On the positive parts of second order symmetric pseudodifferential operators
type: Item
instance of: scholarly article
title: On the positive parts of second order symmetric pseudodifferential operators
zbMATH Open document ID: 1184.35317
author: Marco Mughetti
author: Fabio Nicola
published in: Integral Equations and Operator Theory
publication date: 2009-11-13T00:00:00Z
review text: The authors consider a spectral decomposition of self-adjoint pseudodifferential operators \(P(x,D)\) on \(\mathbb R^n\) of Hörmander class. Considering a degenerate elliptic operator of second order, they assume three condtions. First, the principal symbol vanishes exactly of order two on a symplectic submanifold \(\Sigma\subset T^* \mathbb R^n\). Secondly, the eigenvalues of the fundamental matrix are distinctive at a point \(\rho\in\Sigma\). Thirdly, considering a kind of indicial polynomial determined by the subprincipal symbol of \(P(x,D)\), all the length \(|\alpha|\) of zero points \(\alpha\) should be either odd or even, at this point \(\rho\in\Sigma\). Under these assumptions, they prove that one can microlocally construct the positive part operator \(\Pi^+\) and negative part operator \(\Pi^-\) of \(P(x,D)\), and proves a sharp Garding inequality using \(\Pi^{\pm}\). This problem was previously studied by D.~Fujiwara for first order pseudodifferential operators, and the present article is an attempt of its generalization.
reviewed by: Keisuke Uchikoshi
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 47G30
Mathematics Subject Classification ID: 35J47
zbMATH DE Number: 5633112
zbMATH Keywords: pseudodifferential operators
zbMATH Keywords: spectral projections
zbMATH Keywords: multiple characteristics
zbMATH Keywords: Fefferman-Phong decomposition
MaRDI profile type: Publication
full work available at URL: s00020-009-1699-5
OpenAlex ID: W2031080042
DOI: 10.1007/S00020-009-1699-5
description: scientific article; zbMATH DE number 5633112