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

TL;DR

This paper investigates the sharpness of the weak (1,1) boundedness order for Fourier integral operators with real non-degenerate phase functions, extending previous results and confirming optimality for elliptic operators.

Contribution

It proves that the order -(n-1)/2 for weak (1,1) boundedness is sharp for elliptic Fourier integral operators and related canonical relations, extending prior work.

Findings

Weak (1,1) inequality order is sharp for elliptic Fourier integral operators.

Confirmed sharpness for operators satisfying additional rank conditions.

Extended previous results to broader classes of canonical relations.

Abstract

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 $I^{\mu}_1(X,Y;\Lambda),$ $\mu\leq -(n-1)/2,$ are bounded from $H^1$ to $L^1.$ The sharpness of the order $-(n-1)/2,$ 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 $I^{\mu}_1(X,Y;\Lambda),$ $\mu\leq -(n-1)/2,$ 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)/2$ is sharp for any elliptic Fourier integral operator, as well as its versions for canonical relations satisfying additional rank conditions.