Hyperbolic operators in spaces of generalized distributions
type: Item
instance of: scholarly article
title: Hyperbolic operators in spaces of generalized distributions
zbMATH Open document ID: 0619.46041
DOI: 10.1216/RMJ-1986-16-3-535
author: Abdullah Saleh
published in: Rocky Mountain Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: The main result of the paper is a characterization of hyperbolic operators in the space \({\mathcal D}'_{\omega}\) of Beurling generalized distributions. The notion of hyperbolicity with respect to \(t>0\) and \(t<0\) has been extended to convolution operators in \({\mathcal D}'_{\omega}\), and the following theorem which generalizes a result of Ehrenpreis has been proved. Let S be a convolution operator in \({\mathcal D}'_{\omega}\), the following are equivalent: (i) \(S\) is \(\omega\)-hyperbolic with respect to \(t>0.\) (ii) \(S\) is invertible and there exist constants \(C\), \(A\), \(M\) and \(A_ 1\) so that \[ \Im \tau \leq A(| \Im z| +\frac{1}{2(3+2MA_ 1)}\omega (\Re z,\Re \tau)) \] for all \((z,\tau)\in {\mathbb{C}}^ n\times {\mathbb{C}}\); \(\hat S(z,\tau)=0.\) (iii) There exist positive constants \(C\), \(a\) and \(A\) so that \[ | \hat S(z,\tau)| \geq C\exp (-a[| Im z| +| Im \tau | +\omega (Re z,Re \tau)], \] whenever Im \(\tau > A[| \Im z| +\omega(z,\tau)]\); \((z,\tau)\in {\mathbb{C}}^ n\times {\mathbb{C}}.\)
Mathematics Subject Classification ID: 46F10
Mathematics Subject Classification ID: 46F05
Mathematics Subject Classification ID: 58J45
Mathematics Subject Classification ID: 44A35
Mathematics Subject Classification ID: 35L99
zbMATH DE Number: 4004932
zbMATH Keywords: characterization of hyperbolic operators
zbMATH Keywords: Beurling generalized distributions
zbMATH Keywords: convolution operators
MaRDI profile type: Publication
full work available at URL: rmj-1986-16-3-535
OpenAlex ID: W2059809416
description: scientific article; zbMATH DE number 4004932