Endpoint regularity of general Fourier integral operators

Let n 1, 0<<1, \, 1-\ 1 and m₁=-n+ (n-1) \ 12, \+ 1-2. If the amplitude a belongs to the H\"ormander class S^m₁, and ^2 satisfies the strong non-degeneracy condition, then we prove that the following Fourier integral operator T, ₀ defined by align* T, ₀f (x) =ₑ^₍e^i (x, ) a (x, ) f () d, 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 1, \, 1-\, by using some similar techniques in this note, we can get the corresponding theorems which coincide with the known results.