Global Fourier Integral Operators in the Plane and the Square Function

We prove the local smoothing estimate for general Fourier integral operators with phase function of the form ϕ(x,t,ξ)=x·ξ+tq(ξ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (x,t,\xi )=x\cdot \xi + t \, q(\xi )$$\end{document}, with q∈C∞(R2\{0})\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q \in C^\infty ( {\mathbb {R}}^2 \setminus \{0\} )$$\end{document}, homogeneous of degree one, and amplitude functions in the symbol class of order m≤0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \le 0$$\end{document}. The result is global in the space variable, and also improves our previous work in this direction (Manna et al (in: Georgiev et al., Advances in harmonic analysis and partial differential equations, Trends in Mathematics. Birkhäuser, Cham, pp. 1–35, 2020)). The approach involves a reduction to operators with amplitude function depending only on the covariable, and a new estimate for square function based on angular decomposition.