Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 88, pp. 1-7.
Title: A class of generalized integral operators
Authors: Samir Bekkara (Univ. d'Oran, Oran, Algeria)
Bekkai Messirdi (Univ. d'Oran, Es-Senia, Algeria)
Abderrahmane Senoussaoui (Univ. d'Oran, Es-Senia, Algeria)
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.