On $L^p-$boundedness of pseudo-differential operators of Sjöstrand's class

Abstract

We extended the known result that symbols from modulation spaces $M^{\infty ,1}({\mathbb{R}}^{2n})$, also known as the Sjöstrand's class, produce bounded operators in $L^2({\mathbb{R}}^n)$, to general $L^p$ boundedness at the cost of lost of derivatives. Indeed, we showed that pseudo-differential operators acting from $L^p$-Sobolev spaces $L_s^p({\mathbb{R}}^n)$ to $L^p({\mathbb{R}}^n)$ spaces with symbols from the modulation space $M^{\infty ,1}({\mathbb{R}}^{2n})$ are bounded, whenever $s\ge n|1/p-1/2|.$ This estimate is sharp for all $1\le p\le \infty$.