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## Isroil A. Ikromov and Detlef Müller

Print publication date: 2016

Print ISBN-13: 9780691170541

Published to University Press Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691170541.001.0001

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date: 05 June 2020

# Improved Estimates by Means of Airy-Type Analysis

Chapter:
(p.75) Chapter Five Improved Estimates by Means of Airy-Type Analysis
Source:
Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra (AM-194)
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691170541.003.0005

This chapter turns to the proof of a proposition from the previous chapter. Given the operators appearing in that proposition, this chapter establishes the endpoint result thereof by means of Stein's interpolation theorem for analytic families of operators. It constructs analytic families of complex measure μ‎subscript Greek small letter zeta, for ζ‎ in the complex strip Σ‎ given by 0 ≤ Reζ‎ ≤ 1, by introducing complex coefficients in the sums defining the measures ν‎subscript Greek small letter delta,jsuperscript V and ν‎subscript Greek small letter delta,jsuperscript V I, respectively. These coefficients are chosen as exponentials of suitable affine-linear expression in ζ‎ in such a way that, in particular, μ‎subscript Greek small letter theta subscript c = ν‎subscript Greek small letter delta,jsuperscript V I, respectively, μ‎subscript Greek small letter theta subscript c = ν‎subscript Greek small letter delta,jsuperscript V I. As it turns out, the main problem consists in establishing suitable uniform bounds for the measure μ‎subscript Greek small letter zeta when ζ‎ lies on the right boundary line of Σ‎.

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