Document Type
Article
Publication Date
1-1-1990
Abstract
The Lévy Laplacian ΔF(ξ) = limN→∞N-1∑n = 1N 〈F″(ξ),en⊗ en〉 is shown to be equal to (i) ∝TF″s″(ξ;t)dt, where Fs″ is the singular part of F″, and (ii) 2limρ{variant}→0ρ{variant}-2(MF(ξ,ρ{variant})-F(ξ)), where MF is the spherical mean of F. It is proved that regular polynomials are Δ-harmonic and possess the mean value property. A relation between the Lévy Laplacian Δ and the Gross Laplacian ΔGF(ξ) = ∑n = 1∞=〈F″(ξ),en⊗ en〉 is obtained. An application to white noise calculus is discussed. © 1990.
Publication Source (Journal or Book title)
Journal of Functional Analysis
First Page
74
Last Page
92
Recommended Citation
Kuo, H., Obata, N., & Saitô, K. (1990). Lévy Laplacian of generalized functions on a nuclear space. Journal of Functional Analysis, 94 (1), 74-92. https://doi.org/10.1016/0022-1236(90)90028-J