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Zbl 0699.35027
Balabane, Mikhaël
On a regularizing effect of Schrödinger type groups. (English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 6, No.1, 1-14 (1989). ISSN 0294-1449

From the author's introduction: If one considers the one parameter group of operators exp(it $\Delta)$, because of the group law, and because these operators map isomorphically any $H\sp s({\bbfR}\sp n)$ onto itself, no regularizing effects can be expected in the $H\sp s({\bbfR}\sp n)$ framework. Nevertheless it can be shown that, for any f which belongs to $L\sp 2({\bbfR}\sp n)$, something more than being an $L\sp 2({\bbfR}\sp n)$ function can be asserted about exp((it $\Delta)$f. \par The aim of this paper is to show that even regularity can be asserted. To prevent obstruction due to the group law, one considers Cauchy data in $L\sp 1({\bbfR}\sp n)$ and what is proved is regularity of $W\sp{r,\infty}({\bbfR}\sp n)$ type (with $r>2)$. An interesting remark is that the higher is the order of a pseudo-differential operator P(D) the more regularizing is exp(it $\Delta)$. Furthermore this regularization is more effective in high space dimension. This translates the dispersion of the waves. \par A weaker result was quoted by the author and {\it H. A. Emami Rad} [Trans. Am. Math. Soc. 292, 357-373 (1985; Zbl 0588.35029)] and here we make use of many of the same tools as in that article. These tools are direct computations and estimates using the stationary phase lemma. In the above cited work we emphasized boundedness whereas here we give regularity results. These are essential in view of nonlinear Schrödinger type evolution equations.
[N.D.Kazarinoff]

MSC 2000:: *35B65 Smoothness of solutions of PDE
35G10 Initial value problems for linear higher-order PDE
35K25 Higher order parabolic equations, general
35Q99 PDE of mathematical physics and other areas
47D03 (Semi)groups of linear operators

Keywords: nonlinear Schrödinger equation; one parameter group of operators; regularity; Cauchy data

Citations: Zbl 0588.35029

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