00-263 Tomio Umeda

Action of $\sqrt{-\Delta}$ on distirbutions (45K, AMS-TeX) Jun 11, 00

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Abstract. Action of $\sqrt{-\Delta}$ on distributions is examined in the context of Sobolev spaces, weighted $L^2$ spaces and weighted Sobolev spaces, respectively. The results obtained are as follows: Let $k$ be a real number. (1) If $f$ is in $H^{k}({\bold R}^n)$, then $\sqrt{-\Delta}f$ is in $H^{k-1}({\bold R}^n)$; \ (2) If $f$ is in $\Cal S({\bold R}^n)$, the space of rapidly decreasing functions, then $f$ is in $L^{2, \, s}({\bold R}^n)$ for any $s < n/2 +1$; \ (3) If $f$ is in $H^{k,\, s}({\bold R}^n)$ for some $s > -n / 2 -1$, then $\sqrt{-\Delta}f$ is in ${\Cal S}^{\prime}({\bold R}^n)$, the space of tempered distributions. Also, it is shown in the one dimensional case that there exists a $\varphi_0$ in ${\Cal S}(\bold R)$ such that $\sqrt{-\Delta} \varphi_0$ does not belong to $L^{2, \, s}(\bold R)$ for any $s \ge 3/2$.

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