Oscillatory integrals with polynomial phases
type: Item
instance of: scholarly article
title: Oscillatory integrals with polynomial phases
zbMATH Open document ID: 0829.42014
DOI: 10.1007/BF01231323
author: D. H. Phong
author: Elias M. Stein
published in: Inventiones Mathematicae
publication date: 1993-05-16T00:00:00Z
full work available at URL: https://eudml.org/doc/144041
review text: Let \(S(x, y)\) be a homogeneous polynomial in \((x,y)\in \mathbb{R} \times \mathbb{R}\) of degree \(n\) and define the operator \(T\) by \[ (T\varphi) (x)= \int e^{i\lambda S(x, y)} \chi(y) \varphi(y) dy, \qquad \varphi\in C_0^\infty (\mathbb{R}), \] where \(\chi\) is a fixed \(C^\infty\) cut-off function on \(\mathbb{R}\) with compact support. The main theorem of the paper is: The operator \(T\) extends to a bounded operator on \(L^2 (\mathbb{R})\) with operator norm \(|T|\leq C|\lambda |^{-1/n}\) when \(|\alpha_{(n/2) -1}|+ \cdots+ |\alpha_0 |\neq 0\) for \(n\) even and \(|\alpha_{(n-1) /2-1} |+ \cdots+ |\alpha_0 |\neq 0\) for \(n\) odd. Here \(\alpha_j\) are the coefficients of \(S(x,y)\).
reviewed by: Niels Jacob
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 47G30
Mathematics Subject Classification ID: 35S30
zbMATH DE Number: 167385
zbMATH Keywords: oscillatory integrals
zbMATH Keywords: \(L^ 2\)-bounds
zbMATH Keywords: homogeneous polynomial
MaRDI profile type: Publication
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OpenAlex ID: W1978921923
description: scientific article; zbMATH DE number 167385