On $L^p-$boundedness of pseudo-differential operators of Sj\"ostrand's class

TL;DR

This paper proves that pseudo-differential operators with symbols in Sj"ostrand's class are bounded on $L^p$ spaces with a specific loss of derivatives, extending known $L^2$ results to all $p$ with sharp estimates.

Contribution

It extends the $L^2$ boundedness of Sj"ostrand's class symbols to $L^p$ spaces, establishing sharp bounds with derivative loss depending on $p$.

Findings

Boundedness of operators from $L^p_s$ to $L^p$ for $s \,\geq\, n|1/p - 1/2|$

Sharpness of the derivative loss estimate for all $p$

Generalization of known $L^2$ results to all $p$

Abstract

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 $1\leq p\leq\infty$.