这本书提供了一个连贯的，自给自足的介绍，在自然几何环境中分析偏微分方程的中心主题。主要主题是分析偏微分方程的相空间分析，以及分布和超函数的傅里叶-布罗斯-伊格尔尼泽（FBI）变换，以及对存在性和正则性问题的应用。
本书从建立解析偏微分方程的基本性质开始，首先从柯西-科瓦列夫斯卡亚定理开始，然后通过解析泛函对超函数的方法进行综合概述，首先是在欧几里德空间中，然后在分析流形上布置几何背景。进一步的主题包括Lojaciewicz不等式的证明和解析函数对分布的划分，Frobenius叶理和长野叶理的详细描述，以及eikonal方程对合系统的Hamilton–Jacobi解。然后，读者通过伪微分学进入微观局部分析领域，伪微分学在基本层面上介绍，然后是傅里叶积分算子，包括具有复相函数的算子（a la Sjöstrand）。这最终导致了对主要类型的解析微分（以及后来的伪微分）方程（分布或超函数）解的存在性和规律性的深入讨论，举例说明了之前介绍的所有概念和工具的有用性。最后三章讨论了将结果推广到这些方程的超（或欠）定系统的可能性——一个开放问题的聚宝盆。
这本书提供了以前仅限于研究文章的大量材料的统一展示。与现有的专著相比，本书的方法是解析的，而不是代数的，并且像层上同调、解析变种的分层理论和辛几何这样的工具很少使用，并根据需要进行介绍。本书的前半部分主要是教学性的，供高级研究生和博士后使用，而第二部分，更专业的部分是供研究人员参考。
Analytic Partial Differential Equations by François Treves
This book provides a coherent, self-contained introduction to central topics of Analytic Partial Differential Equations in the natural geometric setting. The main themes are the analysis in phase-space of analytic PDEs and the Fourier–Bros–Iagolnitzer (FBI) transform of distributions and hyperfunctions, with application to existence and regularity questions.
The book begins by establishing the fundamental properties of analytic partial differential equations, starting with the Cauchy–Kovalevskaya theorem, before presenting an integrated overview of the approach to hyperfunctions via analytic functionals, first in Euclidean space and, once the geometric background has been laid out, on analytic manifolds. Further topics include the proof of the Lojaciewicz inequality and the division of distributions by analytic functions, a detailed description of the Frobenius and Nagano foliations, and the Hamilton–Jacobi solutions of involutive systems of eikonal equations. The reader then enters the realm of microlocal analysis, through pseudodifferential calculus, introduced at a basic level, followed by Fourier integral operators, including those with complex phase-functions (à la Sjöstrand). This culminates in an in-depth discussion of the existence and regularity of (distribution or hyperfunction) solutions of analytic differential (and later, pseudodifferential) equations of principal type, exemplifying the usefulness of all the concepts and tools previously introduced. The final three chapters touch on the possible extension of the results to systems of over- (or under-) determined systems of these equations—a cornucopia of open problems.
This book provides a unified presentation of a wealth of material that was previously restricted to research articles. In contrast to existing monographs, the approach of the book is analytic rather than algebraic, and tools such as sheaf cohomology, stratification theory of analytic varieties and symplectic geometry are used sparingly and introduced as required. The first half of the book is mainly pedagogical in intent, accessible to advanced graduate students and postdocs, while the second, more specialized part is intended as a reference for researchers.
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