A survey on explicit representation formulae for fundamental solutions of linear partial differential operators
type: Item
instance of: scholarly article
title: A survey on explicit representation formulae for fundamental solutions of linear partial differential operators
zbMATH Open document ID: 0889.35020
DOI: 10.1023/A:1005784017770
author: Peter Wagner
author: Norbert Ortner
published in: Acta Applicandae Mathematicae
publication date: 1998-05-27T00:00:00Z
review text: The first general existence proofs for fundamental solutions of linear partial differential operators with constant coefficients were presented in 1953 by B. Malgrange and, independently, by L. Ehrenpreis. Their method of proof was based on the Hahn-Banach theorem but the question for explicit representation of the fundamental solution remained open. Thus the goal of the authors is to give a survey on such explicit representation formulas. In Chapter 2 they use a method which goes back to Hörmander for the construction of fundamental solutions with at most exponential growth as a finite sum of \(C^\infty\)-functions multiplied by the inverse Fourier transforms of \(L^\infty\)-functions. In Chapter 3 they study regular fundamental solutions by using temperate weight functions \(k=k (\xi)\) and the well-known \(B^{p,k}\)-spaces. The authors do not treat the existence of temperate fundamental solutions. This problem, construction of temperate fundamental solutions for general differential polynomials (Treves 1967), seems to be unsolved up to now.
Mathematics Subject Classification ID: 35E05
Mathematics Subject Classification ID: 46F10
Mathematics Subject Classification ID: 35C15
zbMATH DE Number: 1014488
zbMATH Keywords: regularity properties
zbMATH Keywords: \(B^{p,k}\)-spaces
reviewed by: Michael Reissig
MaRDI profile type: Publication
full work available at URL: a:1005784017770
OpenAlex ID: W1538280145
description: scientific article; zbMATH DE number 1014488