Sharpness of Seeger-Sogge-Stein orders for the weak (1,1) boundedness of Fourier integral operators

Let $X$ and $Y$ be two smooth manifolds of the same dimension. It was proved by Seeger, Sogge and Stein in \cite{SSS} that the Fourier integral operators with real non-degenerate phase functions in the class $I_1^{\mu }(X,Y;\mathrm{\Lambda }),$ $\mu \le -(n-1)\mathrm{/}2,$ are bounded from $H^1$ to $L^1\mathrm{.}$ The sharpness of the order $-(n-1)\mathrm{/}2,$ for any elliptic operator was also proved in \cite{SSS} and extended to other types of canonical relations in \cite{Ruzhansky1999}. That the operators in the class $I_1^{\mu }(X,Y;\mathrm{\Lambda }),$ $\mu \le -(n-1)\mathrm{/}2,$ satisfy the weak (1,1) inequality was proved by Tao \cite{Tao:weak11}. In this note, we prove that the weak (1,1) inequality for the order $-(n-1)\mathrm{/}2$ is sharp for any elliptic Fourier integral operator, as well as its versions for canonical relations satisfying additional rank conditions.