Title

An extension of weyl’s lemma to infinite dimensions

Document Type

Article

Publication Date

1-1-1974

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(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 C00 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. © 1974 American Mathematical Society.

Publication Name

Transactions of the American Mathematical Society

Volume Number

194

First Page

301

Last Page

324

DOI

10.1090/S0002-9947-1974-0343022-0

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