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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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Reduction to Restriction Estimates near the Principal Root Jet

Reduction to Restriction Estimates near the Principal Root Jet

Chapter:
(p.50) Chapter Three Reduction to Restriction Estimates near the Principal Root Jet
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/9780691170541.003.0003

This chapter shows that one may reduce the desired Fourier restriction estimate to a piece Ssubscript Greek small letter psi of the surface S lying above a small, “horn-shaped” neighborhood Dsubscript Greek small letter psi of the principal root jet ψ‎, on which ∣x₂ − ψ‎(x₁)∣ ≤ ε‎xᵐ₁. Here, ε‎ > 0 can be chosen as small as one wishes. The proof then provides the opportunity to introduce some of the basic tools which will be applied frequently, such as dyadic domain decompositions, rescaling arguments based on the dilations associated to a given edge of the Newton polyhedron, in combination with Greenleaf's restriction and Littlewood–Paley theory, hence summing the estimates that have been obtained for the dyadic pieces.

Keywords:   principal root jet, dyadic domain decompositions, Newton polyhedron, Greenleaf's restriction, Littlewood–Paley theory, Fourier restriction estimate

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