The weak (1,1) boundedness of Fourier integral operators with complex phases

Let $T$ be a Fourier integral operator of order $-(n-1)\mathrm{/}2$ associated with a canonical relation locally parametrised by a real-phase function. A fundamental result due to Seeger, Sogge, and Stein proved in the 90’s, gives the boundedness of $T$ from the Hardy space $H^1$ into $L^1\mathrm{.}$ Additionally, it was shown by T. Tao the weak (1,1) type of $T$. In this work, we establish the weak (1,1) boundedness of a Fourier integral operator $T$ of order $-(n-1)\mathrm{/}2$ when it has associated a canonical relation parametrised by a complex phase function. This result in the complex-valued setting, cannot be derived from its counterpart in the real-valued case.