An extension of Weyl’s lemma to infinite dimensions

Elson, Constance M.

doi:10.1090/S0002-9947-1974-0343022-0

An extension of Weyl’s lemma to infinite dimensions
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Trans. Amer. Math. Soc. 194 (1974), 301-324 Request permission

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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