A Hardy space for Fourier integral operators
type: Item
instance of: scholarly article
title: A Hardy space for Fourier integral operators
zbMATH Open document ID: 1031.42020
DOI: 10.1007/BF02921717
author: Hart F. Smith
published in: The Journal of Geometric Analysis
publication date: 2000-06-22T00:00:00Z
review text: The author introduces a new subspace of \(L^1(\mathbb{R}^n)\), denoted by \({\mathcal H}^1_{\text{FIO}}(\mathbb{R}^n)\), on which the algebra of Fourier integral operators of order \(0\), associated to local canonical transformations, acts continuously. This space is in many ways analogous to the local version of the real Hardy space, which can be characterized as the largest subspace of \(L^1(\mathbb{R}^n)\) that is preserved by order \(0\) pseudodifferential operators. A key role is played by the atomic and molecular decomposition theorems for \({\mathcal H}^1_{\text{FIO}}(\mathbb{R}^n)\). The boundedness of Fourier integral operators on \({\mathcal H}^1_{\text{FIO}}(\mathbb{R}^n)\) is established by showing that such an operator maps a molecule centered at one point in \(S^*(\mathbb{R}^n)= \mathbb{R}^n\times S^{n-1}\) to a molecule centered at the image of that point under the associated canonical transformation. The author also establishes the embedding theorem \[ {\mathcal H}^1(\mathbb{R}^n) @>\langle D\rangle^{-{n-1\over 2}}>>{\mathcal H}^1_{\text{FIO}}(\mathbb{R}^n)@> I>>{\mathcal H}^1(\mathbb{R}^n), \] where \[ \|f\|_{{\mathcal H}^1(\mathbb{R}^n)}= \|(1- r(D))f\|_{H^1(\mathbb{R}^n)}+ \|r(D) f\|_{L^1(\mathbb{R}^n)} \] and \(r\) is a function in \(C^\infty_c(\mathbb{R}^n)\) such that \(r(\xi)= 1\) if \(|\xi|\leq 1\).
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 35S30
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 47G30
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 1463793
zbMATH Keywords: Littlewood-Paley theory
zbMATH Keywords: Fourier integral operators
zbMATH Keywords: canonical transformations
zbMATH Keywords: Hardy space
zbMATH Keywords: pseudodifferential operators
reviewed by: Hiroki Tanabe
MaRDI profile type: Publication
cites work: 𝐿^{𝑝} boundedness of Fourier integral operators
cites work: Q2548398
cites work: Wave packets and fourier integral operators
cites work: A note on spherical summation multipliers
cites work: \(H^p\) spaces of several variables
cites work: \(L^p\) estimates for the waves equation
cites work: Regularity properties of Fourier integral operators
cites work: Q5607484
cites work: Q3142876
cites work: Q3756738
description: scientific article; zbMATH DE number 1463793