$H^1$–$L^1$ Boundedness of Pseudo-differential Operators with Forbidden Amplitudes

Abstract

In this note, we consider a pseudo-differential operator $T_a$ defined as

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

If $0\le \rho ,\delta <1$ and $a(x,y,\xi )$ satisfies that

$$\begin{aligned} \sup _{y,\xi \in {\mathbb {R}}^{n}}(1+|\xi |)^{n(1-\rho )+\rho N-\delta M}\Vert \nabla ^{N}_{\xi }\nabla ^{M}_{y}a(\cdot ,y,\xi )\Vert _{L^\infty ({\mathbb {R}}^n)}<+\infty \end{aligned}$$

for any $N\le n+1$ and $M\le 1$, then we show that the pseudo-differential operator $T_{a}$ is bounded from $H^1$ to $L^1$. However, this result is not true if $\rho =1$ or $\delta =1$.