We extended the known result that symbols from modulation spaces $M^{ \infty,1}(\mathbb{ R}^{ 2n})$, also known as the Sj\"{ o}strand'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 $L^p_s(\mathbb{ R}^n)$ to $L^p(\mathbb{ R}^n)$ spaces with symbols from the modulation space $M^{ \infty,1}(\mathbb{ R}^{ 2n})$ are bounded, whenever $s\geq n|1/p-1/2|.$ This estimate is sharp for all $1\leq p\leq\infty$.
Categories: math.FA
