Pluricomplex energy
Pluricomplex energy
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DOI：
10.1007/bf02392899
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发表时间：
1998
期刊：
影响因子：
3.7
通讯作者：
U. Cegrell
中科院分区：
文献类型：
--
作者：
U. Cegrell
本文研究了复Monge-Schwarf ~ re算子（ddC）n。设~是C ′ ~的开有界子集。如果ujCC 2（~），l~ j~ n，则Monge-Schwarz-re算子对（u1，.，un）并等于ddeulA... AddCun，其中d= O+ O且d~= i（OO）。如果每个uj也是多重次调和的，则dd~ u1 A…地址是一个积极的措施。该算子在多势理论中非常重要，它在多势理论中起着类似于经典势理论中的拉普拉斯算子的作用。拉普拉斯算子是一个线性的二阶微分算子，因此定义在所有分布上的ft，而复杂的蒙赫安培算子是非线性的，不能定义在所有plurisubharmonic函数上的gt，参见。[14]，[20]和[8]。此外，该算子在弱 *-拓扑中是不连续的，参见。[9]的文件。另一方面，贝德福德和Taylor [2]证明了（dde）n在所有局部有界的多重次调和函数上是良好定义的.将定义域扩展到PSHNL~ o以外并描述相应范围的问题已由几位作者研究过：[2]，[3]，[8]，[13]，[15]，[16]和[17]。有关多能理论的调查，请参阅[1]。特别地，该论文的W包含了对（dd~）n定义域的讨论.本文定义了一类多重次调和函数类gp和9 Cp，并研究了其上的复Monge-Schwarf ~ re算子（ddC）.
This paper is a study of the complex Monge-Amp~ re operator (ddC) n. Let~ be an open and bounded subset of C'~. If ujCC2 (~), l~ j~ n, then the Monge-Amp~ re operator operates on (ul,..., un) and equals ddeulA... AddCun, where d= O+ O and d~= i (OO).If also each uj is plurisubharmonic, then dd~ ulA... Addr is a positive measure. This operator is of great importance in pluripotential theory, where it plays a role similar to that of the Laplace operator in classical potential theory. The Laplace operator is a linear, second-order differential operator and thus is defined on all distributions on ft, while the complex Monge Ampere operator is non-linear and cannot be defined on all plurisubharmonic functions on gt, cf.[14],[20] and [8]. Moreover, the operator is discontinuous in the weak*-topology, cf.[9]. On the other hand, it was shown by Bedford and Taylor [2] that (dde) n is welldefined on all locally bounded plurisubharmonic functions. The problem of extending the domain of definition beyond PSHNL~ oe and describing the corresponding range has been studied by several authors:[2],[3],[8],[13],[15],[16] and [17]. See [1] for a survey on pluripotential theory. In particular, w of that paper contains a discussion of the domain of definition for (dd~) n. In this paper, we define certain classes gp and 9Cp of plurisubharmonic functions, and study the complex Monge-Amp~ re operator (ddC) on them.