Gårding inequality in large dimension
type: Item
instance of: scholarly article
title: Gårding inequality in large dimension
zbMATH Open document ID: 1323.35236
author: Richard Lascar
author: J. Nourrigat
published in: Israel Journal of Mathematics
publication date: 2014-06-25T00:00:00Z
review text: The main result of the paper is an analog of the sharp Gårding inequality for some classes of semiclassical pseudodifferential operators. The authors prove the following result. Given nonnegative real numbers \(\rho_j\) and \(\delta_j\), \(0\leq j\leq n\), let \(F\) be a bounded nonnegative real-valued continuous function defined on \(\mathbb R^{2n}\) satisfying for a constant \(M>0\), for all \((\alpha ,\beta )\in \mathbb N^{2n}\), \(0\leq \alpha_j\leq m\), \(0\leq \beta_j\leq m\), \(0\leq j\leq n\), for all \((x,\xi )\in \mathbb R^{2n}\), \(| \partial_x^\alpha\partial_\xi^\beta F(x,\xi )| \leq M\rho^\alpha\delta^\beta\). Then, for all \(f\in L^2(\mathbb R^n)\), \[ \langle Op_h^w(F)f,f\rangle \geq -225\pi Kh\times \rho \cdot \delta \times \prod\limits_{j=1}^n(1+225\pi Kh\rho_j\delta_j)\times\| f\|^2, \] where\ \(K=\sup_{j\leq n}\max \{ 1,(\rho_j\delta_j)^3\} \) and for \(h>0\), \(Op_h^w(F)\) denotes the operator formally defined by \[ Op_h^w(F)(\varphi )(x)=(2\pi h)^{-n}\int\limits_{\mathbb R^{2n}}e^{ \frac{i}{h}(x-y)\cdot \xi}F(\frac{x+y}{2},\xi )\varphi (y)dyd\xi,\quad\varphi \in S. \]
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 47G30
zbMATH DE Number: 6308415
zbMATH Keywords: pseudodifferential operator
zbMATH Keywords: Gårding inequality
reviewed by: Chikh Bouzar
MaRDI profile type: Publication
full work available at URL: s11856-014-0008-4
OpenAlex ID: W2140085288
cites work: On bounded pseudodifferential operators in Wiener spaces
cites work: A Class of Bounded Pseudo-Differential Operators
cites work: Q3941791
cites work: Q3736112
cites work: The L 2 -Boundedness of Pseudodifferential Operators
cites work: Q3898933
DOI: 10.1007/S11856-014-0008-4
description: scientific article; zbMATH DE number 6308415