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

Let T be a Fourier integral operator of order - (n-1) /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 H¹ into L¹. 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) /2 when it has associated a canonical relation parametrised by a complex phase function.