Fine topology, Šilov boundary, and (ddc)n
Fine topology, Šilov boundary, and (ddc)n
复制标题
精细拓扑、Šilov 边界和 (ddc)n
DOI：
10.1016/0022-1236(87)90087-5
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发表时间：
1987
影响因子：
1.7
通讯作者：
Brian D. Taylor
中科院分区：
文献类型：
--
作者：
Eric Bedford;Brian D. Taylor
在经典位势理论中，精细拓扑提供了一种有用的和自然的方法来给出许多结果的精确陈述。这同样适用于与多重次调和（PSH）函数相关的势理论。在第二节中，我们给出了一个简短的讨论的（多）精细拓扑的psh函数。几乎所有的结果都是相同的经典精细拓扑，即使有相同的证明，所以我们省略了所有的证明在这一节。然而，有一个关键的区别：在psh势理论中，“薄集”和“极集”的概念是不等价的。精细拓扑的使用使我们能够给出[BT]中证明的一些收敛定理的尖锐陈述。例如，如果ui是psh函数的一致有界序列，其单调收敛于psh函数U，则在线拓扑中（MU;）"+（d&u）”弱 *。也就是说，如果$是一个有界的，有紧支集的连续函数，那么j Ij/（d&u，）"+ j Ijl（dd 'u）"。在第3节中给出了这个结果的一个更严格的版本，其中还证明了“balayage”的一个版本对于细闭集是可能的（推论3.4）。在第4节中，我们讨论了（dd”）"的定义。算子（dd '）”收敛于有界psh函数的单调极限，这可以用来证明（dd”）”从光滑psh函数的扩展（参见。[BT]）。(We注意（dd”）”在非单调极限下表现不好，如Cegrell [Ce]和Lelong CL]所示。对于有界的，psh U，Oberguggenberger [0]已经证明，如果计算外积d&u A.一个dd 'u，使用Colombeau的分布代数，可以得到与前面相同的（dd'）”。我们在这里展示了这枚硬币-
In classical potential theory, the fine topology provides a useful and natural way to give sharp statements of many results. The same is true for the case of the potential theory associated with plurisubharmonic (psh) functions. In Section 2 we give a short discussion of the (pluri-) fine topology of psh functions. Almost all the results are the same as for the classical fine topology, even with the same proof, so we have omitted all proofs in this section. However, there is one crucial difference-the notions of “thin set” and “polar set” are not equivalent in psh potential theory. The use of the fine topology allows us to give sharp statements of some convergence theorems proved in [BT]. For example, if ui is a uniformly bounded sequence of psh functions which converges monotonically ae to a psh function U, then (MU;)”+(d&u)” weak* in the line topology. That is, if $ is a bounded, tine continuous function with compact support, then j Ij/(d&u,)”+ j Ijl (dd’u)“. A more relined version of this result is given in Section 3, where it is also shown that a version of “balayage” is possible for fine closed sets (Corollary 3.4).In Section 4, we discuss the definition of (dd”)“. The operator (dd’)” converges for monotone limits of bounded, psh functions, and this may be used to justify the extension of (dd”)” from smooth, psh functions (cf.[BT]).(We note that (dd”)” does not behave well under nonmonotone limits, as was shown by Cegrell [Ce] and Lelong CL].) For bounded, psh U, Oberguggenberger [0] has shown that if one computes the exterior product d&u A... A dd’u, using the algebra of distributions of Colombeau, one obtains the same (dd’)” as before. Here we shown that this coin-