Positive-Homogeneous Operators, Heat Kernel Estimates and the Legendre-Fenchel Transform
Positive-Homogeneous Operators, Heat Kernel Estimates and the Legendre-Fenchel Transform
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正齐次算子、热核估计和勒让德-芬切尔变换
DOI：
10.1007/978-3-319-59671-6
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
Saloff-Coste, Laurent
中科院分区：
文献类型：
--
作者：
Randles, Evan;Saloff-Coste, Laurent
我们考虑了有限维向量空间上的一类齐次偏微分算子，并研究了它们的相关热核。这类算子的热核被看作是方格上复值函数卷积幂的极限对象，就像经典的热核在(局部)中心极限定理中出现一样。这些所谓的正齐次算子在定义与坐标无关的意义上推广了半椭圆算子类。更一般地，我们引入了一类变系数算子，每个变系数算子都与一个正齐次算子一致可比，并研究了相应的热方程的柯西问题。在这类算子具有Hölder连续系数的假设下，我们用Friedman和Eidelman的Levi方法构造了它的热方程的基本解。虽然我们在这个方向上的结果被Eidelman for-抛物系统早已知道的结果所暗示，但我们的重点是强调勒让德-芬切尔变换在热核估计中所起的作用。具体地说，我们证明了基本解满足一个非对角线估计，即热核估计，它是用算子主符号的勒让德-芬切尔变换写成的--这个估计在许多情况下被认为是尖锐的。
We consider a class of homogeneous partial differential operators on a finite-dimensional vector space and study their associated heat kernels. The heat kernels for this general class of operators are seen to arise naturally as the limiting objects of the convolution powers of complex-valued functions on the square lattice in the way that the classical heat kernel arises in the (local) central limit theorem. These so-called positive-homogeneous operators generalize the class of semi-elliptic operators in the sense that the definition is coordinate-free. More generally, we introduce a class of variable-coefficient operators, each of which is uniformly comparable to a positive-homogeneous operator, and we study the corresponding Cauchy problem for the heat equation. Under the assumption that such an operator has Hölder continuous coefficients, we construct a fundamental solution to its heat equation by the method of Levi, adapted to parabolic systems by Friedman and Eidelman. Though our results in this direction are implied by the long-known results of Eidelman for-parabolic systems, our focus is to highlight the role played by the Legendre-Fenchel transform in heat kernel estimates. Specifically, we show that the fundamental solution satisfies an off-diagonal estimate, i.e., a heat kernel estimate, written in terms of the Legendre-Fenchel transform of the operator’s principal symbol—an estimate which is seen to be sharp in many cases.