Some Notes on Endpoint Estimates for Pseudo-differential Operators

Abstract

We study the pseudo-differential operator

$$\begin{aligned} T_a f\left( x\right) =\int _{\mathbb {R}^n}e^{ix\cdot \xi }a\left( x,\xi \right) \widehat{f}\left( \xi \right) \,\text {d}\xi , \end{aligned}$$

where the symbol a is in the Hörmander class $S^{m}_{\rho ,1}$ or more generally in the rough Hörmander 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 $0\le \rho \le 1$, 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 $0\le \rho <1$ 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 $L^p(\mathbb {R})$.