LAUSR

Abstract In this note, we consider the Fourier integral operator T ϕ , a ⁢ f ⁢ ( x ) = ∫ R n e i ⁢ ϕ ⁢ ( x , ξ ) ⁢ a ⁢ ( x , ξ ) ⁢ f ^ ⁢ ( ξ ) ⁢ d ξ T_{\phi,a}f(x)=\int_{\mathbb{R}^{n}}e^{i\phi(x,\xi)}a(x,\xi)\hat{f}(\xi)\,d\xi with 𝑎 in the forbidden Hörmander class S ρ , 1 m S^{m}_{\rho,\smash{1}} and ϕ ∈ Φ 2 \phi\in\Phi^{2} satisfying the strong non-degeneracy condition. For 0 ≤ ρ ≤ 1 0\leq\rho\leq 1 , set m ⁢ ( n , ρ , p ) = − ( n − ρ ) ⁢ ( 1 2 − 1 p ) + n p ⁢ ( ρ − 1 ) . m(n,\rho,p)=-(n-\rho)\biggl{(}\frac{1}{2}-\frac{1}{p}\biggr{)}+\frac{n}{p}(\rho-1). When 2 ≤ p ≤ ∞ 2\leq p\leq\infty , 1 ≤ q ≤ ∞ 1\leq q\leq\infty and s > m − m ⁢ ( n , ρ , p ) s>m-m(n,\rho,p) , we show that T ϕ , a T_{\phi,a} is bounded from the Besov space B p , q s B_{p,q}^{s} to B p , q s − m + m ⁢ ( n , ρ , p ) B_{p,q}^{s-m+\smash{m(n,\rho,p)}} . This result is a generalization of some theorems proved by Stein, Meyer, Runst and Bourdaud for the pseudo-differential operator T a T_{a} with a ∈ S 1 , 1 m a\in S^{m}_{1,1} , and indices 𝑠 and m ⁢ ( n , ρ , p ) m(n,\rho,p) are sharp in some cases.