Mehler Kernel Approach to Tempered Distributions

Abstract

Using the Mehler kernel $E(x,\xi,t)$, we show that the solution of the Hermite heat equation $({\partial}_{t} - \triangle + |x|^{2})U(x,t) = 0$ in $\mathbf{R}^{n}\times (0,T)$ satisfying $\sup_{x\in{\mathbf{R}}^{n}}|U(x,t)|\leq C(1+ t^{-N})$ for some constants $C$ and $N$ can be expressed as $U(x,t) = \langle u(\xi), E(x,\xi,t)\rangle$ for unique $u$ in ${\mathcal S}^{'}(\mathbf{R}^{n})$. This is a parallel result with the one in (Theorem 1.2, T. Matsuzawa, {\it A calculus approach to hyperfunctions} III, Nagoya Math. J. {\bf 118} (1990), 133--153). Moreover we represent the tempered distributions as initial values of solution of the Hermite heat equation and apply it to generalize a theorem by Strichartz [Theorem 3.2, Trans. Amer. Math. Soc. {\bf 338} (1993), 971--979] in the space of tempered distributions.