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On a regularizing effect of Schrödinger type groups. (English) Zbl 0699.35027
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 \(H^ s({\mathbb{R}}^ n)\) onto itself, no regularizing effects can be expected in the \(H^ s({\mathbb{R}}^ n)\) framework. Nevertheless it can be shown that, for any f which belongs to \(L^ 2({\mathbb{R}}^ n)\), something more than being an \(L^ 2({\mathbb{R}}^ n)\) function can be asserted about exp((it \(\Delta)\)f.
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 \(L^ 1({\mathbb{R}}^ n)\) and what is proved is regularity of \(W^{r,\infty}({\mathbb{R}}^ n)\) 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.
A weaker result was quoted by the author and 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.
Reviewer: N.D.Kazarinoff

MSC:
35B65 Smoothness and regularity of solutions to PDEs
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
35Q99 Partial differential equations of mathematical physics and other areas of application
47D03 Groups and semigroups of linear operators
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References:
[1] Balabane, M.; Rad, H. A. Emami, L^{p} estimates for Schrödinger evolution equations, Trans. Amer. Math. Soc., Vol. 292, n° 1, (1985) · Zbl 0588.35029
[2] J. J. Duistermaat, Fourier Integral Operators, Courant Inst. Math. Sc. N.Y.U., 1973. · Zbl 0272.47028
[3] Ginibre, J.; Velo, G., On the global Cauchy problem for some non linear Schrödinger equations, Ann. Inst. Henri Poincaré, Vol. 1, n° 4, (1984) · Zbl 0569.35070
[4] Ginibre, J.; Velo, G., The global Cauchy problem for the non linear Schrödinger equation revisited, Ann. Inst. Henri Poincaré, Vol. 2, n° 4, (1985) · Zbl 0586.35042
[5] Strichartz, R., Duke math, Journal, t. 44, (1977)
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