Pseudo-differential operators with forbidden symbols on Triebel–Lizorkin spaces

Abstract

In this note, we consider a pseudo-differential operator \(T_a\) defined as

$$\begin{aligned} T_a f(x)=\int _{\mathbb {R}^n}e^{2\pi ix\cdot \xi }a(x,\xi )\widehat{f}(\xi )d\xi . \end{aligned}$$

It is well-known that \(T_a\) is not bounded on \(L^2\) in general when a belongs to the forbidden Hörmander class \(S^{n(\rho -1)/2}_{\rho ,1},0\le \rho \le 1\). In this note, when \(s>0,0\le \rho \le 1,1\le r\le 2\) and \(a\in S^{n(\rho -1)/r}_{\rho ,1}\), we prove that \(T_a\) is bounded on the Triebel-Lizorkin space \(F^s_{p,q}\) if \(r<p,q<\infty \) or \(r<p\le \infty ,q=\infty \). As the most important special example, when \(a\in S^{n(\rho -1)/2}_{\rho ,1}\) and \(s>0\), if \(2<p,q<\infty \) or \(2<p\le \infty ,q=\infty \), then \(T_a\) is bounded on \(F^s_{p,q}\). When \(\rho <1\), this result is entirely new.