Omar Farouk Aid, Abderrahmane Senoussaoui

L2-Hilbert-Schmidtness of Fourier Integral Operators with Weighted Symbols

In this paper, we define a particular class of Fourier Integral Operators with weighted symbols (FIO, for short). These FIO turn out to be bounded on the spaces $S(\mathbb{R}^{n})$ of rapidly decreasing functions (or Schwartz space) and $S^{\prime}(\mathbb{R}^{n})$ of temperate distributions. We also prove that FIO is Hilbert--Schmidt on $L^{2}( \mathbb{R}^{n}) $ when the weight of the symbol $a$ belongs to $L^{2}(\mathbb{R}^{2n})$.

Mathematics Subject Classification: 35S30, 35S05, 47G30

Key words and phrases: Fourier integral operators, symbol and phase, Hilbert-Schmidt operators