A bilinear extension inequality in two dimensions.
type: Item
instance of: scholarly article
title: A bilinear extension inequality in two dimensions.
zbMATH Open document ID: 1050.42006
DOI: 10.1016/S0022-1236(02)00148-9
author: Anthony Carbery
author: Juan Antonio Barceló
author: Jonathan Bennett
published in: Journal of Functional Analysis
publication date: 2003-08-25T00:00:00Z
review text: Let \[ (f d\sigma)\hat{\;} (x) =\int_{{\mathbb{S}}^1}e^{-ixy} f(y)\,d\sigma(y),\qquad x\in {\mathbb{R}}^2, \] where \(d\sigma\) is the surface measure on \({\mathbb{S}}^1\). Suppose that \(\widetilde{f}(x)= f(-x)\). In the paper it is shown that if \(f,g\in L^2({\mathbb{S}}^1)\) with dist(supp(\(f\)),supp(\(g\))) and dist(supp(\(\widetilde{f}\)),supp(\(g\))) bounded below, then the following bilinear estimate \[ \int_{{\mathbb{S}}^1}| (f d\sigma)\hat{\;} (Rx) (g \,d\sigma)\hat{\;}(Rx)| \,d\sigma(x) \leq\frac{c}{R^{5/6}}\| f\| _2\| g\| _2 \] holds for all \(R>0\). Moreover, the decay rate \(R^{-5/6}\) is optimal.
Mathematics Subject Classification ID: 42A38
Mathematics Subject Classification ID: 42B10
zbMATH DE Number: 1970664
zbMATH Keywords: Fourier transforms
zbMATH Keywords: bilinear extension inequality
reviewed by: Alexander Meskhi
MaRDI profile type: Publication
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full work available at URL: s0022-1236(02)00148-9
OpenAlex ID: W1972082643
description: scientific article; zbMATH DE number 1970664