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

Abstract

AbstractLet X and Y be two smooth manifolds of the same dimension. It was proved by Seeger et al. in (Ann Math 134(2): 231–251, 1991) that the Fourier integral operators with real non-degenerate phase functions in the class $$I^{\mu }_1(X,Y;\Lambda ),$$ I 1 μ ( X , Y ; Λ ) , $$\mu \le -(n-1)/2,$$ μ ≤ - ( n - 1 ) / 2 , are bounded from $$H^1$$ H 1 to $$L^1.$$ L 1 . The sharpness of the order $$-(n-1)/2,$$ - ( n - 1 ) / 2 , for any elliptic operator was also proved in (Seeger et al. Ann Math 134(2): 231–251, 1991) and extended to other types of canonical relations in (Ruzhansky Hokkaido Math J 28(2): 357–362, 1992). That the operators in the class $$I^{\mu }_1(X,Y;\Lambda ),$$ I 1 μ ( X , Y ; Λ ) , $$\mu \le -(n-1)/2,$$ μ ≤ - ( n - 1 ) / 2 , satisfy the weak (1,1) inequality was proved by Tao (J Aust Math Soc 76(1):1–21, 2004). In this note, we prove that the weak (1,1) inequality for the order $$ -(n-1)/2$$ - ( n - 1 ) / 2 is sharp for any elliptic Fourier integral operator, as well as its versions for canonical relations satisfying additional rank conditions.