 00263 Tomio Umeda
 Action of $\sqrt{\Delta}$ on distirbutions
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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
$H^{k1}({\bold R}^n)$; \ (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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