This book, "Introduction to Pseudodifferential and Fourier Integral Operators," by François Treves, is part of the University Series in Mathematics. Volume 1 focuses on pseudodifferential operators, while Volume 2 covers Fourier integral operators. The book aims to provide a practical and accessible introduction to these operators, emphasizing their applications and theoretical foundations. In Volume 1, Treves covers the following key topics: 1. **Standard Pseudodifferential Operators**: Discusses parametrices of elliptic and hypoelliptic equations, fundamental solutions of strongly hyperbolic Cauchy problems, and nonhyperbolic forward Cauchy problems. 2. **Special Topics and Applications**: Explores compact pseudodifferential operators and their applications. 3. **Application to Boundary Problems for Elliptic Equations**: Details the construction and properties of operators that transfer boundary problems to interior problems, focusing on elliptic boundary problems. 4. **Pseudodifferential Operators of Type $(\rho, \delta)$**: Introduces these operators and their applications, including the Calderon–Vaillancourt theorem and the sharp Gårding inequality. 5. **Analytic Pseudodifferential Operators**: Covers analyticity in the base and cotangent bundle, pseudoanalytic and analytic amplitudes, and microlocalization. The book is informally written and includes numerous examples and applications to motivate the theoretical concepts. It also provides a detailed background on differential and symplectic geometry, as well as Fourier distributions and global Fourier integral operators. The prerequisites for the material vary by chapter, but generally include standard real and complex analysis, functional analysis, and distribution theory. The book is intended for researchers and students interested in the practical and theoretical aspects of pseudodifferential and Fourier integral operators, with a focus on their applications in various fields such as boundary value problems, microlocal analysis, and geometric optics.This book, "Introduction to Pseudodifferential and Fourier Integral Operators," by François Treves, is part of the University Series in Mathematics. Volume 1 focuses on pseudodifferential operators, while Volume 2 covers Fourier integral operators. The book aims to provide a practical and accessible introduction to these operators, emphasizing their applications and theoretical foundations. In Volume 1, Treves covers the following key topics: 1. **Standard Pseudodifferential Operators**: Discusses parametrices of elliptic and hypoelliptic equations, fundamental solutions of strongly hyperbolic Cauchy problems, and nonhyperbolic forward Cauchy problems. 2. **Special Topics and Applications**: Explores compact pseudodifferential operators and their applications. 3. **Application to Boundary Problems for Elliptic Equations**: Details the construction and properties of operators that transfer boundary problems to interior problems, focusing on elliptic boundary problems. 4. **Pseudodifferential Operators of Type $(\rho, \delta)$**: Introduces these operators and their applications, including the Calderon–Vaillancourt theorem and the sharp Gårding inequality. 5. **Analytic Pseudodifferential Operators**: Covers analyticity in the base and cotangent bundle, pseudoanalytic and analytic amplitudes, and microlocalization. The book is informally written and includes numerous examples and applications to motivate the theoretical concepts. It also provides a detailed background on differential and symplectic geometry, as well as Fourier distributions and global Fourier integral operators. The prerequisites for the material vary by chapter, but generally include standard real and complex analysis, functional analysis, and distribution theory. The book is intended for researchers and students interested in the practical and theoretical aspects of pseudodifferential and Fourier integral operators, with a focus on their applications in various fields such as boundary value problems, microlocal analysis, and geometric optics.