An extension of weyl’s lemma to infinite dimensions
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.
Transactions of the American Mathematical Society
Elson, Constance M., "An extension of weyl’s lemma to infinite dimensions" (1974). Faculty Articles Indexed in Scopus. 2731.