On 𝐿𝑝 boundedness of rough Fourier integral operators

Abstract

In this paper, we deal with the L p L^{p} boundedness of rough Fourier integral operators T a , φ T_{a,\varphi} with amplitude a ⁢ ( x , ξ ) ∈ L ∞ ⁢ S ρ m a(x,\xi)\in L^{\infty}S_{\rho}^{m} and phase function φ ⁢ ( x , ξ ) ∈ L ∞ ⁢ Φ 2 \varphi(x,\xi)\in{L^{\infty}}{\Phi^{2}} which satisfies a measure condition. We show that T a , φ T_{a,\varphi} is bounded on L p L^{p} for 1 ≤ p ≤ ∞ 1\leq p\leq\infty if m < n ⁢ ( ρ − 1 ) p − ρ ⁢ ( n − 1 ) 2 ⁢ p m<\frac{n(\rho-1)}{p}-\frac{\rho(n-1)}{2p} when 1 ≤ p ≤ 2 1\leq p\leq 2 or m < n ⁢ ( ρ − 1 ) 2 − ρ ⁢ ( n − 1 ) 2 ⁢ ( 1 − 1 p ) m<\frac{n(\rho-1)}{2}-\frac{\rho(n-1)}{2}(1-\frac{1}{p}) when 2 ≤ p ≤ ∞ 2\leq p\leq\infty . Our main results extend and improve some known results about L p L^{p} boundedness of Fourier integral operators.