A class of multi-parameter Fourier integral operators: endpoint Hardy space bounds

In this paper we study a class of Fourier integral operators, whose symbols lie in the multi-parameter H\"ormander class $S^{\vec m}( \mathbb{R}^\vn)$, where ~$\vec m=(m_1,m_2,\dots,m_d)$ is the order. We show that if in addition the phase function $\Phi(x,\xi)$ can be written as $\Phi(x,\xi)=\sum_{i=1}^d\Phi_i(x_i,\xi_i)$, and each $\Phi_i(x_i,\xi_i)$ satisfies the non-degeneracy condition, then such Fourier integral operators with order ~$\vec m=(-(n_1-1)/2, -(n_2-1)/2,\dots, -(n_d-1)/2)$ are actually bounded from rectangular Hardy space $H_{rect}^1(\mathbb{R}^\vn)$ to $L^1( \mathbb{R}^n )$.