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

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.