Plurisubharmonic Functions on Algebraic Varieties

文摘：经典的Phragman-Lindelof定理将极大值原理推广到无界解析函数，证明了一个在上半平面上满足渐近指数界而在实轴上一致上界的解析函数实际上在上半平面上满足一致指数界。过去三十年的研究表明，n维复欧氏空间中解析函数的相似性质估计的有效性实际上等价于线性常系数偏微分算子的某些性质。这些算子的一些性质是实解析函数空间或Gevrey类上的满射性、基本解的缺陷性、线性解算子的存在性以及齐次方程解的连续性(S)。虽然这些估计与算子的不同性质有不同的集合，但它们在精神上都是相似的。这项工作的目的是发展一种方法，它给出了满足给定Phradman-Lindelof条件的代数族的几何刻画。如果成功，这项工作还将深入了解有关偏微分方程的问题，例如具有锥形凹陷的基本解的存在性。这项工作集中在开发复分析中的工具，这些工具可以用来回答关于线性偏微分方程组的基本问题。在20世纪50年代的S，Laurent Schwartz提出了一般线性偏微分方程解的基本问题。它们总是可以解决的吗？如果是这样的话，解决方案能否像解决问题的数据一样顺畅呢？是否存在根本性的解决方案？方程式能不能用“公式”来解，这样答案就能线性地依赖于问题的数据呢？这些问题中的大多数在1950年的《S》中被埃伦普里斯和马尔格兰热回答了。然而，当数据是真正的解析时，解是否可以被选择为真正的解析的问题一直是悬而未决的，直到20世纪60年代末，S给出了第一个反例。1973年，Hormander利用解析函数在给出微分方程的多项式的零点集上的某些不等式的有效性，给出了具有这一性质的方程的特征。1990年，Taylor、Meise和Vogt回答了关于解的“公式”的存在的问题，并证明了它们也用一些类似的不等式来刻画。这个项目的目的是开发工具，允许人们决定所需的估计对给定的偏微分方程式是否有效。我们相信，有可能开发出一种算法，它将进行验证，并进一步解释使不等式满意所必需的相关多项式的零集的几何。

ABSTRACT: The classical Phragmen-Lindelof theorem extends the maximum principleto unbounded analytic functions by showing that an analytic function thatsatisfies an asymptotic exponential bound in the upper half plane and auniform bound on the real axis in fact satisfies a uniform exponentialbound in the upper half plane. Research of the past three decades hasshown that the validity of estimates of a similar character for analyticfunctions on algebraic varieties in n-dimensional complex Euclidean spaceare in fact equivalent to certain properties of linear constantcoefficient partial differential operators. Some such properties of theoperators are surjectivity on the space of real analytic functions orGevrey classes, the existence of lacuna in fundamental solutions, theexistence of linear solution operators, and continuation properties ofsolutions of the homogeneous equation(s). While there are different setsof these estimates associated to the different properties of the operator,they all are similar in spirit. The aim of this work is to developmethods that give a geometric characterization of the algebraic varietiesfor which a given Phragmen-Lindelof condition is satisfied. Ifsuccessful, the work should also give insight into questions about thepartial differential equations such as the existence of fundamentalsolutions with cone-shaped lacuna. This work is focused on developing tools in complex analysis thatcan be used to answer basic questions about linear partial differentialequations. In the 1950's, Laurent Schwartz formulated such fundamentalproblems for general linear partial differential equations. Are theyalways solvable? If so, can the solution be chosen as smooth as the datain the problem? Do fundamental solutions exist? Can the equations besolved with a "formula", so that the answer depends linearly on the dataof the problem? Most of these questions were answered in the 1950's byEhrenpreis and Malgrange. However, the question of whether the solutioncould be chosen to be real analytic when the data is real analytic wasopen until the late 1960's when the first counter examples were given. In1973, Hormander gave a characterization of the equations with thisproperty in terms of the validity of certain inequalities for analyticfunctions on the zero set of the polynomial giving the differentialequation. In 1990, Taylor, Meise, and Vogt answered the question aboutthe existence of "formulas" for the solution and showed they were alsocharacterized in terms of some similar inequalities. The aim of thisproject is to develop tools that allow one to decide whether or not therequired estimates are valid for a given partial differential equation. Webelieve that it is possible to develop an algorithm that will make theverification, and further, will explain the geometry of the zero set ofthe associated polynomial that is necessary for the inequalities to besatisfied.