What is van der Corput's lemma in higher dimensions?
type: Item
instance of: scholarly article
title: What is van der Corput's lemma in higher dimensions?
zbMATH Open document ID: 1045.42005
author: Anthony Carbery
author: James Wright
published in: Publicacions Matemàtiques
publication date: 2002-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/41813
review text: The authors survey, clarify and place in context a number of multilinear estimates involving a real-valued phase function \(u(x_1,\ldots,x_n)\) of several variables on a domain \(\Omega\). More precisely, the authors are interested in multilinear sublevel estimates of the form \[ \left| \int_{\Omega \cap \{ | u| \leq t \}} f_1(x_1) \dots f_n(x_n)\;dx_1 \dots dx_n\right| \leq C t^\gamma \prod_{i=1}^n \| f_i \| _{L^{p_i}} \] and multilinear van der Corput type estimates of the form \[ \left| \int_\Omega e^{i\lambda u(x)} f_1(x_1) \ldots f_n(x_n)\;dx_1 \ldots dx_n\right| \leq C | \lambda| ^{-\gamma} \prod_{i=1}^n \| f_i \| _{L^{p_i}}. \] Here \(\lambda\) should be thought of as a large real parameter, and \(t\) as a small real parameter, and the objective is to obtain exponents \(\gamma\) which are as good as possible. Ideally we wish the exponent to give a scale-invariant result, but the authors show in most cases that this is not feasible except in a number of simple cases. Related to the goal of scale invariance is the goal of minimizing the number of assumptions on the phase \(u\); typically one only asks for some lower bound on a certain mixed derivative of \(u\), as well as some mild convexity conditions on \(\Omega\) (e.g., that \(\Omega\) is a rectangle). The authors point out that it is also often necessary to enforce an additional complexity bound on \(u\), for instance in controlling how often \(u\) or some derivative of \(u\) can change sign. The authors review a number of existing results in these directions, notably those from [\textit{A. Carbery}, \textit{M. Christ}, and \textit{J. Wright}, J. Am. Math. Soc. 12, No. 4, 981--1015 (1999; Zbl 0938.42008)].
Mathematics Subject Classification ID: 42B20
zbMATH DE Number: 1881282
zbMATH Keywords: stationary phase
zbMATH Keywords: oscillatory integral operators
reviewed by: Terence Tao
MaRDI profile type: Publication
Wikidata QID: Q124852602
OpenAlex ID: W2154950626
DOI: 10.5565/PUBLMAT_ESCO02_01
description: scientific article; zbMATH DE number 1881282