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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A correction and some additions to: ``Fundamental solutions for differential equations associated with the number operator''


Author: Yuh Jia Lee
Journal: Trans. Amer. Math. Soc. 276 (1983), 621-624
MSC: Primary 35R15; Secondary 28C20, 46G99, 58D20, 60J99
Original Article: Trans. Amer. Math. Soc. 268 (1981), 467-476.
MathSciNet review: 688965
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Abstract: Let $ (H,B)$ be an abstract Wiener pair and $ \mathfrak{N}$ the operator defined by $ \mathfrak{N}u(x) = - {\text{trace}}_H{D^2}u(x) + (x,Du(x))$, where $ x \in B$ and $ (\cdot, \cdot )$ denotes the $ B$-$ B^{\ast}$ pairing. In this paper, we point out a mistake in the previous paper concerning the existence of fundamental solutions of $ {\mathfrak{N}^k}$ and intend to make a correction. For this purpose, we study the fundamental solution of the operator $ {(\mathfrak{N} + \lambda I)^k}\,(\lambda > 0)$ and investigate its behavior as $ \lambda \to 0$. We show that there exists a family $ \{{Q_\lambda}(x,dy)\} $ of measures which serves as the fundamental solution of $ {(\mathfrak{N} + \lambda I)^k}$ and, for a suitable function $ f$, we prove that the solution of $ {\mathfrak{N}^k}u = f$ can be represented by $ u(x) = {\lim _{\lambda \to 0}}\int_B f(y){Q_\lambda}(x,dy) + C $, where $ C$ is a constant.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1983-0688965-9
PII: S 0002-9947(1983)0688965-9
Keywords: Abstract Wiener spaces, fundamental solution, number operator
Article copyright: © Copyright 1983 American Mathematical Society