The weak-type $(1,1)$ of Fourier integral operators of order $-(n-1)/2$

Abstract

Let $T$ be a Fourier integral operator on $\R^n$ of order $-(n-1)/2$. It was shown by Seeger, Sogge, and Stein that $T$ mapped the Hardy space $H^1$ to $L^1$. In this note we show that $T$ is also of weak-type $(1,1)$. The main ideas are a decomposition of $T$ into non-degenerate and degenerate components, and a factorization of the non-degenerate portion.