Symbolic calculus of Hermite operators

Abstract

Like a pseudo-differential operator, a Hermite operator on $R^n$ is defined as a linear mapping of $C_0^{\infty}(R^n)$ into $C^{\infty}(R^{n+1})$ by using an amplitude which is a $C^{\infty}$ -function satisfying a certain growth condition and with the concept of oscillating integral. In this paper we show that a Hermite operator of degree m on $R^n$ can be extended as a continuous linear mapping of $H_C^s(R^n)$ into $H_{loc}^{s-m}(R^{n+1})$. Also, we can compose Hermite operators and pseudo-differential operators modulo regularizing operators and compute symbols for each case; if K and K`` are Hermite operators on $R^n$ and A and B are pseudo-differential operators on $R^{n+1}$ and $R^n$ respectively, then $K^*\Cdot K``$ is a pseudo differential operator on $R^n$ and $A\Cdot K$ and $A\Cdot B$ are Hermite operators on $R^n$.