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 Hörmander class $S^{\vec{m}}({\mathbb{R}}^{\text{vn}})$, where ~$\vec{m}=(m_1,m_2,\dots ,m_d)$ is the order. We show that if in addition the phase function $\mathrm{\Phi }(x,\xi )$ can be written as $\mathrm{\Phi }(x,\xi )={\sum }_{i=1}^d{\mathrm{\Phi }}_i(x_i,{\xi }_i)$, and each ${\mathrm{\Phi }}_i(x_i,{\xi }_i)$ satisfies the non-degeneracy condition, then such Fourier integral operators with order ~$\vec{m}=(-(n_1-1)\mathrm{/}2,-(n_2-1)\mathrm{/}2,\dots ,-(n_d-1)\mathrm{/}2)$ are actually bounded from rectangular Hardy space $H_{rect}^1({\mathbb{R}}^{\text{vn}})$ to $L^1({\mathbb{R}}^n)$.