Fourier integral operators on Hardy spaces with Hormander class

In this note, we consider a Fourier integral operator defined by align* T, ₀f (x) =ₑ^₍e^i (x, ) a (x, ) f () d, align* where a is the amplitude, and is the phase. Let 0 1, n 2 or 0<1, n=1 and mₚ=-np+ (n-1) \ 12, \. If a belongs to the forbidden H\"ormander class S^mₚ, ₁ and ^2 satisfies the strong non-degeneracy condition, then for any nn+1<p 1, we can show that the Fourier integral operator T, ₀ is bounded from the local Hardy space hᵖ to Lᵖ. Furthermore, if a has compact support in variable x, then we can extend this result to 0<p 1. As S^mₚ, S^mₚ, ₁ for any 0 1, our result supplements and improves upon recent theorems proved by Staubach and his collaborators for a S^m, when is close to 1. As an important special case, when n 2, we show that T, ₀ is bounded from H¹ to L¹ if a S^ (1-n) /2₁, ₁ which is a generalization of the well-known Seeger-Sogge-Stein theorem for a S^ (1-n) /2₁, ₀. This result is false when n=1 and a S^0₁, ₁.