Boundedness Of Fourier Integral Operators On Lebesgue Space

Let Tφ,a be a Fourier integral operator(FIO in short)with symbol a(x,ξ)and phase functionφ(x,ξ).Whenφ(x,ξ)=<x,ξ>,the FIO referred to a pseudodifferential operator and the phase function satisfies the RND condition.This operator has a lot of applications in the field of harmonic analysis and is connected to many issues that come up with partial differential equations.On the other hand,a specific class of FIO also includes the maximal wave operator(?).The fact that φ(x,ξ)=<x,ξ>+t|ξ| ∈ L∞ Φ2 is obvious,but it does not satisfy the RND condition when 1 ≡t(x)∈ L∞。because the rank of the Hessian in question|ξ| is n-1.The main aims of this dissertation plan are to investigate the problems of Lp-boundedness of FIOs with RA a ∈ L∞ Sρm and RPFs φ ∈ L∞Φ2 for all 1 ≤p≤∞.In this Ph.D dissertation entitled "Boundedness of Fourier integral operators on Lebesgue space",carries out mainly the study of global boundedness of FIOs,with rough amplitude in Hormander class and rough phase function satisfies certain non-degeneracy conditions,on the Lebesgue space such as Lp(Rn)spaces for 1≤p≤∞.The content of this dissertation is arranged as follows.In Chapter 1,we provide a background and statement of the problem,the review of the literature,recall some definitions and terminology such as RA and RPFs,SND and RND condition and then give the tools in proving Lpboundedness of FIOs.In Chapter 2,we study the boundedness of FIO with RA a ∈ L∞Spm and RPFs φ ∈ L∞Φ2.We prove the global L1-boundedness for Tφ,a when 1/2<ρ≤1 and m<ρ-(n+1)/2.Additionally,we investigate the L1 boundedness of FIO Tφ,a with RA a ∈ L/∞Sρm and a novel class of RP φ(x,ξ).In this class,we extend the L∞Φ2 and non-degeneracy conditions to some generalized derivative estimation and some measure condition respectively.Moreover,the result proved can be used to establish the boundedness of the maximal wave operator.In chapter 3,we establish the Lp-boundedness of FIOs Tφ,a with rough symbol and RPF,which satisfies the novel class of RND condition.In this study,based on the conditions a ∈ L∞Sρm and φ ∈ L∞Φ2 in section 3.2 we prove the L1-boundedness if m<-n(1-ρ)-ρ when 0≤ρ≤1/2 or m<-(n+1)/2 when 1/2≤ρ≤1,section 3.3 is devoted to the proof of L2,section 3.4 is to study the L∞ and finally when 0≤ρ≤1/2,we show that Tφ,a is bounded on Lp provided m<n/2(ρ-1)-(n-1)/4-1/4(1-2/p)for 2 ≤p≤∞.In chapter 4,we summarize the conclusion of the above chapters studied in boundedness of FIOs on Lebesgue space.