On an interpretation of Hörmander's condition of local solvability
type: Item
instance of: scholarly article
title: On an interpretation of Hörmander's condition of local solvability
zbMATH Open document ID: 0918.35150
DOI: 10.1016/S0019-3577(97)87571-4
author: Vladimir Tulovsky
published in: Indagationes Mathematicae. New Series
publication date: 1999-08-19T00:00:00Z
review text: The author re-considers the problem of the local solvability, by observing that local solvability of a partial differential operator \(P\) is essentially equivalent to the existence of a complete set of eigenfunctions for \(P\). As concrete example of this interesting point of view, the author first reviews the case of the operators with constant coefficients, the eigenfunctions being given in this case by \(\exp(ix\xi)\). As second example, attention is fixed on first-order partial differential operators of the form \[ P= p(x,D)= D_{x_n}- i \sum^{n-1}_{k=1} b_k(x) D_{x_k} \] with \(b_k(x)\) analytic real-valued functions. The author shows that, when the sufficient condition of Hörmander \(\{\text{Re p,Im p}\}< 0\) is satisfied in a neighborhood \(V\) of \((x_0,\xi_0)\), a parametrix for \(P\) can be constructed in the preceding order of ideas, microlocally in \(V\), by using eigenfunctions of the form \(\exp(if(x))\).
reviewed by: Luigi Rodino
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 35A27
zbMATH DE Number: 1251255
zbMATH Keywords: local solvability
zbMATH Keywords: set of eigenfunctions
zbMATH Keywords: parametrix
MaRDI profile type: Publication
cites work: SUBELLIPTIC OPERATORS
cites work: A global construction for pseudo-differential operators with non- involutive characteristics
cites work: Q3736112
cites work: Q3752751
cites work: Fourier integral operators with complex phase functions and parametrix for an interior boundary value problem
cites work: Solvability of a first order linear partial differential equation
cites work: On local solvability of linear partial differential equations—part II. Sufficient conditions
cites work: Operators of principal type with interior boundary conditions
full work available at URL: s0019-3577(97)87571-4
OpenAlex ID: W2050299096
Recommended article: Local Solvability of Nonsmooth Hörmander’s Operators
Recommended article: Local solvability on \({\mathbb{H}_1}\): non-homogeneous operators
Recommended article: Sur le théorème local de Harnack
Recommended article: On local solvability of certain differential complexes
Recommended article: A note on the local solvability of the Cauchy problem
Recommended article: Q2937299
Recommended article: Local solvability of the $\overline{\partial}$-equations on $l^{2}$
Recommended article: Local solvability of the \(k\)-Hessian equations
Recommended article: Q3124312
Recommended article: Q3353404
description: scientific article; zbMATH DE number 1251255