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

TL;DR

This paper develops a criterion to extend local weak (1,1) estimates of integral operators to global ones and applies it to Fourier integral operators, broadening understanding of their boundedness properties.

Contribution

It introduces a natural sufficient condition for extending local weak (1,1) bounds to global bounds for integral operators, including Fourier integral operators.

Findings

Established global weak (1,1) type for a class of Fourier integral operators.

Provided a criterion for extending local estimates to global estimates.

Extended known results from Tao for specific Fourier integral operators.

Abstract

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