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Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra (AM-194)$
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Isroil A. Ikromov and Detlef Müller

Print publication date: 2016

Print ISBN-13: 9780691170541

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691170541.001.0001

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(p.1) Chapter One Introduction
Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra (AM-194)

Isroil A. Ikromov

Detlef Müller

Isroil A. Ikromov

Detlef Müller

Princeton University Press

This chapter discusses the Fourier restriction. The Fourier restriction problem presents one important instance of a wide circle of related problems, such as the boundedness properties of Bochner Riesz means, dimensional properties of Kakeya type sets, smoothing effects of averaging over time intervals for solutions to the wave equation (or more general dispersive equations), or the study of maximal averages along hypersurfaces. The common question underlying all these problems asks for the understanding of the interplay between the Fourier transform and properties of thin sets in Euclidean space, for instance geometric properties of subvarieties. The chapter builds on previous discussions on Fourier restrictions, and presents a brief overview of the succeeding chapters.

Keywords:   Fourier restriction, Fourier restriction problem, Fourier transform, thin sets, Newton polyhedra, smooth hypersurfaces, r-height

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