Electron. J. Diff. Eqns., Vol. 2009(2009), No. 88, pp. 1-7.

A class of generalized integral operators

Samir Bekkara, Bekkai Messirdi, Abderrahmane Senoussaoui

Abstract:
In this paper, we introduce a class of generalized integral operators that includes Fourier integral operators. We establish some conditions on these operators such that they do not have bounded extension on $L^{2}(\mathbb{R}^{n})$. This permit us in particular to construct a class of Fourier integral operators with bounded symbols in $S_{1,1}^{0}(\mathbb{R}^{n}\times \mathbb{R}^{n})$ and in $\bigcap_{0<\rho <1}S_{\rho ,1}^{0} (\mathbb{R}^{n}\times \mathbb{R}^{n})$ which cannot be extended to bounded operators in $L^{2}(\mathbb{R}^{n})$.

Submitted February 12, 2009. Published July 27, 2009.
Math Subject Classifications: 35S30, 35S05, 47A10, 35P05.
Key Words: Integral operators; L2-boundedness; unbounded Fourier integral operators.

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Samir Bekkara
Université des Sciences et de la Technologie d'Oran
Faculté des Sciences, Département de Mathématiques, Oran, Algeria
email: sbekkara@yahoo.fr
Bekkai Messirdi, Abderrahmane Senoussaoui
Université d'Oran Es-Sénia, Faculté des Sciences
Département de Mathématiques. B.P. 1524 El-Mnaouer, Oran, Algeria
email: bmessirdi@univ-oran.dz
Abderrahmane Senoussaoui
Université d'Oran Es-Sénia, Faculté des Sciences
Département de Mathématiques. B.P. 1524 El-Mnaouer, Oran, Algeria
email: senoussaoui.abdou@univ-oran.dz

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