Tomio Umeda
Action of $\sqrt{-\Delta}$ on distirbutions
(45K, AMS-TeX)

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 $s \ge 3/2$.