Lp Boundedness of Fourier Integral Operators in the Class S0,0

We first prove the L2-boundedness of a Fourier integral operator where it’s symbol a∈S12,120(ℝn×ℝn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a \in S_{{1 \over 2},{1 \over 2}}^0\left( {{\mathbb{R}^n} \times {\mathbb{R}^n}} \right)$$\end{document} 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 Lp-boundedness of Fourier integral operators if (1) the symbol a∈Λkm0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a \in \Lambda _k^{{m_0}}$$\end{document} and Supp a = E × ℝn, with E a compact set of ℝn(m0=−|1p−12|n,1<p≤2,k>n2;2<p<∞,k>np)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^n}\left( {{m_0} = - \left| {{1 \over p} - {1 \over 2}} \right|n,\,1 < p \le 2,\,\,k > {n \over 2};\,2 < p < \infty ,\,k > {n \over p}} \right)$$\end{document}, (2) the symbol a∈Λ0,k,k′m0(m0=−|1p−12|n,1<p≤2,k>n2,k′>np;2<p<∞,k>np,k′>n2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a \in \Lambda _{0,k,{k^\prime }}^{{m_0}}\left( {{m_0} = - \left| {{1 \over p} - {1 \over 2}} \right|n,\,1 < p \le 2,\,\,k > {n \over 2},\,{k^\prime } > {n \over p};\,2 < p < \infty ,k > {n \over p},\,\,{k^\prime } > {n \over 2}} \right)$$\end{document} with the phase function S(x, ξ) = xξ + h(x, ξ),x,ξ ∈ ℝn non-degenerate, satisfying certain conditions and ∂ξh ∈ S1,00(ℝn × ℝn), or (3) the symbol a∈Λ0,k,k′m0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a \in \Lambda _{0,k,{k^\prime }}^{{m_0}}$$\end{document}, the requirements for m0, k, k′ are the same as in (2), and ∂ξh is not in S1,00(ℝn × ℝn) but the phase function S is non-degenerate, satisfies certain conditions and is positively homogeneous in ξ-variables.