Cambridge University Press eBooks· 2017· Book chapter
Fourier Integrals in Classical Analysis
- 517citations
- 2017year
Short summary
This advanced monograph explores the connection between singularity propagation for the wave equation and classical analysis, using microlocal analysis and the half-wave operator to study maximal functions and Riesz means.
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Key points
- Connects wave equation singularity propagation with classical analysis via microlocal methods.
- Utilizes the half-wave operator to analyze maximal functions and Riesz means.
- Includes a self-contained review of Fourier analysis and microlocal analysis tools.
- Features new chapters on Hörmander's propagation of singularities theorem and Kakeya conjecture results.
AI-generated from the title and abstract; the full text is not read.
Abstract
This advanced monograph is concerned with modern treatments of central problems in harmonic analysis. The main theme of the book is the interplay between ideas used to study the propagation of singularities for the wave equation and their counterparts in classical analysis. In particular, the author uses microlocal analysis to study problems involving maximal functions and Riesz means using the so-called half-wave operator. To keep the treatment self-contained, the author begins with a rapid review of Fourier analysis and also develops the necessary tools from microlocal analysis. This second edition includes two new chapters. The first presents Hörmander's propagation of singularities theorem and uses this to prove the Duistermaat–Guillemin theorem. The second concerns newer results related to the Kakeya conjecture, including the maximal Kakeya estimates obtained by Bourgain and Wolff.
The authors' abstract, as published at the source. Cambridge University Press eBooks, 2017 · DOI ↗
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Field: Applied Mathematics
Applied MathematicsMathematics