Special Fourier integral operators of types I and II with function-variable symbols: definition, relation to metaplectic transform, and Heisenberg’s uncertainty principles

This study devotes to Heisenberg’s uncertainty principles for Fourier integral operators of types I and II with function-variable symbols, i.e., the symbol σ∈S1⊗vs∞(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma \in S_{1\otimes v_s}^{\infty }(\mathbb {R}^N)$$\end{document} of the type I Fourier integral operator is only w\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{w}$$\end{document}-dependent and the symbol τ∈S1⊗vs∞(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau \in S_{1\otimes v_s}^{\infty }(\mathbb {R}^N)$$\end{document} of the type II Fourier integral operator is only y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{y}$$\end{document}-dependent. These two special Fourier integral operators are abbreviated as the FIO-I-FV and FIO-II-FV, respectively. We disclose an equivalence relation between the FIO-I-FV and the classical metaplectic transform, as well as the FIO-II-FV and the metaplectic transform, based on which we employ various versions of Heisenberg’s uncertainty principles for the metaplectic transform, ranging from the general metaplectic transform of real-valued functions to some specific (e.g., the orthogonal, the orthonormal, the minimum eigenvalue commutative, the maximum eigenvalue commutative) metaplectic transforms of complex-valued functions, in the establishment of the corresponding results for the FIO-I-FV and FIO-II-FV.