\(L^ \infty\)-decay estimations of the spherical mean value on symmetric spaces
type: Item
instance of: scholarly article
title: \(L^ \infty\)-decay estimations of the spherical mean value on symmetric spaces
zbMATH Open document ID: 0826.58039
DOI: 10.1007/BF02108299
author: Paul Günther
published in: Annals of Global Analysis and Geometry
publication date: 1995-12-04T00:00:00Z
review text: Let \(M_t[\varphi] (x)\) be the spherical mean value operator applied to a function \(\varphi\) on a symmetric Riemannian space \(M\) of the non- compact type. The following inequality is proved: For \(\forall t > 0\), \(\forall x \in M\), \[ |(\partial^k / \partial t^k) M_t [\varphi] |_l (x) \leq cM_t [|\varphi|_{l + k}] (x). \] Here \(|\cdot |_l\) denotes a suitable pointwise norm for the spatial Lie derivatives of order \(\leq l\). For rank one spaces the inequality is proved also for the covariant derivatives. Then it is shown that \(M_t [\varphi] (x)\) has decay rate \(1/(1 + m(t)^\sigma)\), \(\sigma \in (0,1]\), provided that \(\varphi\) has a decay rate \(1/(1 + m(t)^\rho)\) with \(\rho > \sigma\), where \(m(t)\) denotes the area of the sphere of radius \(t\). For spaces where Huygens principle hold as were discovered by Helgason, this gives the decay estimate for the solution of the Cauchy problem of the wave equation. For real hyperbolic space \(H^n (\mathbb{R})\) of odd dimension, it is shown that the decay estimate can be improved to an exponential one.
Mathematics Subject Classification ID: 58J45
Mathematics Subject Classification ID: 35L10
zbMATH DE Number: 729755
zbMATH Keywords: symmetric space
zbMATH Keywords: non-compact type
zbMATH Keywords: spherical mean value operator
zbMATH Keywords: \(L^ \infty\)-decay estimations
zbMATH Keywords: wave equations
zbMATH Keywords: Huygens' principle
reviewed by: Akira Kaneko
MaRDI profile type: Publication
cites work: Q4040088
cites work: Q3897665
cites work: Q3331943
cites work: Huygens' principle for wave equations on symmetric spaces
cites work: Weighted L∞ and L1 estimates for solutions to the classical wave equation in three space dimensions
cites work: Decay estimates for the solutions of the non-euclidean wave-equations
cites work: Q5845092
cites work: Q5840383
cites work: Wave propagation on Riemannian symmetric spaces
cites work: On the existence of global solutions of nonlinear wave equations
full work available at URL: https://doi.org/10.1007/bf02108299
OpenAlex ID: W1979742082
description: scientific article; zbMATH DE number 729755