Weighted restriction theorems for space curves
type: Item
instance of: scholarly article
title: Weighted restriction theorems for space curves
zbMATH Open document ID: 1122.42002
author: Jong-Guk Bak
author: Sanghyuk Lee
author: Jung Jin Lee
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-07-19T00:00:00Z
review text: Let \(W(L^p,l^q)(\mathbb R^n)\) be an amalgam space with the norm \[ \| f\| _{W(L^p,l^q)}=\left(\sum_{k\in\mathbb Z^n}\left(\int_{[0,1)^n+k}| f(x)| ^p\,dx\right)^{q/p}\right)^{1/q} \] if \(1\leq p,q<\infty\) (and with usual modification when \(p\) or \(q\) is infinite). There are three main results in this paper. The first main result of the paper under review is the following Theorem 1.1. Let \(\gamma(t)\) be a non-degenerate \(C^n\) curve in \(\mathbb R^n,\) \(t\in I,\) where \(I\) is a finite interval (a typical example of such curve is \((t,t^2,\dots,t^n)\)). Let \(1\leq p<1+2/(n^2+n)\) and \(1/p+[2/(n^2+n)]/q\geq1.\) Suppose that \(\nu\) is a nonnegative (measurable) function such that \(\nu^{-1/(p-1)}\in W(L^1,l^\infty).\) Then there exists a constant \(C,\) independent of \(f\) and \(\nu\) and depending only on \(\gamma,\) \(n,\) and \(I,\) such that \[ \left(\int_I | \widehat f(\gamma(t))| ^q\,dt\right)^{1/q}\leq C\left\| \nu^{-1/(p-1)}\right\| ^{1-1/p}_{W(L^1,l^\infty)}\left(\int_{\mathbb R^n}| f(x)| ^p\nu(x)\,dx\right)^{1/p}. \] Here, as usual, \(\widehat f\) is the Fourier transform of \(f.\) Then this result is generalized to a class of oscillatory integral operators related to restriction estimates. In the last theorem restriction estimates for curves with two weight functions are established.
reviewed by: Elijah Liflyand
Mathematics Subject Classification ID: 42B10
Mathematics Subject Classification ID: 42B35
zbMATH DE Number: 5173018
zbMATH Keywords: Fourier restriction theorem
zbMATH Keywords: oscillatory integral operator
zbMATH Keywords: amalgam space
zbMATH Keywords: weighted norm inequality
MaRDI profile type: Publication
full work available at URL: j.jmaa.2007.01.039
OpenAlex ID: W2057056306
cites work: Q3869424
cites work: Trigonometric sums in number theory and analysis. Transl. from the Russian
cites work: Estimates for an oscillatory integral operator related to restriction to space curves
cites work: Q4462240
cites work: Restrictions of Fourier transforms to curves
cites work: Degenerate curves and harmonic analysis
cites work: Fourier restriction theorems for curves with affine and Euclidean arclengths
cites work: Fourier restriction theorems for degenerate curves
cites work: Amalgams of 𝐿^{𝑝} and 𝑙^{𝑞}
cites work: Harmonic Analysis on Amalgams of LP and lq
cites work: Weighted Restriction for Curves
cites work: Weighted Norm Inequalities for the Fourier Transform
cites work: A Note on Two Weight Function Conditions for a Fourier Transform Norm Inequality
DOI: 10.1016/J.JMAA.2007.01.039
description: scientific article; zbMATH DE number 5173018