Fourier integral operators on $$L^p({\mathbb {R}}^{n})$$ when $$2<p\le \infty $$
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In this paper, we deal with the L p L^{p} boundedness of rough Fourier integral operators T a , Ο T_{a,\varphi} with amplitude a β’ ( x , ΞΎ ) β L β β’ S Ο m a(x,\xi)\in L^{\infty}S_{\rho}^{m} and phase function Ο β’ ( x , ΞΎ ) β L β β’ Ξ¦ 2 \varphi(x,\xi)\in{L^{\infty}}{\Phi^{2}} which satisfies a measure condition. We show that T a , Ο T_{a,\varphi} is bounded on L p L^{p} for 1 β€ p β€ β 1\leq p\leq\infty if m < n β’ ( Ο β 1 ) p β Ο β’ ( n β 1 ) 2 β’ p m<\frac{n(\rho-1)}{p}-\frac{\rho(n-1)}{2p} when 1 β€ p β€ 2 1\leq p\leq 2 or m < n β’ ( Ο β 1 ) 2 β Ο β’ ( n β 1 ) 2 β’ ( 1 β 1 p ) m<\frac{n(\rho-1)}{2}-\frac{\rho(n-1)}{2}(1-\frac{1}{p}) when 2 β€ p β€ β 2\leq p\leq\infty . Our main results extend and improve some known results about L p L^{p} boundedness of Fourier integral operators.
In this paper, we deal with the L p L^{p} boundedness of rough Fourier integral operators T a , Ο T_{a,\varphi} with amplitude a β’ ( x , ΞΎ ) β L β β’ S Ο m a(x,\xi)\in L^{\infty}S_{\rho}^{m} and phase function Ο β’ ( x , ΞΎ ) β L β β’ Ξ¦ 2 \varphi(x,\xi)\in{L^{\infty}}{\Phi^{2}} which satisfies a measure condition. We show that T a , Ο T_{a,\varphi} is bounded on L p L^{p} for 1 β€ p β€ β 1\leq p\leq\infty if m < n β’ ( Ο β 1 ) p β Ο β’ ( n β 1 ) 2 β’ p m<\frac{n(\rho-1)}{p}-\frac{\rho(n-1)}{2p} when 1 β€ p β€ 2 1\leq p\leq 2 or m < n β’ ( Ο β 1 ) 2 β Ο β’ ( n β 1 ) 2 β’ ( 1 β 1 p ) m<\frac{n(\rho-1)}{2}-\frac{\rho(n-1)}{2}(1-\frac{1}{p}) when 2 β€ p β€ β 2\leq p\leq\infty . Our main results extend and improve some known results about L p L^{p} boundedness of Fourier integral operators.
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