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

Let (Formula presented.) be a Fourier integral operator of order (Formula presented.) 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 (Formula presented.) from the Hardy space (Formula presented.) into (Formula presented.). Additionally, it was shown by T. Tao the weak (1,1) type of (Formula presented.). In this work, we establish the weak (1,1) boundedness of a Fourier integral operator (Formula presented.) of order (Formula presented.) 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.