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)\,\textrm{d}\xi, \end{equation*} where the symbol $a$ is in the H\"{o}rmander class $S^{m}_{\rho,1}$ or more generally in the rough H\"{o}rmander class $L^{\infty}S^{m}_{\rho}$ 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 $0\leq \rho\leq 1$ we construct a symbol $a\in S^{n(\rho-1)}_{\rho,1}$ such that $T_a$ is unbounded on $L^1$ 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 $0\leq \rho<1$ and construct a symbol $a\in S^0_{1,1}$ such that $T_a$ is unbounded from $H^1$ to $L^1$. Finally, as a complement, for any $1<p<\infty$ we give an example $a\in S^{-1/p}_{0,1}$ such that $T_a$ is unbounded on $L^p(\mathbb{R})$.