A correction and some additions to: “Fundamental solutions for differential equations associated with the number operator”
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- by Yuh Jia Lee PDF
- Trans. Amer. Math. Soc. 276 (1983), 621-624 Request permission
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 $B\text {-}B^{\ast }$ 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.References
- Hui Hsiung Kuo, Potential theory associated with Uhlenbeck-Ornstein process, J. Functional Analysis 21 (1976), no. 1, 63–75. MR 0391285, DOI 10.1016/0022-1236(76)90029-x
- Yuh Jia Lee, Fundamental solutions for differential equations associated with the number operator, Trans. Amer. Math. Soc. 268 (1981), no. 2, 467–476. MR 632538, DOI 10.1090/S0002-9947-1981-0632538-9
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 276 (1983), 621-624
- MSC: Primary 35R15; Secondary 28C20, 46G99, 58D20, 60J99
- DOI: https://doi.org/10.1090/S0002-9947-1983-0688965-9
- MathSciNet review: 688965