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On $L^p-$boundedness of pseudo-differential operators of Sjöstrand's class (1504.04087v2)

Published 16 Apr 2015 in math.FA

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 $L2(\mathbb{R}n)$, to general $Lp$ boundedness at the cost of lost of derivatives. Indeed, we showed that pseudo-differential operators acting from $Lp$-Sobolev spaces $Lp_s(\mathbb{R}n)$ to $Lp(\mathbb{R}n)$ 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 $1\leq p\leq\infty$.

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