Plurisubharmonic Functions on Algebraic Varieties

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

ABSTRACT: The classical Phragmen-Lindelof theorem extends the maximum principle to unbounded analytic functions by showing that an analytic function that satisfies an asymptotic exponential bound in the upper half plane and a uniform bound on the real axis in fact satisfies a uniform exponential bound in the upper half plane. Research of the past three decades has shown that the validity of estimates of a similar character for analytic functions on algebraic varieties in n-dimensional complex Euclidean space are in fact equivalent to certain properties of linear constant coefficient partial differential operators. Some such properties of the operators are surjectivity on the space of real analytic functions or Gevrey classes, the existence of lacuna in fundamental solutions, the existence of linear solution operators, and continuation properties of solutions of the homogeneous equation(s). While there are different sets of these estimates associated to the different properties of the operator, they all are similar in spirit. The aim of this work is to develop methods that give a geometric characterization of the algebraic varieties for which a given Phragmen-Lindelof condition is satisfied. If successful, the work should also give insight into questions about the partial differential equations such as the existence of fundamental solutions with cone-shaped lacuna. This work is focused on developing tools in complex analysis that can be used to answer basic questions about linear partial differential equations. In the 1950's, Laurent Schwartz formulated such fundamental problems for general linear partial differential equations. Are they always solvable? If so, can the solution be chosen as smooth as the data in the problem? Do fundamental solutions exist? Can the equations be solved with a "formula", so that the answer depends linearly on the data of the problem? Most of these questions were answered in the 1950's by Ehrenpreis and Malgrange. However, the question of whether the solution could be chosen to be real analytic when the data is real analytic was open until the late 1960's when the first counter examples were given. In 1973, Hormander gave a characterization of the equations with this property in terms of the validity of certain inequalities for analytic functions on the zero set of the polynomial giving the differential equation. In 1990, Taylor, Meise, and Vogt answered the question about the existence of "formulas" for the solution and showed they were also characterized in terms of some similar inequalities. The aim of this project is to develop tools that allow one to decide whether or not the required estimates are valid for a given partial differential equation. We believe that it is possible to develop an algorithm that will make the verification, and further, will explain the geometry of the zero set of the associated polynomial that is necessary for the inequalities to be satisfied.