Some notes on endpoint estimates for pseudo-differential operators

We study the pseudo-differential operator \begin{equation*} T_a f\left(x\right)=\int_{$\mathbb{R}$^n}e^{ix\cdot\xi}a\left(x,\xi\right)\widehat{f}\left(\xi\right),$\text{d}$\xi, \end{equation*} where the symbol $a$ is in the H"{o}rmander class $S_{\rho ,1}^m$ or more generally in the rough H"{o}rmander class $L^{\mathrm{\infty }}{S}_{\rho }^m$ with $m\in \mathbb{R}$ and $\rho \in [0,1]$. It is known that $T_a$ is bounded on $L^1({\mathbb{R}}^n)$ for $m<n(\rho -1)$. In this paper we mainly investigate its boundedness properties when $m$ is equal to the critical index $n(\rho -1)$. For any $0\le \rho \le 1$ we construct a symbol $a\in S^{n(\rho-1)}{\rho,1}$suchthat$T_a$isunboundedon$L^1$ and furthermore it is not of weak type $(1,1)$ if $\rho =0$. On the other hand we prove that $T_a$ is bounded from $H^1$ to $L^1$ if $0\le \rho <1$ and construct a symbol $a\in S_{1,1}^0$ such that $T_a$ is unbounded from $H^1$ to $L^1$. Finally, as a complement, for any $1<p<\mathrm{\infty }$ we give an example $a\in S^{-1/p}{0,1}$suchthat$T_a$isunboundedon$L^p($\mathbb{R}$)$.