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 \(L^p(\mathbb {R})\).