Research Catalog

Title

Fourier integrals in classical analysis / Christopher D. Sogge.

Author

Sogge, Christopher D. (Christopher Donald), 1960-

Publication

Cambridge ; New York : Cambridge University Press, 1993.

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Holdings

Details

Description

x, 236 pages; 24 cm.

Summary

• Fourier Integrals in Classical Analysis is an advanced monograph 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. Using microlocal analysis, the author studies problems involving maximal functions and Riesz means using the so-called half-wave operator.
• This self-contained book starts with a rapid review of important topics in Fourier analysis. The author then presents the necessary tools from microlocal analysis, and goes on to give a proof of the sharp Weyl formula which he then modifies to give sharp estimates for the size of eigenfunctions on compact manifolds. Finally, at the end, the tools that have been developed are used to study the regularity properties of Fourier integral operators, culminating in the proof of local smoothing estimates and their applications to singular maximal theorems in two and more dimensions. This book will be of vital interest to advanced graduate students and research mathematicians working in analysis.

Series Statement

Cambridge tracts in mathematics ; 105

Uniform Title

Cambridge tracts in mathematics ; 105.

Subject

• Fourier series
• Fourier integral operators

Note

• Errata slip included.

Bibliography (note)

• Includes bibliographical references and indexes.

Contents

• 0. Background. 0.1. Fourier Transform. 0.2. Basic Real Variable Theory. 0.3. Fractional Integration and Sobolev Embedding Theorems. 0.4. Wave Front Sets and the Cotangent Bundle. 0.5. Oscillatory Integrals -- 1. Stationary Phase. 1.1. Stationary Phase Estimates. 1.2. Fourier Transform of Surface-carried Measures -- 2. Non-homogeneous Oscillatory Integral Operators. 2.1. Non-degenerate Oscillatory Integral Operators. 2.2. Oscillatory Integral Operators Related to the Restriction Theorem. 2.3. Riesz Means in R[superscript n]. 2.4. Kakeya Maximal Functions and Maximal Riesz Means in R[superscript 2] -- 3. Pseudo-differential Operators. 3.1. Some Basics. 3.2. Equivalence of Phase Functions. 3.3. Self-adjoint Elliptic Pseudo-differential Operators on Compact Manifolds -- 4. The Half-wave Operator and Functions of Pseudo-differential Operators. 4.1. The Half-wave Operator. 4.2. The Sharp Weyl Formula. 4.3. Smooth Functions of Pseudo-differential Operators.
• 5. L[superscript p] Estimates of Eigenfunctions. 5.1. The Discrete L[superscript 2] Restriction Theorem. 5.2. Estimates for Riesz Means. 5.3. More General Multiplier Theorems -- 6. Fourier Integral Operators. 6.1. Lagrangian Distributions. 6.2. Regularity Properties. 6.3. Spherical Maximal Theorems: Take 1 -- 7. Local Smoothing of Fourier Integral Operators. 7.1. Local Smoothing in Two Dimensions and Variable Coefficient Kakeya Maximal Theorems. 7.2. Local Smoothing in Higher Dimensions. 7.3. Spherical Maximal Theorems Revisited -- Appendix: Lagrangian Subspaces of T*R[superscript n].

ISBN

0521434645

LCCN

92024678

OCLC

• 26259514
• ocm26259514

Owning Institutions

Columbia University Libraries