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.