摘要
We first prove the L~2-boundedness of a Fourier integral operator where it’s symbol a ∈S_(1/2,1/2)~0(R~n× R~n) and the phase function S is non-degenerate,satisfies certain conditions and may not be positively homogeneous in ξ-variables.Then we use the above property,Paley’s inequality,covering lemma of Calderon and Zygmund etc.,and obtain the L~p-boundedness of Fourier integral operators if(1) the symbol a ∈ Λ_(k)^(m_(0)) and Supp a = E×R~n,with E a compact set of R~n(m_(0) =-|1/p-1/2|n,1<p≤2,k>n/2;2<p<∞,k>n/p),(2) the symbol a ∈ Λ_(0,k,k’)^(m_(0))(m_(0) =-|1/p-1/2|n,1<p ≤2,k>n/2,k’>n/p;2<p<∞,k>n/p,k’>n/2) with the phase function S(x,ξ) = xξ + h(x,ξ),x,ξ ∈ R~n non-degenerate,satisfying certain conditions and ?ξ h ∈ S_(1,0)~0(R~n× R~n),or(3) the symbol a ∈ Λ_(0,k,k’)^(m_(0)),the requirements for m_(0),k,k’ are the same as in(2),and ?_(ξ)h is not in S_(1,0)~0(R~n× R~n) but the phase function S is non-degenerate,satisfies certain conditions and is positively homogeneous in ξ-variables.
