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Transactions of the American Mathematical Society

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An extension of Weyl's lemma to infinite dimensions


Author: Constance M. Elson
Journal: Trans. Amer. Math. Soc. 194 (1974), 301-324
MSC: Primary 46G05; Secondary 46F10
DOI: https://doi.org/10.1090/S0002-9947-1974-0343022-0
MathSciNet review: 0343022
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Abstract: 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(\Delta f) = 0$, where $ \Delta f$ fis 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 $ {U^C}$. 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^\infty }$ 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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0343022-0
Keywords: Infinite dimensions, abstract Wiener spaces, harmonic, generalized Laplacian, test functions, distributions, regularity of generalized solutions of Laplace's equation
Article copyright: © Copyright 1974 American Mathematical Society

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