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

Abstract

In this paper we study a class of Fourier integral operators, whose symbols lie in the multi-parameter Hörmander class $S^{\vec m}( \mathbb{R}^\vn)$, where ~$\vec{m}=(m_1,m_2,\dots ,m_d)$ is the order. We show that if in addition the phase function $\hat{I}¦(x,\hat{I}¾)$ can be written as $\hat{I}¦(x,\hat{I}¾)={\sum }_{i=1}^d\hat{I}{¦}_i(x_i,\hat{I}{¾}_i)$, and each $\hat{I}{¦}_i(x_i,\hat{I}{¾}_i)$ 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 $L^1({\mathbb{R}}^n)$.