On \(L^2\)-boundedness of \(h\)-pseudodifferential operators (Q2023027)

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