A local-to-global weak (1,1) type argument and applications to Fourier integral operators

In this work we provide a criterion for the global weak (1,1) type of integral operators which are known to be locally uniformly of weak (1,1) type. As an application, we establish the global weak (1,1) type for a class of Fourier integral operators. While the local result is known from the work of Tao [33] for Fourier integral operators of order $-(n-1)/2,$ we give natural sufficient conditions in order to extend it to the corresponding global estimate
Comment: Unfortunately, we have found a mistake in the proof of Theorem 1.2, that is the main theorem of this work. We claim in page 10, below of (2.5) that the uniform estimate of the weak L^1 norm of T at any atom is enough for concluding the proof, but this is not the case, because the weak L^1 space is not normable