*Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, Stephen Wainger, Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, and Stephen Wainger (eds)*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159416
- eISBN:
- 9781400848935
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159416.003.0013
- Subject:
- Mathematics, Numerical Analysis

This chapter presents three sets of problems and explains how these questions can be answered in an (almost) complete way in terms of Newton polyhedra associated to the given surface S (here, a ...
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This chapter presents three sets of problems and explains how these questions can be answered in an (almost) complete way in terms of Newton polyhedra associated to the given surface S (here, a smooth, finite type hypersurface in R³ with Riemannian surface measure dσ). The first problem is a classical question about estimates for oscillatory integrals, and there exists a huge body of results on it, in particular for convex hypersurfaces. The other two problems had first been formulated by Stein: the study of maximal averages along hypersurfaces has been initiated in Stein's work on the spherical maximal function, and also the idea of Fourier restriction goes back to him.Less

This chapter presents three sets of problems and explains how these questions can be answered in an (almost) complete way in terms of Newton polyhedra associated to the given surface *S* (here, a smooth, finite type hypersurface in **R**³ with Riemannian surface measure *dσ*). The first problem is a classical question about estimates for oscillatory integrals, and there exists a huge body of results on it, in particular for convex hypersurfaces. The other two problems had first been formulated by Stein: the study of maximal averages along hypersurfaces has been initiated in Stein's work on the spherical maximal function, and also the idea of Fourier restriction goes back to him.

*Jean Bourgain*

*Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, and Stephen Wainger (eds)*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159416
- eISBN:
- 9781400848935
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159416.003.0003
- Subject:
- Mathematics, Numerical Analysis

This chapter discusses the progress made towards problems originating from Stein's seminal paper, “Some problems in harmonic analysis.” It is by now well-known that the mapping properties of Fourier ...
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This chapter discusses the progress made towards problems originating from Stein's seminal paper, “Some problems in harmonic analysis.” It is by now well-known that the mapping properties of Fourier restriction operators to hypersurfaces in Rn and their variable coefficient generalizations are intimately related to questions of a combinatorial nature. Over recent years there has been quite a bit of research around these underlying issues. In some way, it became interdisciplinary with connections towards geometric measure theory, the theory of finite fields, incidence geometry, and mathematical computer science. While the central original problems remain unsolved, this line of research has produced many new results of independent interest, though the chapter focuses primarily on developments around the theory of oscillatory integrals.Less

This chapter discusses the progress made towards problems originating from Stein's seminal paper, “Some problems in harmonic analysis.” It is by now well-known that the mapping properties of Fourier restriction operators to hypersurfaces in **R**^{n} and their variable coefficient generalizations are intimately related to questions of a combinatorial nature. Over recent years there has been quite a bit of research around these underlying issues. In some way, it became interdisciplinary with connections towards geometric measure theory, the theory of finite fields, incidence geometry, and mathematical computer science. While the central original problems remain unsolved, this line of research has produced many new results of independent interest, though the chapter focuses primarily on developments around the theory of oscillatory integrals.

*Isroil A. Ikromov and Detlef Müller*

- Published in print:
- 2016
- Published Online:
- October 2017
- ISBN:
- 9780691170541
- eISBN:
- 9781400881246
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691170541.003.0002
- Subject:
- Mathematics, Geometry / Topology

This chapter compiles various auxiliary results; including variants of van der Corput-type estimates for one-dimensional oscillatory integrals and related sublevel estimates through “integrals of ...
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This chapter compiles various auxiliary results; including variants of van der Corput-type estimates for one-dimensional oscillatory integrals and related sublevel estimates through “integrals of sublevel type.” It also derives a straightforward variant of a beautiful real interpolation method that has been devised by Bak and Seeger and that will allow in some cases the replacement of the more classical complex interpolation methods in the proof of Stein–Tomas-type Fourier restriction estimates by substantially shorter arguments. Last, this chapter derives normal forms for phase functions φ of linear height < 2 for which no linear coordinate system adapted to φ does exist.Less

This chapter compiles various auxiliary results; including variants of van der Corput-type estimates for one-dimensional oscillatory integrals and related sublevel estimates through “integrals of sublevel type.” It also derives a straightforward variant of a beautiful real interpolation method that has been devised by Bak and Seeger and that will allow in some cases the replacement of the more classical complex interpolation methods in the proof of Stein–Tomas-type Fourier restriction estimates by substantially shorter arguments. Last, this chapter derives normal forms for phase functions φ of linear height < 2 for which no linear coordinate system adapted to φ does exist.