| Description |
1 online resource |
| Series |
Annals of mathematics studies ; number 194 |
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Annals of mathematics studies ; no. 194.
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| Contents |
880-01 Frontmatter -- Contents -- Chapter 1. Introduction -- Chapter 2. Auxiliary Results -- Chapter 3. Reduction to Restriction Estimates near the Principal Root Jet -- Chapter 4. Restriction for Surfaces with Linear Height below 2 -- Chapter 5. Improved Estimates by Means of Airy-Type Analysis -- Chapter 6. The Case When h -- Chapter 7. How to Go beyond the Case h -- Chapter 8. The Remaining Cases Where m = 2 and B = 3 or B = 4 -- Chapter 9. Proofs of Propositions 1.7 and 1.17 -- Bibliography -- Index |
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880-01/Grek Cover -- Title -- Copyright -- Dedication -- Contents -- Chapter 1 Introduction -- 1.1 Newton Polyhedra Associated with ϕ, Adapted Coordinates, and Uniform Estimates for Oscillatory Integrals with Phase ϕ -- 1.2 Fourier Restriction in the Presence of a Linear Coordinate System That Is Adapted to ϕ -- 1.3 Fourier Restriction When No Linear Coordinate System Is Adapted to ϕ-the Analytic Case -- 1.4 Smooth Hypersurfaces of Finite Type, Condition (R), and the General Restriction Theorem -- 1.5 An Invariant Description of the Notion of r-Height. -- 1.6 Organization of the Monograph and Strategy of Proof -- Chapter 2 Auxiliary Results -- 2.1 Van der Corput-Type Estimates -- 2.2 Airy-Type Integrals -- 2.3 Integral Estimates of van der Corput Type -- 2.4 Fourier Restriction via Real Interpolation -- 2.5 Uniform Estimates for Families of Oscillatory Sums -- 2.6 Normal Forms of ϕ under Linear Coordinate Changes When hlin(ϕ) <2 -- Chapter 3 Reduction to Restriction Estimates near the Principal Root Jet -- Chapter 4 Restriction for Surfaces with Linear Height below 2 -- 4.1 Preliminary Reductions by Means of Littlewood-Paley Decompositions -- 4.2 Restriction Estimates for Normalized Rescaled Measures When 2̂2j δ3 ≲ 1 -- Chapter 5 Improved Estimates by Means of Airy-Type Analysis -- 5.1 Airy-Type Decompositions Required for Proposition 4.2(c) -- 5.2 The Endpoint in Proposition 4.2(c): Complex Interpolation -- 5.3 Proof of Proposition 4.2(a), (b): Complex Interpolation -- Chapter 6 The Case When hlin(ϕ) ≥ 2: Preparatory Results -- 6.1 The First Domain Decomposition -- 6.2 Restriction Estimates in the Transition Domains El When hlin(ϕ) ≥ 2 -- 6.3 Restriction Estimates in the Domains Dl, l <lpr, When hlin(ϕ)≥2 -- 6.4 Restriction Estimates in the Domain Dpr When hlin(ϕ)≥ 5 |
| Summary |
This is the first book to present a complete characterization of Stein-Tomas type Fourier restriction estimates for large classes of smooth hypersurfaces in three dimensions, including all real-analytic hypersurfaces. The range of Lebesgue spaces for which these estimates are valid is described in terms of Newton polyhedra associated to the given surface. Isroil Ikromov and Detlef Müller begin with Elias M. Stein's concept of Fourier restriction and some relations between the decay of the Fourier transform of the surface measure and Stein-Tomas type restriction estimates. Varchenko's ideas relating Fourier decay to associated Newton polyhedra are briefly explained, particularly the concept of adapted coordinates and the notion of height. It turns out that these classical tools essentially suffice already to treat the case where there exist linear adapted coordinates, and thus Ikromov and Müller concentrate on the remaining case. Here the notion of r-height is introduced, which proves to be the right new concept. They then describe decomposition techniques and related stopping time algorithms that allow to partition the given surface into various pieces, which can eventually be handled by means of oscillatory integral estimates. Different interpolation techniques are presented and used, from complex to more recent real methods by Bak and Seeger. Fourier restriction plays an important role in several fields, in particular in real and harmonic analysis, number theory, and PDEs. This book will interest graduate students and researchers working in such fields |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Print version record |
| Subject |
Hypersurfaces.
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Polyhedra.
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Surfaces, Algebraic.
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Fourier analysis.
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Fourier Analysis
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polyhedra.
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MATHEMATICS -- Geometry -- General.
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Fourier analysis
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Hypersurfaces
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Polyhedra
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Surfaces, Algebraic
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| Form |
Electronic book
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| Author |
Müller, Detlef, 1954- author.
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| ISBN |
9781400881246 |
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1400881242 |
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