Hilbert-Schmidtness of Fourier integral operators in $\mathbf{SG}$ classes

Document Type : Original Article


Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Ahmed BEN BELLA. B.P. 1524 El M'naouar, Oran, Algeria



In this paper, we define a particular class of Fourier integral operators with $\mathbf{SG}$-symbol. These class of operators turn out to be bounded on the spaces $\mathcal{S}\left(\mathbb{R}^{n}\right)$ of rapidly decreasing functions and turn out to be Hilbert-Schmidt on $L^2\left(\mathbb{R}^{n}\right)$. Mainly, we prove that the Fourier integral operators with $\mathbf{SG}$-symbol are a Hilbert-Schmidt operators.