On domination conditions for systems of minimal differential operators acting in the space \(L_\infty(\mathbb R^n)\)
type: Item
instance of: scholarly article
title: On domination conditions for systems of minimal differential operators acting in the space \(L_\infty(\mathbb R^n)\)
zbMATH Open document ID: 1065.35104
published in: Mathematical Notes
publication date: 2004-12-07T00:00:00Z
review text: Let \(l=(l_1,\dots,l_n)\), \(n \geq 2\), let \(l_i\) be natural, let \(P(D)\) be a differential operator of the form \(P(D) = \sum\limits_{| \alpha:l| \leq 1} a_\alpha D^\alpha\). The main result of the paper under review is the following: (i) if the operator \(P(D)\) is quasielliptic then it subordinates in \(L_\infty(\mathbb R^n)\) all differential operators \(Q(D)\) of the form \[ Q(D) = \sum\limits_{| \beta:l| <1} b_\beta D^\beta; \eqno(1) \] (ii) if \(n \geq 3\) and an operator subordinates all differential operators of the form (1) then this operator is quasielliptic. This assertion generalizes the result obtained by \textit{K. De Leeuw} and \textit{H. Mirkil} [Ill. J. Math. 8, 112--124 (1964; Zbl 0131.33202)] for elliptic operators. As noted by the author, (i) was proved by \textit{M. M. Malamud} [Trans. Mosc. Math. Soc. 1995, 159--202 (1995; Zbl 0871.47036)] in a more general form.
Mathematics Subject Classification ID: 35H30
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 2121425
zbMATH Keywords: elliptic and quasielliptic systems
zbMATH Keywords: Eberlein lemma
zbMATH Keywords: differential polynomials
author: D. V. Limanskiy
reviewed by: G. V. Demidenko
MaRDI profile type: Publication
full work available at URL: b:matn.0000030988.11731.5b
OpenAlex ID: W1978801134
DOI: 10.1023/B:MATN.0000030988.11731.5B
description: scientific article; zbMATH DE number 2121425