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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 Case When hlin(φ‎) ≥ 2: Preparatory Results

The Case When hlin(φ‎) ≥ 2: Preparatory Results

(p.105) Chapter Six The Case When hlin(φ‎) ≥ 2: Preparatory Results
Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra (AM-194)

Isroil A. Ikromov

Detlef Müller

Princeton University Press

This chapter turns to the case where hsubscript lin(φ‎) ≥ 2. In a first step, the chapter performs a decomposition of the remaining piece Ssubscript Greek small letter psi of the surface S. Then, in the domains Dₗ the chapter once again applies dyadic decomposition techniques in combination with rescaling arguments, making use of the dilations associated with the weight κ‎ₗ. But serious new problems arise, caused by the nonlinear change from the coordinates (x₁, x₂) to the adapted coordinates (y₁, y₂). Therefore, the chapter takes a closer look at the domain Dsubscript pr and devises a further decomposition of the domain Dsubscript pr into various subdomains of “type” Dsubscript (l) and Esubscript (l).

Keywords:   preparatory results, dyadic decompositions, restriction estimates, transition domains, stopping-time algorithm, open cases

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