An extension of Weyl’s lemma to infinite dimensions
An extension of Weyl’s lemma to infinite dimensions
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韦尔引理向无限维的扩展
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发表时间：
1974
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通讯作者：
Constance M. Elson
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作者：
Constance M. Elson
一个理论的分布类似于施瓦茨分布理论制定可分的Banach空间，使用抽象的维纳空间技术。一个分布T是开集U上的调和分布，如果对U上的任意测试函数f，T（Af）= 0，其中Af是f的广义拉普拉斯算子。证明了U上的调和分布可以表示为U的任何子集上的唯一测度，该子集与UC有正距离。在空间是有限维的情况下，从外尔引理可以得出，测度实际上是由C'函数表示的。这种功能表示不能预期在无限维，但它表明，措施具有光滑性类似于无限可微的功能。
A theory of distributions analogous to Schwartz distribution theory is formulated for separable Banach spaces, using abstract Wiener space techniques. A distribution T is harmonic on an open set U if for any test function f on U, T(Af) = 0, where Afis the generalized Laplacian of f. We prove that a harmonic distribution on U can be represented as a unique measure on any subset of U which is a positive distance from UC. In the case where the space is finite dimensional, it follows from Weyl's lemma that the measure is in fact represented by a C' function. This functional representation cannot be expected in infinite dimensions, but it is shown that the measure has smoothness properties analogous to infinite differentiability of functions.