The Poincaré inequality for vector fields satisfying Hörmander's condition
type: Item
instance of: scholarly article
title: The Poincaré inequality for vector fields satisfying Hörmander's condition
zbMATH Open document ID: 0614.35066
DOI: 10.1215/S0012-7094-86-05329-9
published in: Duke Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: From author's summary: The purpose of this paper is to prove a Poincaré-type inequality of the form \[ \int_{B(r)}| f- f_{B(r)}|^ 2\leq Cr^ 2\int_{B(r)}\sum^{m}_{i=1}| X_ if|^ 2,\text{ for all } f\in C^{\infty}(\overline{B(r)}), \] where \(X_ 1,...,X_ m\) denote vector fields on \(R^ d\) satisfying Hörmander's condition, B(r) denotes a ball of radius r with respect to a natural metric associated with \(X_ 1,...,X_ m\) and \(f_{B(r)}\) denotes the average value of f on the ball B(r). This inequality is the same as finding a lower bound \(1/Cr^ 2\) of the least nonzero eigenvalue in the Neumann problem for \(L=\sum^{m}_{i=1}X^*_ iX_ i\) of B(r).
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 35J99
Mathematics Subject Classification ID: 35B99
zbMATH DE Number: 3994138
zbMATH Keywords: Poincaré-type inequality
zbMATH Keywords: Hörmander's condition
zbMATH Keywords: average
zbMATH Keywords: lower bound
zbMATH Keywords: least nonzero eigenvalue
zbMATH Keywords: Neumann problem
author: David Jerison
MaRDI profile type: Publication
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full work available at URL: s0012-7094-86-05329-9
OpenAlex ID: W2057335654
description: scientific article; zbMATH DE number 3994138