Lévy Laplacian of generalized functions on a nuclear space

Hui Hsiung Kuo, Nobuaki Obata, Kimiaki Saitô

Research output: Contribution to journalArticlepeer-review

49 Citations (Scopus)


The Lévy Laplacian ΔF(ξ) = limN→∞N-1n = 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.

Original languageEnglish
Pages (from-to)74-92
Number of pages19
JournalJournal of Functional Analysis
Issue number1
Publication statusPublished - 1990 Nov


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