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The Fourier transform on \(\mathbb {R}^n\) plays a central role in the development of microlocal analysis. Schwartz functions and tempered distributions are the most natural setting for it; they are introduced in Sections 2.1–2.2. The most important function spaces for us will be the \(L^2\) -based Sobolev spaces \(H^s(\mathbb {R}^n)\) , \(s\in \mathbb {R}\) ; these are recalled in Section 2.3. Distributions arise in this book not only as solutions of PDEs but also as integral kernels of linear operators.

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Schwartz Functions and Tempered Distributions

  • 章节

  • 发表日期：

    2025年

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  • Peter Hintz

摘要

The Fourier transform on \(\mathbb {R}^n\) plays a central role in the development of microlocal analysis. Schwartz functions and tempered distributions are the most natural setting for it; they are introduced in Sections 2.1–2.2. The most important function spaces for us will be the \(L^2\) -based Sobolev spaces \(H^s(\mathbb {R}^n)\) , \(s\in \mathbb {R}\) ; these are recalled in Section 2.3. Distributions arise in this book not only as solutions of PDEs but also as integral kernels of linear operators.