Function Theory on Varieties

品种函数论

• 批准号: 0555789
• 负责人: B. Alan Taylor
• 金额: $ 16.77万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0555789&HistoricalAwards=false
• 关键词: Function; Theory; Varieties

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.

摘要：该项目的主要目标是找到止痛变化的条件，以保证分析函数在实际点处的变化的良好估计值。特别地，它涉及类似于Phragman-Lindelof定理的估计，该定理给出了复欧氏空间实点上有界且满足渐近线性增长率的多重次调和函数的一致上界。提出了一个新的几何条件，即簇是近双曲的，从而刻画了这些性质。新的条件给出了一种精确的意义，即这种变化有许多实点，并且在强度上介于拥有全维实点集和允许在实点上有实纤维的投影图之间。这个项目的目的是确定欧氏空间中代数和解析簇的几何性质，这些几何性质决定了簇上的解析函数何时具有类似于欧几里德空间上的整函数的性质。它是由常系数偏微分算子组的解算子的性质和整体光滑函数上的卷积算子的性质所推动的。目标是给出判定特定算子是否允许解和解算子在无穷可微函数空间和分布空间上的算法。这项研究使用了多势理论的方法并贡献了新的结果，多势理论与多个复变量的解析函数的关系是经典位势理论与单复变量解析函数理论的关系的类比。