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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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The Remaining Cases Where m = 2 and B = 3 or B = 4

The Remaining Cases Where m = 2 and B = 3 or B = 4

Chapter:
(p.181) Chapter Eight The Remaining Cases Where m = 2 and B = 3 or B = 4
Source:
Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra (AM-194)
Author(s):

Isroil A. Ikromov

Detlef Müller

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691170541.003.0008

This chapter discusses the remaining cases for l = 1. With the same basic approach as in Chapter 5, the chapter again performs an additional dyadic frequency domain decomposition related to the distance to a certain Airy cone. This is needed in order to control the integration with respect to the variable x₁ in the Fourier integral defining the Fourier transform of the complex measures ν‎subscript Greek small letter delta superscript Greek small letter lamda. It first applies a suitable translation in the x₁-coordinate before performing a more refined analysis of the phase Φ‎superscript Music sharp sign. The chapter then treats the case where λ‎ρ‎(̃‎δ‎) ≲ 1 and hereafter deals with the case where λ‎ρ‎(̃‎δ‎) ≲ 1 and B = 4. Finally, the chapter turns to the case where B = 3.

Keywords:   dyadic decompositions, refined Airy-type analysis, Airy-type analysis, Fourier transform, Fourier integral, Airy cone, open cases

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