Some Notes on Endpoint Estimates for Pseudo-differential Operators

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