具有超平面奇点的微分算子的支撑性质和Holmgren唯一性定理
Support properties and Holmgren's uniqueness theorem for differential operators with hyperplane singularities
摘要：
设W是线性作用于$R^n$上的有限Coxeter群.本文研究了具有实解析系数的W-不变偏微分算子D在$\R^n$上的支撑性质。我们假设D的主象征有一个特殊的形式，与对应于W的根系有关。特别地，假定主象征的零点位于由W中的反射固定的超平面上。我们证明了conv（suppDf）= conv（suppf）对所有紧支撑的光滑函数f成立，使得conv（suppf）是W-不变的.证明的主要工具是Holmgren唯一性定理和一些初等凸几何。与根系相关的特殊函数理论的几个例子和应用。
英文摘要：
Let W be a finite Coxeter group acting linearly on $R^n$. In this article we study support properties of W-invariant partial differential operator D on $\R^n$ with real analytic coefficients. Our assumption is that the principal symbol of D has a special form, related to the root system corresponding to W. In particular the zeros of the principal symbol are supposed to be located on hyperplanes fixed by reflections in W. We show that conv(supp Df) = conv(supp f) holds for all compactly supported smooth functions f so that conv(supp f) is W-invariant. The main tools in the proof are Holmgren's uniqueness theorem and some elementary convex geometry. Several examples and applications linked to the theory of special functions associated with root systems are presented.
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