Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
MathSciNet review: 0343022
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46G05, 46F10

Retrieve articles in all journals with MSC: 46G05, 46F10

Additional Information

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