TY - JOUR
T1 - Lévy Laplacian of generalized functions on a nuclear space
AU - Kuo, Hui Hsiung
AU - Obata, Nobuaki
AU - Saitô, Kimiaki
N1 - Funding Information:
* Supported by Research Center Bielefeld-Bochum-Stochastics (BiBoS). + Supported by the Japan Society for the Promotion of Science (International Research Program, 1988-1989).
PY - 1990/11
Y1 - 1990/11
N2 - 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.
AB - 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.
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U2 - 10.1016/0022-1236(90)90028-J
DO - 10.1016/0022-1236(90)90028-J
M3 - Article
AN - SCOPUS:0003073673
SN - 0022-1236
VL - 94
SP - 74
EP - 92
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 1
ER -