**Omar Farouk Aid, Abderrahmane Senoussaoui**

##
*L*^{2}-Hilbert-Schmidtness of Fourier Integral Operators with
Weighted Symbols

**abstract:**

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