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 loss of derivatives. Indeed, we showed that pseudo-differential operators acting from $L^p$-Sobolev spaces $L^p_s(\mathbb {R}^n)$ 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 $1< p<\infty$.