Fourier integral operators on Hardy spaces with Hormander class

In this note, we consider a Fourier integral operator defined by {align*} Tφ,af(x)=∫_{Rⁿ}eiφ(x,ξ)a(x,ξ){f}(ξ)dξ, {align*} where a is the amplitude, and φ is the phase. Let 0≤ρ≤ 1,n≥ 2 or 0≤ρ<1,n=1 and mₚ=ρ-n/p+(n-1)min\ 12,ρ\. If a belongs to the forbidden H\"{o}rmander class Smₚρ,1 and φ∈ Φ² satisfies the strong non-degeneracy condition, then for any n/n+1<p≤ 1, we can show that the Fourier integral operator Tφ,a 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 Smₚρ,δ⊂ Smₚρ,1 for any 0≤ δ≤ 1, our result supplements and improves upon recent theorems proved by Staubach and his collaborators for a∈ Sᵐρ,δ when δ is close to 1. As an important special case, when n≥ 2, we show that Tφ,a 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∈ S⁰1,1.