributions. The study uses methods from and contributes newresults to pluri-potential theory, the area whose relationship toanalytic functions of several complex variables is the analogue of therelationship of classical potential theory to the theory of analyticfunctions of one complex variable.