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Advances in AnalysisThe Legacy of Elias M. Stein$
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Advances in Analysis: The Legacy of Elias M. Stein

Charles Fefferman, Alexandru D. Ionescu, D.H. Phong, and Stephen Wainger

Print publication date: 2014

Print ISBN-13: 9780691159416

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691159416.001.0001

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Problems of Harmonic Analysis Related to Finite-Type Hypersurfaces in R3, and Newton Polyhedra Detlef Müller

Problems of Harmonic Analysis Related to Finite-Type Hypersurfaces in R3, and Newton Polyhedra Detlef Müller

Chapter:
(p.303) Chapter Thirteen Problems of Harmonic Analysis Related to Finite-Type Hypersurfaces in R3, and Newton Polyhedra Detlef Müller
Source:
Advances in Analysis
Author(s):
Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, Stephen Wainger, Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, Stephen Wainger
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691159416.003.0013

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.

Keywords:   harmonic analysis, finite-type hypersurfaces, Newton polyhedra, oscillatory integrals, convex hypersurfaces, maximal averages, spherical maximal function, Fourier restriction

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