The endpoint estimates for pseudo-differential operators

Let \(T_{a}\) be a pseudo-differential operator with symbol a. When \(a\in S^m_{\rho ,1},m=n(\rho -1)\), it is well known that \(T_{a}\) is not always bounded on \({L^1}({\mathbb {R}^n})\). However, under extra assumptions on a, we prove that \(T_{a}\) is bounded on \({L^p}({\mathbb {R}^n})\) for \(1 \le p \le \infty \) when \(a \in {L^\infty }S_\rho ^{n(\rho - 1)}(\omega )\).