The Boundedness Of Some Classes Of Fourier Integral Operators On Lebesgue Space

In this dissertation,we focus on the study of some classes of Fourier integral operator’s boundedness on Lebesgue space.And mainly consider the following four questions:When a ∈ L∞Sρm,φ∈ L∞ Φ2 and φ satisfies RND condition,we consider the L2 boundesness of Tφ,a;When a ∈ L∞Sρm,φ∈ L∞Φ2 and we assume φ satisfies a more general condition than RND condition,we consider the L2 boundesness of Tφ,a;When a ∈ L∞Sρm,φ(?) L∞Φ2,φ satisfies some certain inhomogeneous condition and measure condition,we consider this class of inhomogeneous Fourier integral operator’s L2 and L∞ boundesness;When a ∈ Sρδm,φ(?) Φ2,φ satisfies some smooth inhomogeneous condition and SND condition,we consider this class of smooth inhomogeneous Fourier integral operator’s L2 boundedness.The dissertation is divided into five chapters.In Charter 1,we introduce the background and main results of the dissertation.In Chapter 2,when a ∈ L∞Sρm,φ∈ L∞Φ2 and φ satisfies the RND condition,we get an L∞ boundedness result,then by the”Fefferman-Stein" interpolation theorem,we also get the boundedness of on Lp(2≤p≤∞),which improves some results of Dos Santos Ferreira and Staubach.In Chapter 3,assuming the phase function φ satisfies a certain more general condition than RND condition.That is,when a ∈ L∞Sρm,φ∈ L∞Φ2 and for any ξ,y∈ Rn,r>0,φsatisfies |{x:|▽ξφ(x,ξ)-y|≤r}|≤C(rn-1+rn).For this class of Fourier integral operator,we get an L2 boundedness result.Moreover,the almost everywhere convergence of wave operator can be associated with this class of Fourier integral operator.In Chapter 4,we consider a class of inhomogeneous rough Fourier integral operator.For this class of Fourier integral operator,the phase function φ no longer belongs to L∞Φ2,and φ satisfies some certain inhomogeneous condition,that is,for any k≥ 2 and some δ ∈(0,1],φ satisfies supξ∈Rn\{0} |ξ|k-δ‖▽ξkφ(·,ξ)‖L∞<∞ and some certain measure condition.When a ∈ L∞ Sρm and φ satisfies the above conditions,we establish the L2 boundedness of this class of Fourier integral operator.One motivation to study inhomogeneous rough Fourier integral operator is to study fraction Schrodinger operator.In fact,the almost everywhere convergence of fraction Schrodinger operator can be associated with this class of inhomogeneous rough Fourier integral operator.Moreover,we also get an L∞ boundedness result,then by the " Fefferman-Stein" interpolation theorem,we get the Lp(2≤p≤∞)boundedness result.In Chapter 5,we consider a class of inhomogeneous smooth Fourier integral operator.When a∈Sρ,δm,and for any non-negative integer k,s,k+s≥ 2,and some ε∈(0,1],φsatisfies sup(x,ξ)∈Rn×Rn\{0}|ξ|k-ε|▽ξk▽xsφ(x,ξ)|<∞ and SND condition,we prove the L2 boundedness.For a ∈ L∞Sρm,φ satisfies the same conditions,we also get the L2 boundedness.