Fourier integral operators on noncompact symmetric spaces of real rank one
type: Item
instance of: scholarly article
title: Fourier integral operators on noncompact symmetric spaces of real rank one
zbMATH Open document ID: 0962.43004
author: Alexandru D. Ionescu
published in: Journal of Functional Analysis
publication date: 2001-06-07T00:00:00Z
review text: Let \(X=G/K\) be a noncompact symmetric space of rank one, \(g\) and \(k\) be the Lie algebras of \(G\) and \(K\), respectively, \(g=k\oplus p\) the Cartan decomposition, \(a\) be a maximal abelian subspace of \(p\), \(a^*_c\) the complex dual of \(a\). A Fourier transform on the symmetric space \(X\) associates to a smooth compactly supported function \(f\) on \(X\) a function \(\widetilde f\) on \(a^*_c \times K/M\), where \(M\) is the centralizer of \(a\) in \(K\). Then every bounded function \(m:R\to C\) defines a bounded operator \(T_m\) given by \(\widetilde T_mf(\lambda,b) =m(\lambda) \widetilde f(\lambda,b)\). Using explicit expressions of the Harish-Chandra function \(c(\lambda)\) and elementary spherical functions \(\Phi_\lambda(s)\) on the symmetric spaces of rank one, the author obtains \(L^p\) boundedness properties of radial Fourier integral operators defined by the multipliers \(m_\tau (\lambda) \cos( \lambda \tau)\) and \(m_\tau(\lambda) \lambda^{-1} \sin(\lambda \tau)\), from which he derives an estimate of the solution \(u_\tau\) of the wave equation on \(X\) by its initial datas \(f\) and \(g\): \[ \|u_\tau\|_{l^p (X)}\leq c_p(\tau) \bigl(\|f\|_{L^p_{b_p (X)}}+ (1+\tau) \|g\|_{L^p_{b_p-1}(X)} \bigr). \] The second part deals with an analog of E. M. Stein's maximal spherical averages, i.e., let \(d\sigma_\tau\) be a normalized spherical measure, the author then obtains exponential estimates on the \(L^p\) norm of the maximal operator \(\sup_{T\leq\tau \leq(T+1)} |f*d \sigma_\tau (z)|\) for a compactly supported function \(f\).
reviewed by: Fuliu Zhu
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 1496042
zbMATH Keywords: \(L^p\) boundedness properties
zbMATH Keywords: symmetric space
zbMATH Keywords: Fourier transform
zbMATH Keywords: bounded operator
zbMATH Keywords: Harish-Chandra function
zbMATH Keywords: spherical functions
zbMATH Keywords: Fourier integral operators
zbMATH Keywords: multipliers
zbMATH Keywords: wave equation
zbMATH Keywords: maximal spherical averages
zbMATH Keywords: normalized spherical measure
zbMATH Keywords: exponential estimates
zbMATH Keywords: maximal operators
MaRDI profile type: Publication
full work available at URL: jfan.2000.3572
OpenAlex ID: W2016211693
cites work: \(\mathbb{L}_ p\) Fourier multipliers on Riemannian symmetric spaces of the noncompact type
cites work: Multiplicateurs Sur Certains Espaces Symetriques
cites work: Averages in the plane over convex curves and maximal operators
cites work: Q4307544
cites work: L p -multipliers for Noncompact Symmetric Spaces
cites work: \(L^ p\)-\(L^ q\) estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces. I
cites work: \(H^p\) spaces of several variables
cites work: Q3472639
cites work: Q4301588
cites work: Q3874679
cites work: Q3870038
cites work: Q4213401
cites work: \(L^p\) estimates for the waves equation
cites work: Regularity properties of Fourier integral operators
cites work: Propagation of singularities and maximal functions in the plane
cites work: Averages over hypersurfaces. Smoothness of generalized Radon transforms
cites work: Expansions for spherical functions on noncompact symmetric spaces
cites work: Maximal functions: Spherical means
cites work: Q3142876
cites work: AN EXAMPLE IN THE THEORY OF THE SPECTRUM OF A FUNCTION
DOI: 10.1006/JFAN.2000.3572
Recommended article: Endpoint Results for Fourier Integral Operators on Noncompact Symmetric Spaces
Recommended article: Vector-valued Fourier multipliers on symmetric spaces of the noncompact type
Recommended article: Hardy's theorem for the helgason Fourier transform on noncompact rank one symmetric spaces
Recommended article: Fourier multipliers on non-Riemannian symmetric spaces
Recommended article: Q3485482
Recommended article: Some Estimates for the Fourier Transform on Rank 1 Symmetric Space
Recommended article: Q3830090
Recommended article: Fourier transforms of Dini-Lipschitz functions on rank 1 symmetric spaces
Recommended article: Q5293703
Recommended article: \(\mathbb{L}_ p\) Fourier multipliers on Riemannian symmetric spaces of the noncompact type
description: scientific article; zbMATH DE number 1496042