On L 2 -Boundedness of h -Pseudodifferential Operators

Abstract

Let $T_a^h$ be the $h$ -pseudodifferential operators with symbol $a$ . When $a\in S_{\rho ,1}^m$ and $m=n\left(\rho -1\right)/2$ , it is well known that $T_a^h$ is not always bounded in $L^2\left({ℝ}^n\right)$ . In this paper, under the condition $a\left(x,\xi \right)\in L^{\infty }{S}_{\rho }^{n\left(\rho -1\right)/2}\left(\omega \right)$ , we show that $T_a^h$ is bounded on $L^2$ .