Endpoint regularity of general Fourier integral operators

Let n≥ 1,0<ρ<1, max\ρ,1-ρ\≤ δ≤ 1 and m₁=ρ-n+(n-1)min\ 12,ρ\+1-δ/2. If the amplitude a belongs to the H\"{o}rmander class Sm₁ρ,δ and φ∈ Φ² satisfies the strong non-degeneracy condition, then we prove that the following Fourier integral operator Tφ,a defined by {align*} Tφ,af(x)=∫_{Rⁿ}eiφ(x,ξ)a(x,ξ){f}(ξ)dξ, {align*} is bounded from the local Hardy space h¹(Rⁿ) to L¹(Rⁿ). As a corollary, we can also obtain the corresponding Lᵖ(Rⁿ)-boundedness when $1<p<2$. These theorems are rigorous improvements on the recent works of Staubach and his collaborators. When 0≤ ρ≤ 1,δ≤ max\ρ,1-ρ\, by using some similar techniques in this note, we can get the corresponding theorems which coincide with the known results.