Boundedness of Fourier Integral Operators with complex phases on Fourier Lebesgue spaces

In this paper, we develop boundedness estimates for Fourier integral operators on Fourier Lebesgue spaces when the associated canonical relation is parametrised by a complex phase function. Our result constitutes the complex analogue of those obtained for real canonical relations by Rodino, Nicola, and Cordero. We prove that, under the spatial factorization condition of rank $\mathrm{\varkappa }$, the corresponding Fourier integral operator is bounded on the Fourier Lebesgue space $\mathcal{F}{L}^p,$ provided that the order $m$ of the operator satisfies that $m\le -\mathrm{\varkappa }\mid \frac{1}{p}-\frac{1}{2}\mid ,1\le p\le \mathrm{\infty }\mathrm{.}$ This condition on the order $m$ is sharp.