Lewy's operator and its ramifications
type: Item
instance of: scholarly article
title: Lewy's operator and its ramifications
zbMATH Open document ID: 0638.35002
DOI: 10.1016/0022-1236(86)90102-3
author: Leon Ehrenpreis
published in: Journal of Functional Analysis
publication date: 1986-01-01T00:00:00Z
review text: The Cauchy-Kowalewski theorem shows that linear partial differential operators with analytic coefficients map analytic functions locally onto analytic functions. Malgrange and Ehrenpreis proved that local surjectivity also holds in the space of \(C^{\infty}\) functions for operators with constant coefficients. However, Lewy discovered a first order operator L with linear coefficients for which local surjectivity at the \(C^{\infty}\) level does not hold. In fact, \(L=\partial /\partial x+i(\partial /\partial y)-2i(x+iy)\partial /\partial t.\) In this paper two new proofs are presented for the solvability of L. The first proof uses complex Fourier analysis and leads to a theory of linear operators with linear coefficients. The second proof uses the theory of representations of Lie groups and the fact that L can be considered as a left invariant operator on the Heisenberg group.
Mathematics Subject Classification ID: 35A07
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 35A10
Mathematics Subject Classification ID: 35F05
Mathematics Subject Classification ID: 35A20
zbMATH DE Number: 4040373
zbMATH Keywords: Lewy's operator
zbMATH Keywords: ramifications
zbMATH Keywords: Schwartz distributions
zbMATH Keywords: Fourier transforms
zbMATH Keywords: method of characteristics
zbMATH Keywords: Lie algebra
zbMATH Keywords: solvability
zbMATH Keywords: complex Fourier analysis
zbMATH Keywords: linear operators
zbMATH Keywords: linear coefficients
zbMATH Keywords: representations of Lie groups
zbMATH Keywords: Heisenberg group
reviewed by: Alan J. Pryde
MaRDI profile type: Publication
full work available at URL: 0022-1236(86)90102-3
OpenAlex ID: W2005539991
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description: scientific article; zbMATH DE number 4040373