Hyperfunction fundamental solutions of surjective convolution operators on real analytic functions
type: Item
instance of: scholarly article
title: Hyperfunction fundamental solutions of surjective convolution operators on real analytic functions
zbMATH Open document ID: 0841.46025
author: Michael Langenbruch
published in: Journal of Functional Analysis
publication date: 1996-03-20T00:00:00Z
review text: Let \(\mu\in A(\mathbb{R})\). The author characterizes the surjectivity of the convolution operator \(T_\mu:= \mu*\) on real analytic functions by two equivalent conditions: (1) \(T_\mu\) admits hyperfunctional elementary solutions \(E_+\) (and \(E_-\)), which are analytic on an angular neighbourhood of \(]- \infty, C[\) (respectively, \([- C, \infty[\)) for some \(C\leq 0\). (2) The Fourier transform \(\widehat\mu\) of \(\mu\) satisfies a slowly decreasing condition and the following alternative. There is \(\delta> 0\) such that \[ |\text{Im } z|\geq \delta|\text{Re } z|\quad\text{or}\quad |\text{Im } z|= o(|z|)\quad\text{for any}\quad z\text{ with }\widehat\mu(z)= 0. \] This latter condition was obtained by \(\mu\in A(\{0\})'\) by Napalkov/Rudakov and Meyer.
Mathematics Subject Classification ID: 46F10
Mathematics Subject Classification ID: 46F12
Mathematics Subject Classification ID: 46F15
zbMATH DE Number: 847459
zbMATH Keywords: surjectivity of the convolution operator
zbMATH Keywords: hyperfunctional elementary solutions
zbMATH Keywords: Fourier transform
MaRDI profile type: Publication
full work available at URL: jfan.1995.1083
OpenAlex ID: W1992398646
DOI: 10.1006/JFAN.1995.1083
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description: scientific article; zbMATH DE number 847459